Even if all that one wishes were to happen, this could still only be called the Grace of God. There is no logical association between one's will and the world to guarantee it. The physical association one presumes to exist between any and every thing is surely nothing we could will ourselves. Just as only logical necessity exists, so too only logical impossibility exists. For example: that two colors are simultaneously present at the same place in the visual field is in fact logically impossible, since it is ruled out by the logical structure of color.
In physics, this contradiction appears like this: a particle cannot have two velocities at the same time; it cannot be in two places at the same time; particles that are in different places at the same time cannot be identical.
It is clear that the logical product of two elemental propositions can neither be a tautology nor a contradiction. The assertion that a point in the visual field has two different colors at the same time is a contradiction.
Monday, April 7, 2008
The relative position of logic and science.
That an image can be described using a grid with a given form tells us nothing about the image. (A grid works for all such images.) But what does characterize the image is that it can be completely described by a particular grid with a particular mesh size. So too, it tells us nothing about the world that it can be described by Newtonian mechanics or whatever. That it can be described at all, and in a particular way, does tell us something indeed. That one method of theoretical description is simpler than another also tells us something about the world.
Theoretical physics is an attempt to construct all the true propositions that we need to describe the world using a single plan. Throughout their whole logical apparatus, the laws of physics still speak about the objects of the world. We ought not forget that any theoretical description of the world will always be completely general. In mechanics, for example, one never speaks of particular point-masses, but only about any whatsoever.
Although the spots in our image are geometrical figures, it is obvious that geometry can say nothing at all about their actual form and position. The grid, however, is purely geometrical; all its properties can be given a priori. Laws like the principle of sufficient reason, etc. deal with the grid and not with what the grid describes.
If there were a law of causality, one might state it as: "There are laws of nature." But of course that cannot be said: it can be seen. Using Hertz's terminology, one might say: "Only regular correlations are thinkable. Hence the only way we can describe the lapse of time is to rely on some process such as the movement of a chronometer."
Something entirely analogous applies to space. Wherever one says that neither of two exclusive events can occur because there is no reason one should occur rather than the other, one is really dealing with the fact that one cannot describe either without some sort of asymmetry between them. And if such an asymmetry is found, we can regard it as the cause that made one occur and not the other.
Kant's problem about the right hand and the left hand, which cannot be made to coincide, exists already in two dimensions; indeed, even in one-dimensional space. The two congruent figures, a and b, cannot be made to coincide unless they are moved out of this space.
The right hand and the left hand are in fact completely congruent. It is quite irrelevant that they cannot be made to coincide. A right-hand glove could be put on the left hand, if it could be turned round in four-dimensional space.
The procedure of induction consists in accepting as true the simplest law that can be reconciled with our experiences. But this procedure has no logical, only a psychological, justification. There is no reason to believe that the simplest case will in fact be realized. That the sun will rise tomorrow is a hypothesis; we do not know whether it will rise. There is nothing to compel one thing to happen because something else has. There is only logical necessity.
The whole modernist world view is based on the illusion that the laws of nature actually explain natural phenomena. Thus they stand before the laws of nature as something inviolable, just as the ancients did before God and Fate. Both, in fact, are both right and wrong. Nevertheless, the view of the ancients is clearer in so far as they acknowledge it as closure, while the modern system tries to make it seem as if everything were explained.
Theoretical physics is an attempt to construct all the true propositions that we need to describe the world using a single plan. Throughout their whole logical apparatus, the laws of physics still speak about the objects of the world. We ought not forget that any theoretical description of the world will always be completely general. In mechanics, for example, one never speaks of particular point-masses, but only about any whatsoever.
Although the spots in our image are geometrical figures, it is obvious that geometry can say nothing at all about their actual form and position. The grid, however, is purely geometrical; all its properties can be given a priori. Laws like the principle of sufficient reason, etc. deal with the grid and not with what the grid describes.
If there were a law of causality, one might state it as: "There are laws of nature." But of course that cannot be said: it can be seen. Using Hertz's terminology, one might say: "Only regular correlations are thinkable. Hence the only way we can describe the lapse of time is to rely on some process such as the movement of a chronometer."
Something entirely analogous applies to space. Wherever one says that neither of two exclusive events can occur because there is no reason one should occur rather than the other, one is really dealing with the fact that one cannot describe either without some sort of asymmetry between them. And if such an asymmetry is found, we can regard it as the cause that made one occur and not the other.
Kant's problem about the right hand and the left hand, which cannot be made to coincide, exists already in two dimensions; indeed, even in one-dimensional space. The two congruent figures, a and b, cannot be made to coincide unless they are moved out of this space.
The right hand and the left hand are in fact completely congruent. It is quite irrelevant that they cannot be made to coincide. A right-hand glove could be put on the left hand, if it could be turned round in four-dimensional space.
The procedure of induction consists in accepting as true the simplest law that can be reconciled with our experiences. But this procedure has no logical, only a psychological, justification. There is no reason to believe that the simplest case will in fact be realized. That the sun will rise tomorrow is a hypothesis; we do not know whether it will rise. There is nothing to compel one thing to happen because something else has. There is only logical necessity.
The whole modernist world view is based on the illusion that the laws of nature actually explain natural phenomena. Thus they stand before the laws of nature as something inviolable, just as the ancients did before God and Fate. Both, in fact, are both right and wrong. Nevertheless, the view of the ancients is clearer in so far as they acknowledge it as closure, while the modern system tries to make it seem as if everything were explained.
Sunday, April 6, 2008
Mathematics is a method of logic.
Using equations characterizes the essence of the mathematical method. Because this so, every proposition of mathematics must be self explanatory. Mathematics arrives at equations by the method of substitution. Equations express that one can substitute an expression for another and so, starting from a number of equations, we advance to new equations by substituting different expressions as we go along in accordance with them. Thus the proof of the proposition 2 + 2 = 4 runs as follows:
Ω^(2×2')x=(Ω^2)^(2')x=(Ω^2)^(1+1')x=Ω^2'Ω^2'x=Ω^(1+1')Ω^(1+1')x
=(Ω'Ω)'(Ω'Ω)'x=Ω'Ω'Ω'Ω'x=Ω^(1+1+1+1')x=Ω4'x.
Exploring logic exploring everything that is subject to regularity, so everything outside of logic is happenstance. The law of induction cannot possibly be a law of logic, norcan it be an a priori law, since it is obviously a proposition that makes sense. The law of causality is not a law but the form of a law. 'Law of causality' that is the name of a type. Just as mechanics has minimizing principles, such as the law of least action, so too does physics have causal laws, laws of the causal form. In fact, one even surmised that there must be a 'law of least action' before know ing exactly how it went. (As always, what is a priori proves to be purely logical.)
We do not believe the law of conservation a priori, but rather know a priori that such a logical form is possible. All such propositions, including the principle of sufficient reason, tile laws of continuity in nature and of least effort in nature, etc. etc., all these are a priori insights about the forms which the propositions of science can take.
Newtonian mechanics, for example, provides a unified form for describing the world. Let us imagine a white surface with irregular black spots on it. Whatever kind of image these make, one can always approximate its description as closely as one wishes by covering the surface with a sufficiently fine square grid, and then declaring every square black or white. This grid provided a unified form for describing the surface. That form is optional, since I using a triangular or hexagonal grid would have achieved the same result. It could be that a triangular grid would have been simpler; that is to say, that we could describe the surface more accurately with a coarse triangular grid than with a fine square grid (or conversely), and so on. The different grids correspond to different systems for describing the world.
Mechanics determines one particular form of description of the world by saying: All propositions describing the world must be obtained in a given way from a given set of propositions, the axioms of mechanics. It supplies the bricks for building the edifice of science, and says: 'Any building that you want to erect, it must be with these and only these components.' (Just as we must be able to write down any amount we wish using the number system, so we must be able to write down any proposition of physics that we wish using the system of mechanics.)
(Ω^ν)^μ'x=Ω^(ν×μ')x Def.,
Ω^(2×2')x=(Ω^2)^(2')x=(Ω^2)^(1+1')x=Ω^2'Ω^2'x=Ω^(1+1')Ω^(1+1')x
=(Ω'Ω)'(Ω'Ω)'x=Ω'Ω'Ω'Ω'x=Ω^(1+1+1+1')x=Ω4'x.
Exploring logic exploring everything that is subject to regularity, so everything outside of logic is happenstance. The law of induction cannot possibly be a law of logic, norcan it be an a priori law, since it is obviously a proposition that makes sense. The law of causality is not a law but the form of a law. 'Law of causality' that is the name of a type. Just as mechanics has minimizing principles, such as the law of least action, so too does physics have causal laws, laws of the causal form. In fact, one even surmised that there must be a 'law of least action' before know ing exactly how it went. (As always, what is a priori proves to be purely logical.)
We do not believe the law of conservation a priori, but rather know a priori that such a logical form is possible. All such propositions, including the principle of sufficient reason, tile laws of continuity in nature and of least effort in nature, etc. etc., all these are a priori insights about the forms which the propositions of science can take.
Newtonian mechanics, for example, provides a unified form for describing the world. Let us imagine a white surface with irregular black spots on it. Whatever kind of image these make, one can always approximate its description as closely as one wishes by covering the surface with a sufficiently fine square grid, and then declaring every square black or white. This grid provided a unified form for describing the surface. That form is optional, since I using a triangular or hexagonal grid would have achieved the same result. It could be that a triangular grid would have been simpler; that is to say, that we could describe the surface more accurately with a coarse triangular grid than with a fine square grid (or conversely), and so on. The different grids correspond to different systems for describing the world.
Mechanics determines one particular form of description of the world by saying: All propositions describing the world must be obtained in a given way from a given set of propositions, the axioms of mechanics. It supplies the bricks for building the edifice of science, and says: 'Any building that you want to erect, it must be with these and only these components.' (Just as we must be able to write down any amount we wish using the number system, so we must be able to write down any proposition of physics that we wish using the system of mechanics.)
Logic is transcendental.
Logic is not a body of doctrine, but a mirror-image of the world. Logic is transcendental. Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-propositions. A proposition of mathematics does not express a thought.
Indeed in real life a mathematical proposition is never what one needs. Rather, one uses mathematical propositions only to make inferences from propositions that do not belong to mathematics to others that likewise do not belong to mathematics. (In philosophy the question, 'What do we actually use this word or this proposition for?' repeatedly leads to valuable insights.)
The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics. If two expressions are joined by the sign of equality, they can be substituted for one another. But it must be manifest in the two expressions themselves whether this is the case or not. When two expressions can be substituted for one another, that characterizes their logical form.
It is a property of affirmation that it can be construed as double negation. It is a property of '1 + 1 + 1 + 1' that it can be construed as '(1 + 1) + (1 + 1)'.
Frege says that the two expressions have the same meaning but different senses. But the essential point about an equation is that it is not necessary in order to show that the two expressions connected by the sign of equality have the same meaning, since this can be seen from the two expressions themselves. And the possibility of proving the propositions of mathematics means simply that their correctness can be perceived without its being necessary that what they express should itself be compared with the facts in order to determine its correctness.
It is impossible to assert the identity of meaning of two expressions. For in order to be able to assert anything about their meaning, I must know their meaning, and I cannot know their meaning without knowing whether what they mean is the same or different. An equation merely marks the point of view from which I consider the two expressions, it marks their equivalence in meaning.
Intuition is needed to solve mathematical, but language itself provides the necessary intuition. The process of calculating serves to bring about that intuition. Calculation is not an experiment.
Indeed in real life a mathematical proposition is never what one needs. Rather, one uses mathematical propositions only to make inferences from propositions that do not belong to mathematics to others that likewise do not belong to mathematics. (In philosophy the question, 'What do we actually use this word or this proposition for?' repeatedly leads to valuable insights.)
The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics. If two expressions are joined by the sign of equality, they can be substituted for one another. But it must be manifest in the two expressions themselves whether this is the case or not. When two expressions can be substituted for one another, that characterizes their logical form.
It is a property of affirmation that it can be construed as double negation. It is a property of '1 + 1 + 1 + 1' that it can be construed as '(1 + 1) + (1 + 1)'.
Frege says that the two expressions have the same meaning but different senses. But the essential point about an equation is that it is not necessary in order to show that the two expressions connected by the sign of equality have the same meaning, since this can be seen from the two expressions themselves. And the possibility of proving the propositions of mathematics means simply that their correctness can be perceived without its being necessary that what they express should itself be compared with the facts in order to determine its correctness.
It is impossible to assert the identity of meaning of two expressions. For in order to be able to assert anything about their meaning, I must know their meaning, and I cannot know their meaning without knowing whether what they mean is the same or different. An equation merely marks the point of view from which I consider the two expressions, it marks their equivalence in meaning.
Intuition is needed to solve mathematical, but language itself provides the necessary intuition. The process of calculating serves to bring about that intuition. Calculation is not an experiment.
One can describe all true logical propositions in advance.
It is possible, even according to the old conception of logic, to describe all true logical propositions in advance. Hence there can never be surprises in logic.
One can calculate whether a proposition belongs to logic, by calculating the logical properties of the symbol. This is how one proves a logical proposition. For, without bothering about sense or meaning, we construct the logical proposition out of others using only rules that deal with signs. One proves logical propositions by generating them from logical propositions by successively applying certain operations that always generate further tautologies out of the initial ones. (And in fact only tautologies follow from a tautology.) Of course this way of showing that the propositions of logic are tautologies is not at all essential to logic, if only because the propositions from which the proof starts must show without any proof that they are tautologies.
In logic process and result are equivalent. (Hence no surprise.) Proof in logic is merely a mechanical expedient to facilitate the recognition of tautologies in complicated cases. It would be altogether too remarkable if a proposition that had sense could be proved logically from others, and so too could a logical proposition. It is clear from the start that a logical proof of a proposition that makes sense and a proof in logic must be two entirely different things.
A proposition that makes sense states something, which is shown by its proof to be so. In logic every proposition is the form of a proof. Every proposition of logic is a modus ponens represented in signs. (And one cannot express the modus ponens by means of a proposition.) It is always possible to construe logic so that every proposition is its own proof.
All the propositions of logic are of equal status: it is not the case that some of them are essentially derived propositions. Every tautology itself shows that it is a tautology.
The number of primitive propositions of logic is arbitrary, since one could derive logic from a single primitive proposition, e.g. by simply constructing the logical product of Frege's primitive propositions. (Frege would perhaps say that we should then no longer have an immediately self-evident primitive proposition. But it is remarkable that a thinker as rigorous as Frege appealed to the degree of self-evidence as the criterion of a logical proposition.)
One can calculate whether a proposition belongs to logic, by calculating the logical properties of the symbol. This is how one proves a logical proposition. For, without bothering about sense or meaning, we construct the logical proposition out of others using only rules that deal with signs. One proves logical propositions by generating them from logical propositions by successively applying certain operations that always generate further tautologies out of the initial ones. (And in fact only tautologies follow from a tautology.) Of course this way of showing that the propositions of logic are tautologies is not at all essential to logic, if only because the propositions from which the proof starts must show without any proof that they are tautologies.
In logic process and result are equivalent. (Hence no surprise.) Proof in logic is merely a mechanical expedient to facilitate the recognition of tautologies in complicated cases. It would be altogether too remarkable if a proposition that had sense could be proved logically from others, and so too could a logical proposition. It is clear from the start that a logical proof of a proposition that makes sense and a proof in logic must be two entirely different things.
A proposition that makes sense states something, which is shown by its proof to be so. In logic every proposition is the form of a proof. Every proposition of logic is a modus ponens represented in signs. (And one cannot express the modus ponens by means of a proposition.) It is always possible to construe logic so that every proposition is its own proof.
All the propositions of logic are of equal status: it is not the case that some of them are essentially derived propositions. Every tautology itself shows that it is a tautology.
The number of primitive propositions of logic is arbitrary, since one could derive logic from a single primitive proposition, e.g. by simply constructing the logical product of Frege's primitive propositions. (Frege would perhaps say that we should then no longer have an immediately self-evident primitive proposition. But it is remarkable that a thinker as rigorous as Frege appealed to the degree of self-evidence as the criterion of a logical proposition.)
We can do without logical propositions.
The propositions of logic demonstrate the logical properties of propositions in that they combine them to form propositions without content. This could also be called a null method. In a logical proposition, propositions are brought into equilibrium with one another, and that equilibrium then indicates what the logical constitution of these propositions must be.
It follows from this that we can even do without logical propositions because a suitable notation enables one to recognize the formal properties of propositions by inspection. If, for example, two propositions (p) and (q) in the compound proposition (p⊃q) yield a tautology, then it is clear that (q) follows from (p). That (q) follows from (p⊃q.p) is seen from the two propositions themselves, but it is also possible to show it by combining them to form (p⊃q.p:⊃:q), and then showing that this is a tautology.
Logical propositions cannot be confirmed by experience any more than they can be refuted by it. Not only must a proposition of logic be irrefutable by any possible experience, but it must also be unconfirmable by any possible experience. We can postulate the truths of logic in sthat we can postulate an adequate notation. Clearly: the laws of logic cannot themselves be subject to laws of logic. (There is not, as Russell thought, a special law of contradiction for each 'type'; one law is enough instead, since it is not applied to itself.)
The mark of a logical proposition is not general validity, since that only means to be valid for all things by happenstance. An ungeneralized proposition can just as well be tautological as a generalized one.
We could call logical generality essential, in contrast with the accidental generality of such propositions as 'All men are mortal'. Propositions like Russell's 'axiom of reducibility' are not logical propositions, and this explains our feeling that, even if they were true, their truth could only be the result of a fortunate accident.
One can imagine a world in which the axiom of reducibility is not valid. But it is clear that logic has nothing to do with the question of whether our world is really so or not.
Logical propositions describe the structural skeleton of the world. They have no content on their own, but presuppose that names have meaning and elementary propositions make sense; and that connects them to the world. Clearly, it must show something about the world that certain conjunctions of symbols—that in essence have a specific character—are tautologies. This is a decisive point.
Some things are arbitrary in the symbols that we use and some things are not. In logic, only the latter express. That does not mean we express what we wish with the help of signs, but rather that one in which the nature of the absolutely necessary signs speaks for itself. If we know the logical syntax of any symbolic language, then we have already been given all the propositions of logic.
It follows from this that we can even do without logical propositions because a suitable notation enables one to recognize the formal properties of propositions by inspection. If, for example, two propositions (p) and (q) in the compound proposition (p⊃q) yield a tautology, then it is clear that (q) follows from (p). That (q) follows from (p⊃q.p) is seen from the two propositions themselves, but it is also possible to show it by combining them to form (p⊃q.p:⊃:q), and then showing that this is a tautology.
Logical propositions cannot be confirmed by experience any more than they can be refuted by it. Not only must a proposition of logic be irrefutable by any possible experience, but it must also be unconfirmable by any possible experience. We can postulate the truths of logic in sthat we can postulate an adequate notation. Clearly: the laws of logic cannot themselves be subject to laws of logic. (There is not, as Russell thought, a special law of contradiction for each 'type'; one law is enough instead, since it is not applied to itself.)
The mark of a logical proposition is not general validity, since that only means to be valid for all things by happenstance. An ungeneralized proposition can just as well be tautological as a generalized one.
We could call logical generality essential, in contrast with the accidental generality of such propositions as 'All men are mortal'. Propositions like Russell's 'axiom of reducibility' are not logical propositions, and this explains our feeling that, even if they were true, their truth could only be the result of a fortunate accident.
One can imagine a world in which the axiom of reducibility is not valid. But it is clear that logic has nothing to do with the question of whether our world is really so or not.
Logical propositions describe the structural skeleton of the world. They have no content on their own, but presuppose that names have meaning and elementary propositions make sense; and that connects them to the world. Clearly, it must show something about the world that certain conjunctions of symbols—that in essence have a specific character—are tautologies. This is a decisive point.
Some things are arbitrary in the symbols that we use and some things are not. In logic, only the latter express. That does not mean we express what we wish with the help of signs, but rather that one in which the nature of the absolutely necessary signs speaks for itself. If we know the logical syntax of any symbolic language, then we have already been given all the propositions of logic.
Recognizing a Tautology.
In order to recognize a tautology in cases where no generality-sign occurs in it, one can employ the following method.
Instead of 'p', 'q', 'r', etc. one writes elemental propositions as 'WpF', 'WqF', 'WrF', etc. ('Wahr' is 'True' in German.)
One expresses combinations using brackets, e.g.
Lines connect the truth or falsity of the compound proposition with the truth value combinations that are its arguments.
This sign, for instance, represents the proposition (p⊃q) 'p implies q'.
Now, one can examine the proposition ~(p .~p) (the law of contradiction) in order to determine whether it is a tautology.
In our notation the form '~ξ' is written as
and the form 'ξ.η' as:
Hence, one writes the form ~(~p .q) as
When one substitutes 'p' for 'q' and examines how the outermost T and F connect to the innermost, the result will be that the truth of the whole proposition is correlated with all the combinations of its argument, and its falsity with none of them.
Instead of 'p', 'q', 'r', etc. one writes elemental propositions as 'WpF', 'WqF', 'WrF', etc. ('Wahr' is 'True' in German.)
One expresses combinations using brackets, e.g.
Lines connect the truth or falsity of the compound proposition with the truth value combinations that are its arguments.
This sign, for instance, represents the proposition (p⊃q) 'p implies q'.
Now, one can examine the proposition ~(p .~p) (the law of contradiction) in order to determine whether it is a tautology.
In our notation the form '~ξ' is written as
and the form 'ξ.η' as:
Hence, one writes the form ~(~p .q) as
When one substitutes 'p' for 'q' and examines how the outermost T and F connect to the innermost, the result will be that the truth of the whole proposition is correlated with all the combinations of its argument, and its falsity with none of them.
Saturday, April 5, 2008
The propositions of logic are tautologies.
Thus, propositions of logic (analytic propositions) tell one nothing. So theories that let a logical statement appear to have content are always invalid. One might think, for example, that the words 'true' and 'false' signified properties just like others. Remarkably, every proposition has one of these two properties. Now that seems nothing less than obvious, just as, for instance, the proposition, 'All roses are either yellow or red', would sound obvious if it were true. Indeed, the latter acquires all the characteristics of a proposition of natural science - a sure indication that it has been misconstrued. A valid explanation of the propositions of logic must assign them a unique status among all propositions.
It is the unique characteristic of logical propositions that one can recognize that they are true from the symbol alone. In itself, this fact contains the whole philosophy of logic. And so it is also one of the most important facts that the truth or falsity of non-logical propositions can not be recognized from the statement alone.
That the propositions of logic are tautologies, is shown by the formal, logical properties of language, of the world. That the constituents of logic, thus connected, yield a tautology, characterizes the logic of those constituents. Propositions must have certain structural properties to yield a tautology when joined in an appropriate way. When they do yield a tautology, that shows they possess these structural properties.
That, for example, the propositions (p) and (~p) yield a tautology in the combination (~(p . ~p)) shows that they contradict one another. The fact that the propositions (p⊃q), (p), and (q), combined with one another in the form ((p⊃q).(p):⊃:(q)), yield a tautology shows that q follows from p and (p⊃q). That ((x).fx:⊃:fa) is a tautology shows that (fa) follows from ((x).fx) etc.
It is clear that one would reach the same conclusions using contradictions instead of tautologies.
It is the unique characteristic of logical propositions that one can recognize that they are true from the symbol alone. In itself, this fact contains the whole philosophy of logic. And so it is also one of the most important facts that the truth or falsity of non-logical propositions can not be recognized from the statement alone.
That the propositions of logic are tautologies, is shown by the formal, logical properties of language, of the world. That the constituents of logic, thus connected, yield a tautology, characterizes the logic of those constituents. Propositions must have certain structural properties to yield a tautology when joined in an appropriate way. When they do yield a tautology, that shows they possess these structural properties.
That, for example, the propositions (p) and (~p) yield a tautology in the combination (~(p . ~p)) shows that they contradict one another. The fact that the propositions (p⊃q), (p), and (q), combined with one another in the form ((p⊃q).(p):⊃:(q)), yield a tautology shows that q follows from p and (p⊃q). That ((x).fx:⊃:fa) is a tautology shows that (fa) follows from ((x).fx) etc.
It is clear that one would reach the same conclusions using contradictions instead of tautologies.
The General Form of a Truth Function.
Wittgenstein did not fully explain his symbolism in the Tractatus. The following uses text from Russell's introduction.
The general form of a truth function is: [p-, ξ-, N(ξ-)] where
p- stands for all elemental propositions,
ξ- stands for any set of propositions, and
N(ξ-) stands for the negation of all members of ξ-.
This general form of a proposition simply says that every proposition generated by successive negations of elemental propositions. The whole symbol [p-, ξ-, N(ξ-)] is an algorithm:
• select a set of any of the atomic propositions,
• negate them all,
• then take any selection of the set of propositions now obtained,
• together with any of the originals
• -- and so on indefinitely.
This describes a procedure which, when performed on given elemental propositions, can generate all other propositions that are not elemental. The process depends upon:
(a) Sheffer's proof that all truth-functions can be obtained out of simultaneous negation, i.e. out of ``not-p and not-q'';
(b) Wittgenstein's theory of the derivation of general propositions from conjunctions and disjunctions; and
(c) The assertion that a proposition can only occur in another proposition as argument to a truth-function.
If the general form of constructing propositions is given, then how one proposition can be generated from another is also given. Thus, the general form of an operation Ω'(η-) can be written as: [ξ-, N(ξ-)]'(η-) or as [η-,ξ-,N(ξ-)]. This is the most general form of transition from one proposition to another.
To apply this method to arrive at numbers one defines:
0+1+1+1=3 Def.,
etc.
One sees from this that a number is the exponent of an operation.
The concept number is just the general form of a number which is common to all numbers; it is the variable number. Also, the concept numerical equality is the general form of all particular cases of numerical equality.
So the general form of an integer is: [0, ξ, ξ+1].
The theory of classes is completely superfluous in mathematics due to the fact that the generality required in mathematics is not accidental generality.
The general form of a truth function is: [p-, ξ-, N(ξ-)] where
p- stands for all elemental propositions,
ξ- stands for any set of propositions, and
N(ξ-) stands for the negation of all members of ξ-.
This general form of a proposition simply says that every proposition generated by successive negations of elemental propositions. The whole symbol [p-, ξ-, N(ξ-)] is an algorithm:
• select a set of any of the atomic propositions,
• negate them all,
• then take any selection of the set of propositions now obtained,
• together with any of the originals
• -- and so on indefinitely.
This describes a procedure which, when performed on given elemental propositions, can generate all other propositions that are not elemental. The process depends upon:
(a) Sheffer's proof that all truth-functions can be obtained out of simultaneous negation, i.e. out of ``not-p and not-q'';
(b) Wittgenstein's theory of the derivation of general propositions from conjunctions and disjunctions; and
(c) The assertion that a proposition can only occur in another proposition as argument to a truth-function.
If the general form of constructing propositions is given, then how one proposition can be generated from another is also given. Thus, the general form of an operation Ω'(η-) can be written as: [ξ-, N(ξ-)]'(η-) or as [η-,ξ-,N(ξ-)]. This is the most general form of transition from one proposition to another.
To apply this method to arrive at numbers one defines:
x=Ω^(0)'x Def.
andΩ'Ω^(ν)'x=Ω^(ν+1)'x Def.
Now, according to the rules of our notation, one write the series
Now, according to the rules of our notation, one write the series
x, Ω'x, Ω'Ω'x, Ω'Ω'Ω'x,...,
so thatΩ^(0)'x, Ω^(0+1)'x, Ω^(0+1+1)'x, Ω^(0+1+1+1)'x, .....
Therfore, one can rewrite the general form [x, ξ, Ω'ξ]) as
Therfore, one can rewrite the general form [x, ξ, Ω'ξ]) as
[Ω0'x, Ων'x, Ων+1'x]
and define:0+1=1 Def.,
0+1+1=2 Def.,0+1+1+1=3 Def.,
etc.
The concept number is just the general form of a number which is common to all numbers; it is the variable number. Also, the concept numerical equality is the general form of all particular cases of numerical equality.
So the general form of an integer is: [0, ξ, ξ+1].
The theory of classes is completely superfluous in mathematics due to the fact that the generality required in mathematics is not accidental generality.
The microcosm.
I am my universe of discourse; there is no thinking, imagining subject.
If one wrote a book called "The World as I found it", it would include a report on the body that tells which parts were subordinate to the will, and which were not, etc. Now this would be a way to isolate the subject, or rather of showing that in an important sense there is none. For it alone could not be discussed in that book. The subject is not a part of the world, but part of its boundary.
Where in the world is a metaphysical subject to be perceived? One could say that this is just like the eye and the visual field. But really you do not see the eye. And nothing in the visual field allows you to infer that it is seen by an eye.
In this connection, experience is not a priori. All one sees could be different. All one can describe at all could be different. There is no way things should, a priori, be.
Here one sees that strict solipsism coincides with pure realism. The subject of solipsism shrinks to a point without extension, but the reality associated with it remains. Thus there really is a sense in which philosophy can talk about the self in a non-psychological way. The self enters philosophy through the fact that 'the world is my world'.
The philosophical self is not the human being, not the human body, or the human psyche of psychology, but rather the metaphysical subject, the boundary - not a part - of the world.
If one wrote a book called "The World as I found it", it would include a report on the body that tells which parts were subordinate to the will, and which were not, etc. Now this would be a way to isolate the subject, or rather of showing that in an important sense there is none. For it alone could not be discussed in that book. The subject is not a part of the world, but part of its boundary.
Where in the world is a metaphysical subject to be perceived? One could say that this is just like the eye and the visual field. But really you do not see the eye. And nothing in the visual field allows you to infer that it is seen by an eye.
In this connection, experience is not a priori. All one sees could be different. All one can describe at all could be different. There is no way things should, a priori, be.
Here one sees that strict solipsism coincides with pure realism. The subject of solipsism shrinks to a point without extension, but the reality associated with it remains. Thus there really is a sense in which philosophy can talk about the self in a non-psychological way. The self enters philosophy through the fact that 'the world is my world'.
The philosophical self is not the human being, not the human body, or the human psyche of psychology, but rather the metaphysical subject, the boundary - not a part - of the world.
The boundary of my language represents the boundary of my world.
Logic spreads throughout the world, so the boundaries of the world are also its boundaries. One cannot logically say that the world has one thing in it but not the other. This exclusion would mean one can go beyond the boundaries of the world. That, after all, is the only way to view those boundaries from the other side. What we cannot think, we cannot think; so also, we cannot say what we cannot think.
This remark provides the key to deciding how much truth there is to solipsism. What the solipsist means to say is quite correct; only it cannot be said, but proves to be so. That the world is my world is shown by the fact that the limits of language (the only language I understand) comprise the limits of my world. The world and life are one.
This remark provides the key to deciding how much truth there is to solipsism. What the solipsist means to say is quite correct; only it cannot be said, but proves to be so. That the world is my world is shown by the fact that the limits of language (the only language I understand) comprise the limits of my world. The world and life are one.
Elemental Propositions.
We now discuss all possible forms of elemental propositions a priori. An elemental proposition consists of names. However, because we cannot state all the names with different meanings, we cannot state the composition of an elemental proposition.
Our fundamental principle is that deciding anything at all by logic must be straightforward. (If we find ourselves looking to experience for an answer, we are definitely on the wrong track.)
What one needs to understand the logic of something is not how that something is, but rather, that it is. But that is not an experience. Logic is prior to every experience, the experience that something is so. It is prior to 'How?' not prior to 'What?' And if this were not so, how could we apply logic? We might put it in this way: If logic could exist even if there were no world, how then could logic exist given that there is a world?
According to Russell, there are simple relations between different numbers of things (individuals). But between what numbers? And how is this supposed to be decided? By experience? (There is no pre-eminent number.)
Any specific form we use has to be completely arbitrary. For example, one should be able to say a priori whether the sign for a 27-termed relation will be needed order to signify something. But may we even ask such a question? Can one set up the form of a sign unless one knows what it is for? Does it make sense to ask: What must be for something to be the case?
It is clear that, apart from its special logical form, we have a concept of an elemental proposition. But when one can create symbols using a system, the system is what is logically important and not the individual symbols. How, then, can the forms in logic be something one can invent? Rather, one must deal with what enables one to invent them. The forms of elemental propositions cannot have a hierarchy. We can foresee only what we ourselves construct.
Empirical reality is limited by the entirety of objects. This limit appears again in the entirety of elemental propositions. Hierarchies are independent of reality, and necessarily so. If purely logical grounds tell us that elemental propositions must exist, then all that is required to know that is to understand them in their unanalyzed form.
In the event, all the sentences of our everyday language are entirely, logically ordered just as they are. That most simple thing that we now formulate is not a likeness of the truth, but in itself the whole truth. (Our problems are not abstract, but perhaps the most concrete that there are.)
Logic is applied to decide what elemental propositions there are, but logic cannot anticipate what it will be applied to. It is clear that logic must not clash with its application, but must be in touch with it. In any case, logic and its application must not overlap. Since one cannot state elemental propositions a priori, then trying to state them anyway makes no sense.
Our fundamental principle is that deciding anything at all by logic must be straightforward. (If we find ourselves looking to experience for an answer, we are definitely on the wrong track.)
What one needs to understand the logic of something is not how that something is, but rather, that it is. But that is not an experience. Logic is prior to every experience, the experience that something is so. It is prior to 'How?' not prior to 'What?' And if this were not so, how could we apply logic? We might put it in this way: If logic could exist even if there were no world, how then could logic exist given that there is a world?
According to Russell, there are simple relations between different numbers of things (individuals). But between what numbers? And how is this supposed to be decided? By experience? (There is no pre-eminent number.)
Any specific form we use has to be completely arbitrary. For example, one should be able to say a priori whether the sign for a 27-termed relation will be needed order to signify something. But may we even ask such a question? Can one set up the form of a sign unless one knows what it is for? Does it make sense to ask: What must be for something to be the case?
It is clear that, apart from its special logical form, we have a concept of an elemental proposition. But when one can create symbols using a system, the system is what is logically important and not the individual symbols. How, then, can the forms in logic be something one can invent? Rather, one must deal with what enables one to invent them. The forms of elemental propositions cannot have a hierarchy. We can foresee only what we ourselves construct.
Empirical reality is limited by the entirety of objects. This limit appears again in the entirety of elemental propositions. Hierarchies are independent of reality, and necessarily so. If purely logical grounds tell us that elemental propositions must exist, then all that is required to know that is to understand them in their unanalyzed form.
In the event, all the sentences of our everyday language are entirely, logically ordered just as they are. That most simple thing that we now formulate is not a likeness of the truth, but in itself the whole truth. (Our problems are not abstract, but perhaps the most concrete that there are.)
Logic is applied to decide what elemental propositions there are, but logic cannot anticipate what it will be applied to. It is clear that logic must not clash with its application, but must be in touch with it. In any case, logic and its application must not overlap. Since one cannot state elemental propositions a priori, then trying to state them anyway makes no sense.
Propositions occur in each other only as bases of a truth operation.
The general propositional form permits one proposition to contain another only as the basis of a truth operation, even though, at first sight, a different way seems possible.
In certain forms of proposition in psychology, such as 'A believes that p is the case' and 'A has the thought p', etc. it seems as if the proposition p stood in some kind of relation to an object A. In the event, however, 'A believes that p', 'A thinks p', and 'A says p' are actually the form 'p' says 'p'. This last is not an assignment of a fact to an object. Rather, facts are assigned to each other by assigning their objects to each other.
This also shows that the psyche - the subject, etc. - as conceived in Freud's psychoanalysis makes no sense. Namely, a composite soul would no longer be a soul.
The correct explanation of the form of the proposition, 'A concludes that p', must show that one cannot reach a conclusion that does not make sense. (Russell's theory does not satisfy this requirement.) To perceive a complex means to perceive that its constituents are related to one another in a particular way.
This, no doubt. also explains why there are two possible ways of seeing the figures in the drawings of M.C. Escher. We really do see two different facts. Each alternative perception of the drawing, though it is an illusion, is a fact. If I first perceive one and then the other, I have imagined two different images although all else has remained the same.
In certain forms of proposition in psychology, such as 'A believes that p is the case' and 'A has the thought p', etc. it seems as if the proposition p stood in some kind of relation to an object A. In the event, however, 'A believes that p', 'A thinks p', and 'A says p' are actually the form 'p' says 'p'. This last is not an assignment of a fact to an object. Rather, facts are assigned to each other by assigning their objects to each other.
This also shows that the psyche - the subject, etc. - as conceived in Freud's psychoanalysis makes no sense. Namely, a composite soul would no longer be a soul.
The correct explanation of the form of the proposition, 'A concludes that p', must show that one cannot reach a conclusion that does not make sense. (Russell's theory does not satisfy this requirement.) To perceive a complex means to perceive that its constituents are related to one another in a particular way.
This, no doubt. also explains why there are two possible ways of seeing the figures in the drawings of M.C. Escher. We really do see two different facts. Each alternative perception of the drawing, though it is an illusion, is a fact. If I first perceive one and then the other, I have imagined two different images although all else has remained the same.
Friday, April 4, 2008
Expressions.
On occasion, one is tempted to use forms such as (a = a) or (p ⊃ p) and the like. In fact, this happens when discussing prototypes, such as proposition, thing, etc. Thus in Russell's Principles of Mathematics, the phrase 'p is a proposition' or (p ⊃ p) was placed in front of certain propositions as an hypothesis in order to exclude everything but propositions from their arguments. But that makes no sense. A non-proposition as argument does not make the hypothesis false but empty, and the wrong kind of arguments make the proposition itself empty. So it prevents invalid no worse than the empty hypothesis.
The same would apply if one wanted to express 'There are no things' by writing (~(∃x).x =x). But even if this were a proposition, would it not be equally true if in fact 'there were things' but they were not identical with themselves?
The same would apply if one wanted to express 'There are no things' by writing (~(∃x).x =x). But even if this were a proposition, would it not be equally true if in fact 'there were things' but they were not identical with themselves?
Identity.
Wittgenstein's notation does not use a sign of identity, but expresses the identity of an object by using the identical sign for it; if the objects are different, so are the signs. Identity is not a relation between objects. This becomes very clear if one considers the example: "All x such that x satifies the function f and x=a."
By the way: to say two things are identical is absurd, and to say that one thing is identical with itself is to say nothing at all.
In Wittgenstein's notation, one does not write (f(a, b).a = b) but (f(a, a)) or (f(b, b)).
Not (f(a,b).~a = b) but (f(a, b)).
Analogously, one does not write ((∃x, y). f(x, y). x = y), but ((∃x).f(x, x));
and not ((∃x, y).f(x, y)).~x = y), but ((∃x, y).f(x, y)).
So instead of Russels ((∃x,y).f(x,y)) one writes ((∃x,y).f(x,y).∨.(∃x).f(x,x)).
Instead of ((x):fx⊃x = a) one writes ((∃x).fx.⊃:(∃x, y).fx.fy).
And the proposition, 'Only one x satisfies f( )', will read ((∃x).fx:~(∃x, y).fx.fy).
The identity-sign, therefore, So a sign of identity is not essential in a correct conceptual notation, and pseudo-propositions like (a=a), (a=b.b=c.⊃a=c), (x).x = x), ((∃x).x = a), etc. cannot even be written down. This also disposes of all the problems that were connected with them.
All the problems with Russell's 'axiom of infinity' can be solved now by considering that it would be expressed in language by the existence of infinitely many names with different meanings.
((x):fx.⊃.x = a).
This proposition simply says: "Only 'a' satisfies the function 'f', and not that only things that have a certain relation to 'a' satisfy the function. Of course, it might then be said that only 'a' has this relation to 'a'; but in order to express that, we would have to use the identity sign itself. Russell's definition of '=' is inadequate for this purpose, because according to it we cannot say that two objects have all their properties in common. (Even though this proposition is never correct, it still makes sense.)By the way: to say two things are identical is absurd, and to say that one thing is identical with itself is to say nothing at all.
In Wittgenstein's notation, one does not write (f(a, b).a = b) but (f(a, a)) or (f(b, b)).
Not (f(a,b).~a = b) but (f(a, b)).
Analogously, one does not write ((∃x, y). f(x, y). x = y), but ((∃x).f(x, x));
and not ((∃x, y).f(x, y)).~x = y), but ((∃x, y).f(x, y)).
So instead of Russels ((∃x,y).f(x,y)) one writes ((∃x,y).f(x,y).∨.(∃x).f(x,x)).
Instead of ((x):fx⊃x = a) one writes ((∃x).fx.⊃:(∃x, y).fx.fy).
And the proposition, 'Only one x satisfies f( )', will read ((∃x).fx:~(∃x, y).fx.fy).
The identity-sign, therefore, So a sign of identity is not essential in a correct conceptual notation, and pseudo-propositions like (a=a), (a=b.b=c.⊃a=c), (x).x = x), ((∃x).x = a), etc. cannot even be written down. This also disposes of all the problems that were connected with them.
All the problems with Russell's 'axiom of infinity' can be solved now by considering that it would be expressed in language by the existence of infinitely many names with different meanings.
Truth Functions do not Include the Concept All.
Wittgenstein dissociates the concept all from truth-functions. If the values of ξ are all the values of a function fx for all values of x, then its negation negates them all.
Frege and Russell introduced generality in association with logical product or logical sum. This made it difficult to understand the propositions ((∃x). fx) and ((x).fx), which include both ideas.
What is peculiar to the generality-sign is first, that it indicates a logical prototype, and secondly, that it gives prominence to constants. The generality-sign appears as an argument. But if objects are given, then that already gives us all objects. If elementary propositions are given, then all elementary propositions are thereby given.
It is incorrect to render the proposition '(∃x) . fx' in the words, 'fx is possible ' as Russell does. The certainty, possibility, or impossibility of a situation is not expressed by a proposition. Rather we determine which applies by an expression's being a tautology, a proposition that makes sense, or a contradiction. The precedent which we cite must lie in the symbol itself.
One can describe the world completely by means of fully generalized propositions, which means that one need not begin by assigning a name to any particular object. To then arrive at a conventional expression, one must simply follow the expression: " There is one and only one x such that ... " with: "And this x is a".
A fully generalized proposition, like every other proposition, is composite. (This is shown by the fact that in ((∃x, φ).φx) we have to mention 'φ' and 'x' separately. They both, independently, stand in signifying relations to the world, just as is the case in ungeneralized propositions.) It is characteristic of a composite symbol to have something in common with other symbols.
The truth or falsity of every proposition, after all, changes the general construction of the world. And the range that the totality of elementary propositions leaves open for its construction is exactly the same as that which is delimited by entirely general propositions. (If an elementary proposition is true, that means, at any rate, one more elementary proposition i true.)
N(ξ‾) = ~(∃x).fx
Frege and Russell introduced generality in association with logical product or logical sum. This made it difficult to understand the propositions ((∃x). fx) and ((x).fx), which include both ideas.
What is peculiar to the generality-sign is first, that it indicates a logical prototype, and secondly, that it gives prominence to constants. The generality-sign appears as an argument. But if objects are given, then that already gives us all objects. If elementary propositions are given, then all elementary propositions are thereby given.
It is incorrect to render the proposition '(∃x) . fx' in the words, 'fx is possible ' as Russell does. The certainty, possibility, or impossibility of a situation is not expressed by a proposition. Rather we determine which applies by an expression's being a tautology, a proposition that makes sense, or a contradiction. The precedent which we cite must lie in the symbol itself.
One can describe the world completely by means of fully generalized propositions, which means that one need not begin by assigning a name to any particular object. To then arrive at a conventional expression, one must simply follow the expression: " There is one and only one x such that ... " with: "And this x is a".
A fully generalized proposition, like every other proposition, is composite. (This is shown by the fact that in ((∃x, φ).φx) we have to mention 'φ' and 'x' separately. They both, independently, stand in signifying relations to the world, just as is the case in ungeneralized propositions.) It is characteristic of a composite symbol to have something in common with other symbols.
The truth or falsity of every proposition, after all, changes the general construction of the world. And the range that the totality of elementary propositions leaves open for its construction is exactly the same as that which is delimited by entirely general propositions. (If an elementary proposition is true, that means, at any rate, one more elementary proposition i true.)
How is this useful?
How can such specialized rigs and catches be used so that logic acts as the all inclusive the mirror of the world? Well, in that they all hook up in an infinitely fine grid to form that great mirror.
(~p) is true if 'p' is false. Therefore, in the case when the proposition (~p) is true, proposition (p) is false. How then can the prefixed tilde '~' make it agree with reality? In (~p) it is not '~' that negates, but rather what it is that all the signs in this notation that negate p have in common. And that is the common rule that governs the construction of (~p), (~~~p), (~p ∨ ~p), (~p.~p), etc. etc. ad inf. It is this common rule that mirrors negation.
We might say: What is common to all symbols that affirm both p and q is the proposition (p.q). What is common to all symbols that affirm either p or q is the proposition (p∨q). Similarly, we can say that two propositions are opposed to one another if they have nothing in common with one another. And we can also say that every proposition has only one negative, since there is only one proposition that lies completely outside if it. Thus, in Russell's notation as well, we see that (q:p∨~p) says the same thing as 'q', and that (p∨~p) says nothing.
Once a notation has been established, it will contain a rule governing the construction of all propositions that negate p, affirm p, affirm p or q, and so on. These rules are equivalent to the symbols and reflect their sense.
Our symbols themselves must show that compounds created by using '∨', '.', etc. can only be propositions. This is indeed the case, since the symbol in (p) and (q) itself presupposes '∨', '~', etc. If the sign 'p' in (p∨q) could not stand for a complex sign, then it would not make sense in isolation. But in that case, the signs (p∨p), (p.p), etc., which have the same sense as (p), could not make sense either. But if (p∨p) makes no sense, then neither can (p∨q).
Must the sign of a negative proposition be constructed from that of the positive proposition? (Such as: when 'a' does not stand in a certain relation to 'b' then this could say that (aRb) was not the case.) The positive proposition necessarily presupposes the existence of the negative proposition and vice versa.
(~p) is true if 'p' is false. Therefore, in the case when the proposition (~p) is true, proposition (p) is false. How then can the prefixed tilde '~' make it agree with reality? In (~p) it is not '~' that negates, but rather what it is that all the signs in this notation that negate p have in common. And that is the common rule that governs the construction of (~p), (~~~p), (~p ∨ ~p), (~p.~p), etc. etc. ad inf. It is this common rule that mirrors negation.
We might say: What is common to all symbols that affirm both p and q is the proposition (p.q). What is common to all symbols that affirm either p or q is the proposition (p∨q). Similarly, we can say that two propositions are opposed to one another if they have nothing in common with one another. And we can also say that every proposition has only one negative, since there is only one proposition that lies completely outside if it. Thus, in Russell's notation as well, we see that (q:p∨~p) says the same thing as 'q', and that (p∨~p) says nothing.
Once a notation has been established, it will contain a rule governing the construction of all propositions that negate p, affirm p, affirm p or q, and so on. These rules are equivalent to the symbols and reflect their sense.
Our symbols themselves must show that compounds created by using '∨', '.', etc. can only be propositions. This is indeed the case, since the symbol in (p) and (q) itself presupposes '∨', '~', etc. If the sign 'p' in (p∨q) could not stand for a complex sign, then it would not make sense in isolation. But in that case, the signs (p∨p), (p.p), etc., which have the same sense as (p), could not make sense either. But if (p∨p) makes no sense, then neither can (p∨q).
Must the sign of a negative proposition be constructed from that of the positive proposition? (Such as: when 'a' does not stand in a certain relation to 'b' then this could say that (aRb) was not the case.) The positive proposition necessarily presupposes the existence of the negative proposition and vice versa.
Thursday, April 3, 2008
Every truth-function an be obtained by successively negating elemental propositions.
Successive applications of the operator (-----T)(ξ,....) to elemental propositions can generate every truth-function. Wittgenstein calls it the negation of all the propositions in the right-hand pair of brackets.
Let the sign 'ξ' indicate a variable whose terms are propositions. Let ξ‾ indicate the list of all those values where the order of the terms is indifferent. The bar indicates that the variable represents of all its values. The declaration describes the propositions the variable represents. If ξ has the three values P, Q, R, then (ξ‾)=(P, Q, R).
What the values of the terms are are must be fixed, but how the each term of the bracketed expression are described is indifferent. There are three ways that it can be done:
Tt is now clear that this operation may be used to construct propositions, but exactly how it can be done must be made clear as well.
Let the sign 'ξ' indicate a variable whose terms are propositions. Let ξ‾ indicate the list of all those values where the order of the terms is indifferent. The bar indicates that the variable represents of all its values. The declaration describes the propositions the variable represents. If ξ has the three values P, Q, R, then (ξ‾)=(P, Q, R).
What the values of the terms are are must be fixed, but how the each term of the bracketed expression are described is indifferent. There are three ways that it can be done:
- Direct enumeration, in which case we can simply substitute constant values for the variable.
- Stating a function fx whose values for all values of x are the propositions to be described.
- Stating a formal law that governs the construction of the propositions. In that case the bracketed expression has all the terms of a series of forms as its members.
Tt is now clear that this operation may be used to construct propositions, but exactly how it can be done must be made clear as well.
Occam's rule points out that unnecessary signs mean nothing.
Logic must provide for itself, so if a sign is possible it must be able to signify. In logic, whatever can be done is also allowed. ('Socrates is identical' is absurd because 'identical' is not a property. The proposition makes no sense because we have not made an arbitrary assignment and not because the symbol itself is invalid.) In a certain sense, we cannot err in logic. Self-evidence can only be dispensed with in logic because language itself prevents every logical mistake. So logic is a priori because illogical thought is not possible: We cannot give a sign the wrong sense.
Occam's maxim is, of course, not an arbitrary rule, nor is it one justified by practice: It points out that unnecessary signs mean nothing. So signs that serve one purpose are logically equivalent, and signs that serve none are logically meaningless.
Frege says: any legitimately constructed proposition must make sense. Wittgenstein says: Any possible proposition is legitimately constructed, and can only be absurd because we have not given some of its constituents a meaning. (Even if we think that we have done so.) Thus 'Socrates is identical' says nothing because we gave the word 'identical' no meaning as an adjective. For when it appears as a sign for identity, the signifying relation is an entirely different one, so the symbols also are entirely different. In the two cases, the symbols have only the sign in common and that by chance.
The number of necessary fundamental operations depends only on our notation. We need only construct a system of signs with a particular number of dimensions - with a definite mathematical multiplicity. It is clear that this is not a matter of some primitive ideas that need a sign, but rather of expressing a rule.
Occam's maxim is, of course, not an arbitrary rule, nor is it one justified by practice: It points out that unnecessary signs mean nothing. So signs that serve one purpose are logically equivalent, and signs that serve none are logically meaningless.
Frege says: any legitimately constructed proposition must make sense. Wittgenstein says: Any possible proposition is legitimately constructed, and can only be absurd because we have not given some of its constituents a meaning. (Even if we think that we have done so.) Thus 'Socrates is identical' says nothing because we gave the word 'identical' no meaning as an adjective. For when it appears as a sign for identity, the signifying relation is an entirely different one, so the symbols also are entirely different. In the two cases, the symbols have only the sign in common and that by chance.
The number of necessary fundamental operations depends only on our notation. We need only construct a system of signs with a particular number of dimensions - with a definite mathematical multiplicity. It is clear that this is not a matter of some primitive ideas that need a sign, but rather of expressing a rule.
Signs for logical operations are punctuation marks.
When logical signs are introduced properly, then one has in effect also introduced the content of all their combinations; i.e. not only (p ∨ q) but (~(p ∨ q)) as well, etc. etc. Thus it is clear that the actual general primitive signs are not (p ∨ q), ((∃x).fx), etc. but the most general ones required to form them and their combinations. The seemingly unimportant fact that, unlike real relations, the pseudo-relations of logic, such as '∨' and '⊃', need brackets is actually significant. The use of brackets with these signs already indicates that they are not really primitive. And surely no one is going to believe brackets have an independent meaning. So signs for logical operations are punctuation marks.
Clearly, whatever we can say in advance about the form of all propositions, we must be able to say all at once. In the event, an elemental proposition already contains all logical operations in itself. For (fa) says the same thing as
In any composite sentence, there are argument and function, and with these, all the logical constants. One could say that the sole logical constant is that which all propositions, by their very nature, must have in common with one another. But that is the general propositional form. The general propositional form is the essence of the proposition. Stating the essence of the proposition means to state the essence of all description, that is the essence of the world. The description of the most general propositional form is the description of the one and only general primitive sign in logic.
Clearly, whatever we can say in advance about the form of all propositions, we must be able to say all at once. In the event, an elemental proposition already contains all logical operations in itself. For (fa) says the same thing as
((∃x) . fx . x = a).
In any composite sentence, there are argument and function, and with these, all the logical constants. One could say that the sole logical constant is that which all propositions, by their very nature, must have in common with one another. But that is the general propositional form. The general propositional form is the essence of the proposition. Stating the essence of the proposition means to state the essence of all description, that is the essence of the world. The description of the most general propositional form is the description of the one and only general primitive sign in logic.
Logic must be clearly constructed from its primitive signs.
If there are primitive logical signs, then a valid logic must clearly show how they relate to one another and justify their existence. If logic has fundamental ideas, they must be independent of one another. If a fundamental idea has been introduced, it must have been introduced in all the combinations in which it ever occurs. It cannot, therefore, be introduced first for one combination and later reintroduced for another.
For example, once negation has been introduced, we must understand it both in propositions of the form (~p) and in propositions like (~(p∨q)), ((∃x).~fx), etc. We must not introduce it first for one class of cases and then again for another, since it would then be doubtful whether it means the same thing in both cases, and neither would there be any reason to combine the signs in the same way.
In short, Frege's remarks in The Fundamental Laws of Arithmetic about introducing signs by means of definitions also apply, mutatis mutandis, to the introduction of primitive signs.
The introduction of any new aid into the symbolism of logic is always significant. It should not be introduced in brackets or in a footnote - sneaked in, so to speak. But if a new aid is required at a certain point, place in logic must be explained. We must immediately ask ourselves: Where must it be used?
All numbers in logic need justification; or rather: It must become evident that there are no numbers in logic. Thus, there are no privileged numbers in logic; neither are co-ordination or classification; nor degrees of the general or specific. The solutions of the problems of logic must be simple, since they set the standard of simplicity.
We have always had a presentiment that there must be an area of investigation whose results - a priori - lie combined in a self-consistent system. A realm in which it is valid to assert: Simplex sigillum veri.
For example, once negation has been introduced, we must understand it both in propositions of the form (~p) and in propositions like (~(p∨q)), ((∃x).~fx), etc. We must not introduce it first for one class of cases and then again for another, since it would then be doubtful whether it means the same thing in both cases, and neither would there be any reason to combine the signs in the same way.
In short, Frege's remarks in The Fundamental Laws of Arithmetic about introducing signs by means of definitions also apply, mutatis mutandis, to the introduction of primitive signs.
The introduction of any new aid into the symbolism of logic is always significant. It should not be introduced in brackets or in a footnote - sneaked in, so to speak. But if a new aid is required at a certain point, place in logic must be explained. We must immediately ask ourselves: Where must it be used?
All numbers in logic need justification; or rather: It must become evident that there are no numbers in logic. Thus, there are no privileged numbers in logic; neither are co-ordination or classification; nor degrees of the general or specific. The solutions of the problems of logic must be simple, since they set the standard of simplicity.
We have always had a presentiment that there must be an area of investigation whose results - a priori - lie combined in a self-consistent system. A realm in which it is valid to assert: Simplex sigillum veri.
Logical objects or logical constants in Frege's and Russell's sense are superfluous.
We now see that 'logical objects' or 'logical constants' in Frege's and Russell's sense are superfluous. This is because truth operations on truth functions that are one and the same truth function of elemental propositions will yield identical results.
It is self-evident that ∨, ⊃, etc. are relations in a different sense than that in which right and left etc. are. The interdefinability of Frege's and Russell's 'primitive signs' of logic is enough to show that they are not primitive signs, still less signs for relations. And it is obvious that the '⊃' which we define by means of '~' and '∨' is identical with the one by which we define '∨' using '~'. It is also obvious that the latter '∨' is identical with the former; and so on.
On the other hand, it seems implausible one fact p should generate infinitely many others, namely (~~p), (~~~~p), etc. And it is no more plausible that the infinite number of propositions of (mathematical) logic follow from half a dozen 'fundamental laws'. All the propositions of logic say the same thing; namely nothing.
Truth-functions are not material functions. When one can, for example, produce an affirmation by double negation, is then negation - in some sense - part of affirmation? Does (~~p) negate (~p), or does it affirm (p) - or both?
The proposition (~~p) does not treat negation as an object; but the possibility of negation is nevertheless already hinted at in affirmation. And if there were an object called '~', it would follow that (~~p) said something different from what 'p' said, just because the one proposition would then be about '~' and the other would not.
The apparent logical constants also disappear in the case of (~(∃x).~fx) that says the same as ((x).fx), or when((∃x).fx.x = a), says the same as (fa).
If we are given a proposition, then with it we are already given the results of all truth operations that have it as their base as well.
It is self-evident that ∨, ⊃, etc. are relations in a different sense than that in which right and left etc. are. The interdefinability of Frege's and Russell's 'primitive signs' of logic is enough to show that they are not primitive signs, still less signs for relations. And it is obvious that the '⊃' which we define by means of '~' and '∨' is identical with the one by which we define '∨' using '~'. It is also obvious that the latter '∨' is identical with the former; and so on.
On the other hand, it seems implausible one fact p should generate infinitely many others, namely (~~p), (~~~~p), etc. And it is no more plausible that the infinite number of propositions of (mathematical) logic follow from half a dozen 'fundamental laws'. All the propositions of logic say the same thing; namely nothing.
Truth-functions are not material functions. When one can, for example, produce an affirmation by double negation, is then negation - in some sense - part of affirmation? Does (~~p) negate (~p), or does it affirm (p) - or both?
The proposition (~~p) does not treat negation as an object; but the possibility of negation is nevertheless already hinted at in affirmation. And if there were an object called '~', it would follow that (~~p) said something different from what 'p' said, just because the one proposition would then be about '~' and the other would not.
The apparent logical constants also disappear in the case of (~(∃x).~fx) that says the same as ((x).fx), or when((∃x).fx.x = a), says the same as (fa).
If we are given a proposition, then with it we are already given the results of all truth operations that have it as their base as well.
All propositions are the result of truth operators applied to elemental propositions.
A truth operation is how a truth function originates from elemental propositions. According to the nature of truth operations, a truth function turns into a new truth-function just as an elemental proposition turns into its own truth function. Every application of a truth operator to truth functions of elemental propositions generates another truth function of elemental propositions, another proposition. Whenever a truth operator is applied to the results of truth operations on elemental propositions, there is always a single operation on elemental propositions that has the same result. Thus, every proposition is the result of truth operations on elemental propositions.
Truth value tables have a meaning even when 'p', 'q', 'r', etc. are not elemental propositions. And it is easy to see that a propositional sign in truth table form expresses a single truth-function of elemental propositions even when 'p' and 'q' are themselves truth-functions of other propositions. So all truth functions are the result of successive application of a finite number of truth operations to elemental propositions.
Truth value tables have a meaning even when 'p', 'q', 'r', etc. are not elemental propositions. And it is easy to see that a propositional sign in truth table form expresses a single truth-function of elemental propositions even when 'p' and 'q' are themselves truth-functions of other propositions. So all truth functions are the result of successive application of a finite number of truth operations to elemental propositions.
Wednesday, April 2, 2008
The structures of propositions relate internally to one another.
Wittgenstein highlights internal relations in his notation by displaying a proposition as generated by an operation on other propositions, the bases of the operation. An operation makes the relation between the structures of its result and of its bases explicit because it expresses what has to be done to one proposition in order to generate another from it. That will naturally depend on the formal properties of both base and result, on the internal similarity of their forms. The internal relation that orders a series is equivalent to the operation that produces one term from another. It can only appear at the point at which one proposition is generated out of another in a logically meaningful way; where the logical construction of the proposition begins.
Truth functions of simple propositions are results of operations based on simple propositions. (Wittgenstein calls them truth operations.) The sense of a truth function of p is a function of the sense of p. Negation is an operation that reverses the sense of a proposition. Logical addition, logical multiplication, etc. etc. are also operations.
When an operation appears in a variable sentence, it shows how we can get from one form of proposition to the next. It expresses the difference between their forms. The common element between the bases of an operation and its result is just the bases themselves. An operation does not characterize any form, but only characterizes the difference between forms.
Let the operation that produces (q) from (p) also produces (r) from (q), and so on. This requires that 'p', 'q', 'r', etc. be variables that give general expression to particular formal relations. The occurrence of an operation, however, does not characterize the sense of a proposition. For an operation makes no statement, only its result does, which result depends on the bases of the operation.
Operations and functions must not be confused with each other. A function cannot be its own argument, whereas an operation can take one of its own results as its base. (A function can be defined recursively, an operation cannot.) An operation is the only way to proceed from one term of a series of forms to another.
Wittgenstein calls repeated applications of an operation to its own result its successive application. (OOOa) is the result of three successive applications of the operation (Oξ) to 'a'. In a similar sense, he speaks of successive applications of more than one operation to a number of propositions. Accordingly, the general term of the series of forms a, Oa, OOa, ... is written as: [a, x, O'x]. The bracketed expression is a variable where the first term is the beginning of the series of forms, the second is the form of a term (x) arbitrarily selected from the series, and the third is the form of the term that immediately follows (x) in the series.
The concept of successive application of operators is equivalent to 'and so on'. One operation can reverse the effect of another or cancel it. Operations can even vanish. For example, negation in (~~p) : (~~p = p).
Truth functions of simple propositions are results of operations based on simple propositions. (Wittgenstein calls them truth operations.) The sense of a truth function of p is a function of the sense of p. Negation is an operation that reverses the sense of a proposition. Logical addition, logical multiplication, etc. etc. are also operations.
When an operation appears in a variable sentence, it shows how we can get from one form of proposition to the next. It expresses the difference between their forms. The common element between the bases of an operation and its result is just the bases themselves. An operation does not characterize any form, but only characterizes the difference between forms.
Let the operation that produces (q) from (p) also produces (r) from (q), and so on. This requires that 'p', 'q', 'r', etc. be variables that give general expression to particular formal relations. The occurrence of an operation, however, does not characterize the sense of a proposition. For an operation makes no statement, only its result does, which result depends on the bases of the operation.
Operations and functions must not be confused with each other. A function cannot be its own argument, whereas an operation can take one of its own results as its base. (A function can be defined recursively, an operation cannot.) An operation is the only way to proceed from one term of a series of forms to another.
Wittgenstein calls repeated applications of an operation to its own result its successive application. (OOOa) is the result of three successive applications of the operation (Oξ) to 'a'. In a similar sense, he speaks of successive applications of more than one operation to a number of propositions. Accordingly, the general term of the series of forms a, Oa, OOa, ... is written as: [a, x, O'x]. The bracketed expression is a variable where the first term is the beginning of the series of forms, the second is the form of a term (x) arbitrarily selected from the series, and the third is the form of the term that immediately follows (x) in the series.
The concept of successive application of operators is equivalent to 'and so on'. One operation can reverse the effect of another or cancel it. Operations can even vanish. For example, negation in (~~p) : (~~p = p).
Propositions of probability.
Propositions of probability do not have anything special about them. If Tr denotes the number of basic truth values of proposition 'r', and Trs denotes the number of basic truth values of proposition 's' that are also basic truth values of 'r', then we call the ratio Trs / Tr the probability of 'r' given 's'. In a truth table, let Tr be the number of 'T' scores in proposition r, and let Trs, be the number of T scores in proposition s in rows in which the proposition r has T. Then given proposition r, proposition s has the probability value: Trs / Tr.
Propositions with no truth value arguments in common are independent of one another, so two elementary propositions give one another a probability of 1/2. If p follows from q, then the proposition q gives proposition p a probability of 1. The certainty of logical inference is a limiting case of probability. (This can be applied to tautology and contradiction.) A proposition is neither probable nor improbable on its own: Either an event occurs or it does not; the middle is excluded.
An urn contains black and white balls in equal numbers (and none of any other kind). I draw one ball after another, and replace them. With this experiment I can establish that the number of black balls drawn and the number of white balls drawn grow closer as the draw continues. So that is is not a mathematical fact. Now, if I say: "It is equally likely that I will draw a white ball as a black one," this means: All the circumstances that I know of (including the laws of nature assumed as hypotheses) give no more probability to the occurrence of the one event than to that of the other. That is to say, they give each case the probability 1/2 as can easily be gathered from the above definitions. What I confirm by the experiment is that the occurrence of the two events, the circumstances of which I do not know in more detail, is independent.
A normal probability proposition is: Circumstances of which I have no further knowledge give a degree of probability to the occurrence of a particular event. Thus, probability is a generalization, a general description of a propositional form. We use probability only in default of certainty, when our knowledge of a fact is not complete, but we do know something about its form. (A proposition may well be an incomplete picture of a certain situation, but it is always a complete picture.) So a statement of probability can be likened to a synopsis of other propositions.
Propositions with no truth value arguments in common are independent of one another, so two elementary propositions give one another a probability of 1/2. If p follows from q, then the proposition q gives proposition p a probability of 1. The certainty of logical inference is a limiting case of probability. (This can be applied to tautology and contradiction.) A proposition is neither probable nor improbable on its own: Either an event occurs or it does not; the middle is excluded.
An urn contains black and white balls in equal numbers (and none of any other kind). I draw one ball after another, and replace them. With this experiment I can establish that the number of black balls drawn and the number of white balls drawn grow closer as the draw continues. So that is is not a mathematical fact. Now, if I say: "It is equally likely that I will draw a white ball as a black one," this means: All the circumstances that I know of (including the laws of nature assumed as hypotheses) give no more probability to the occurrence of the one event than to that of the other. That is to say, they give each case the probability 1/2 as can easily be gathered from the above definitions. What I confirm by the experiment is that the occurrence of the two events, the circumstances of which I do not know in more detail, is independent.
A normal probability proposition is: Circumstances of which I have no further knowledge give a degree of probability to the occurrence of a particular event. Thus, probability is a generalization, a general description of a propositional form. We use probability only in default of certainty, when our knowledge of a fact is not complete, but we do know something about its form. (A proposition may well be an incomplete picture of a certain situation, but it is always a complete picture.) So a statement of probability can be likened to a synopsis of other propositions.
All deduction is a priori.
One cannot deduce an elementary proposition from another. There is no way to infer the existence of a situation from the existence of another, entirely different situation. There is no causal nexus which would justify such an inference. We cannot infer future events from present events. Belief in a causal nexus is superstition.
Free will consists of not being able to know future actions now. We could only know them if causality were an internal necessity like that of logical inference: The connection between knowledge and what is known is that of logical necessity. ('A knows that p is the case', makes no sense if p is a tautology.) If the truth of a proposition does not follow from the fact that it is self-evident to us, then its self-evidence in no way justifies our belief in its truth.
If one proposition follows from another, then the latter says more than the former, and the former less than the latter.
• If p follows from q and q from p, then they are one and same proposition.
• A tautology follows from all propositions: it says nothing.
• Contradiction is that common factor of propositions which no proposition has in common with another.
• Tautology is the common factor of all propositions that have nothing in common with one another.
• Contradiction, one might say, vanishes outside all propositions: tautology vanishes inside them.
• Contradiction is the outer limit of propositions: tautology is the unsubstantial point at their centre.
Free will consists of not being able to know future actions now. We could only know them if causality were an internal necessity like that of logical inference: The connection between knowledge and what is known is that of logical necessity. ('A knows that p is the case', makes no sense if p is a tautology.) If the truth of a proposition does not follow from the fact that it is self-evident to us, then its self-evidence in no way justifies our belief in its truth.
If one proposition follows from another, then the latter says more than the former, and the former less than the latter.
• If p follows from q and q from p, then they are one and same proposition.
• A tautology follows from all propositions: it says nothing.
• Contradiction is that common factor of propositions which no proposition has in common with another.
• Tautology is the common factor of all propositions that have nothing in common with one another.
• Contradiction, one might say, vanishes outside all propositions: tautology vanishes inside them.
• Contradiction is the outer limit of propositions: tautology is the unsubstantial point at their centre.
Logical Inference.
If all the verifiers common to a number of propositions are also verifiers of a particular proposition, then the truth of that proposition follows from the truth of the others. In particular, the truth of one proposition (p) follows from the truth of another proposition (q) if all the verifiers of (q) are also verifiers of (p). In that case, the sense of (p) is contained in the sense of (q). If a god creates a world in which certain propositions are true, then by that very act he also creates a world in which all the propositions that follow from them come true. And similarly he could not create a world in which the proposition (p) was true without creating all its objects.
A proposition affirms every proposition that follows from it. (p.q) is a proposition that affirms (p) as well as (q).
(TFTF)(p,q) (p)
(TTFF)(p,q) (q)
(TFFF)(p,q) (p.q)
(^___)
Two propositions are opposed to one another if no proposition that makes sense affirms them both. Every proposition that contradicts another negates it.
One can see that the truth of one proposition follows from the truth of others from the structure of the propositions. When that is the case, the relations between the forms of the propositions express that. Furthermore, we need not associate them in a compound proposition. Rather, these relations are internal, an immediate consequence of the existence of the propositions.
When we infer q from (p∨q and ~p), the relation between the propositional forms of (p∨q) and (~p) is masked by our notation.
(TTTF)(p,q) (p∨q)
(FTFT)(p,q) (~p)
(TTFF)(p,q) (q)
(_^__)
But if instead of (p∨q) we write, for example, (p|q . | . p|q), and instead of (~p), (p|p) where (p|q = ~p.~q), then the internal relation becomes apparent.
(FFFT)(p,q) (~p.~q or p|q)
(FTFT)(p,q) (~p)
(FFTT)(p,q) (~q)
(___^)
(TTTF)(p.q) (~(p|p).~(p|p))
(FTFT)(p,q) (p|p)
(TTFF)(p,q) (q)
(_^__)
(That one can start with (x).fx and conclude fa shows that the symbol (x).fx itself is generalizable.)
If p follows from q, I can infer p from q; deduce p from q. The nature of the conclusion can be gathered only from the two propositions; only they themselves can justify the inference. 'Laws of inference', which are supposed to justify inferences, as in the works of Frege and Russell, have no purpose, and would be superfluous.
A proposition affirms every proposition that follows from it. (p.q) is a proposition that affirms (p) as well as (q).
(TFTF)(p,q) (p)
(TTFF)(p,q) (q)
(TFFF)(p,q) (p.q)
(^___)
Two propositions are opposed to one another if no proposition that makes sense affirms them both. Every proposition that contradicts another negates it.
One can see that the truth of one proposition follows from the truth of others from the structure of the propositions. When that is the case, the relations between the forms of the propositions express that. Furthermore, we need not associate them in a compound proposition. Rather, these relations are internal, an immediate consequence of the existence of the propositions.
When we infer q from (p∨q and ~p), the relation between the propositional forms of (p∨q) and (~p) is masked by our notation.
(TTTF)(p,q) (p∨q)
(FTFT)(p,q) (~p)
(TTFF)(p,q) (q)
(_^__)
But if instead of (p∨q) we write, for example, (p|q . | . p|q), and instead of (~p), (p|p) where (p|q = ~p.~q), then the internal relation becomes apparent.
(FFFT)(p,q) (~p.~q or p|q)
(FTFT)(p,q) (~p)
(FFTT)(p,q) (~q)
(___^)
(TTTF)(p.q) (~(p|p).~(p|p))
(FTFT)(p,q) (p|p)
(TTFF)(p,q) (q)
(_^__)
(That one can start with (x).fx and conclude fa shows that the symbol (x).fx itself is generalizable.)
If p follows from q, I can infer p from q; deduce p from q. The nature of the conclusion can be gathered only from the two propositions; only they themselves can justify the inference. 'Laws of inference', which are supposed to justify inferences, as in the works of Frege and Russell, have no purpose, and would be superfluous.
A proposition is a truth function of elemental propositions.
An elemental proposition is a truth function of itself. The truth value arguments of a proposition are elemental propositions.
The arguments of functions are readily confused with the indices of names because both arguments and indices enable me to recognize the meaning of the signs containing them. When Russell writes '+c', the 'c' is an index and the sign as a whole indicates addition of cardinal numbers. But this use is arbitrary and it would be quite possible to choose a simple sign instead of '+c'. In (~p) however, 'p' is not an index but an argument: the sense of (~p) cannot be understood unless the sense of 'p' has been understood already. (An index always describes the object to whose name we attach it. In the name Julius Caesar, 'Julius' is an index: e.g. the Caesar of the Julian gens.) According to Wittgenstein, Frege's theory about the meaning of propositions and functions is based on the confusion between an argument and an index. Frege regarded the propositions of logic as names, and their arguments as the indexes of those names.
Truth functions can be ordered as rows of a table. Probability theory is based on this.
The truth functions of any given number of elemental propositions can be written out in a table like this one for all binary combinations:
(TTTT)(p,q) (If p, then p, and if q, then q.) (p⊃p.q⊃q) Tautology
(FFFF)(p,q) (p and not p, and q and not q.) (p.~p.q.~q) Contradiction
(TFTF)(p,q) (p)
(TTFF)(p,q) (q)
(FTFT)(p,q) (Not p.) (~p)
(FFTT)(p,q) (Not q.) (~q)
(TFFF)(p,q) (p and q.) (p.q)
(FTTT)(p,q) (Not (p and q).) (~(p.q))
(FFFT)(p,q) (Neither p nor q.) (~p.~q or p|q)
(TTTF)(p,q) (p or q.) (p∨q)
(FTTF)(p,q) (p or q, but not both.) (p.~q:∨:q.~p)
(FFTF)(p,q) (p and not q.) (p.~q)
(FTFF)(p,q) (q and not p.) (q.~p)
(TFTT)(p,q) (If q, then p.) (q⊃p)
(TTFT)(p,q) (If p, then q.) (p⊃q)
(TFFT)(p,q) (If p, then q, and if q, then p.) (p≡q)
Wittgenstein calls those truth arguments that make a proposition true its verifiers.
The arguments of functions are readily confused with the indices of names because both arguments and indices enable me to recognize the meaning of the signs containing them. When Russell writes '+c', the 'c' is an index and the sign as a whole indicates addition of cardinal numbers. But this use is arbitrary and it would be quite possible to choose a simple sign instead of '+c'. In (~p) however, 'p' is not an index but an argument: the sense of (~p) cannot be understood unless the sense of 'p' has been understood already. (An index always describes the object to whose name we attach it. In the name Julius Caesar, 'Julius' is an index: e.g. the Caesar of the Julian gens.) According to Wittgenstein, Frege's theory about the meaning of propositions and functions is based on the confusion between an argument and an index. Frege regarded the propositions of logic as names, and their arguments as the indexes of those names.
Truth functions can be ordered as rows of a table. Probability theory is based on this.
The truth functions of any given number of elemental propositions can be written out in a table like this one for all binary combinations:
(TTTT)(p,q) (If p, then p, and if q, then q.) (p⊃p.q⊃q) Tautology
(FFFF)(p,q) (p and not p, and q and not q.) (p.~p.q.~q) Contradiction
(TFTF)(p,q) (p)
(TTFF)(p,q) (q)
(FTFT)(p,q) (Not p.) (~p)
(FFTT)(p,q) (Not q.) (~q)
(TFFF)(p,q) (p and q.) (p.q)
(FTTT)(p,q) (Not (p and q).) (~(p.q))
(FFFT)(p,q) (Neither p nor q.) (~p.~q or p|q)
(TTTF)(p,q) (p or q.) (p∨q)
(FTTF)(p,q) (p or q, but not both.) (p.~q:∨:q.~p)
(FFTF)(p,q) (p and not q.) (p.~q)
(FTFF)(p,q) (q and not p.) (q.~p)
(TFTT)(p,q) (If q, then p.) (q⊃p)
(TTFT)(p,q) (If p, then q.) (p⊃q)
(TFFT)(p,q) (If p, then q, and if q, then p.) (p≡q)
Wittgenstein calls those truth arguments that make a proposition true its verifiers.
Tuesday, April 1, 2008
The general propositional form is the variable: 'It is thus and so'.
We now state the most general form of a sentence. That is, we describe sentences in any notation whatsoever so that every possible sense can be expressed by a symbol fitting that description, and so that every such symbol can make sense, once appropriate meanings of the names are chosen. It is clear that the description of the most general propositional form should only include what is essential - otherwise it would not be the most general form.
That there is a general form of a sentence is proved by the fact that there is no grammatical sentence whose form it did not allow us to foresee (i.e. derive). The general form of a sentence is: It is thus and so.
Suppose that all elemental propositions are given. What propositions can one construct with them? That would be all propositions and that is their limit. Propositions are all that follows from the entirety of elemental propositions (and, of course, from its being all of them). In a certain sense, it could be said that all propositions were generalizations of elemental propositions. The general propositional form is a variable.
That there is a general form of a sentence is proved by the fact that there is no grammatical sentence whose form it did not allow us to foresee (i.e. derive). The general form of a sentence is: It is thus and so.
Suppose that all elemental propositions are given. What propositions can one construct with them? That would be all propositions and that is their limit. Propositions are all that follows from the entirety of elemental propositions (and, of course, from its being all of them). In a certain sense, it could be said that all propositions were generalizations of elemental propositions. The general propositional form is a variable.
Tautology and Contradiction.
We can use a mathematical function to calculate that n elemental propositions produce L(n) groups of truth values. These can be ordered in a row (or added as a column to the truth table).
Among all the possible groups of truth values there are two extreme cases. The first case agrees with all combinations of truth values and we call the proposition a tautology. The second case agrees with none and we call it a contradiction.
A proposition shows what it has to say; tautologies and contradictions show that they have nothing to say. Because a tautology is unconditionally true, it has no truth-conditions; and a contradiction has none because it is never true.
Tautologies and contradictions have no sense; just as a point from which arrows go out in two directions. (For example, I know nothing about the weather when I know that it is either raining or not raining.) Tautologies and contradictions are not, however, absurd. They belong to symbolism; much as zero to the symbolism of arithmetic.
Tautologies and contradictions are not images of reality. They do not represent possible situations, for the former admit all situations, and latter none. In a tautology the conditions of agreement with the world - the embodying relations - cancel one another, so that it does not express reality.
The truth-conditions of a proposition determine the range of the facts. (Interpreted in the negative sense, a proposition, a picture, or a model is like a solid body that restricts the freedom of movement of others. Interpreted in the positive sense, it is like a space bounded by solid substance in which there is room for a body.) A tautology leaves the infinite whole of logical space open to reality. A contradiction fills it, leaving no point of it for reality. Thus neither of them can determine reality in any way.
A tautology is certainly true, a proposition possibly, and a contradiction certainly not. (We have the scale that we need in the theory of probability.) The logical product of a tautology and a proposition says the same thing as the proposition and is therefore identical with the proposition because one cannot change the essence of a symbol without changing its sense.
A particular logical combination of signs corresponds to a particular logical combination of their meanings. Absolutely any combination corresponds to uncombined signs, but only to them. In other words, propositions that are true for every situation cannot be sign combinations at all, for if they were, only particular combinations of objects could correspond to them. (And what is not a logical combination has no combination of objects corresponding to it.)
Tautology and contradiction are the limiting cases of sign combinations - their dissolution. Admittedly the signs are still combined with one another even in tautologies and contradictions. That is, they relate to one another. But these relations have no meaning, they are not essential to the symbol.
Among all the possible groups of truth values there are two extreme cases. The first case agrees with all combinations of truth values and we call the proposition a tautology. The second case agrees with none and we call it a contradiction.
A proposition shows what it has to say; tautologies and contradictions show that they have nothing to say. Because a tautology is unconditionally true, it has no truth-conditions; and a contradiction has none because it is never true.
Tautologies and contradictions have no sense; just as a point from which arrows go out in two directions. (For example, I know nothing about the weather when I know that it is either raining or not raining.) Tautologies and contradictions are not, however, absurd. They belong to symbolism; much as zero to the symbolism of arithmetic.
Tautologies and contradictions are not images of reality. They do not represent possible situations, for the former admit all situations, and latter none. In a tautology the conditions of agreement with the world - the embodying relations - cancel one another, so that it does not express reality.
The truth-conditions of a proposition determine the range of the facts. (Interpreted in the negative sense, a proposition, a picture, or a model is like a solid body that restricts the freedom of movement of others. Interpreted in the positive sense, it is like a space bounded by solid substance in which there is room for a body.) A tautology leaves the infinite whole of logical space open to reality. A contradiction fills it, leaving no point of it for reality. Thus neither of them can determine reality in any way.
A tautology is certainly true, a proposition possibly, and a contradiction certainly not. (We have the scale that we need in the theory of probability.) The logical product of a tautology and a proposition says the same thing as the proposition and is therefore identical with the proposition because one cannot change the essence of a symbol without changing its sense.
A particular logical combination of signs corresponds to a particular logical combination of their meanings. Absolutely any combination corresponds to uncombined signs, but only to them. In other words, propositions that are true for every situation cannot be sign combinations at all, for if they were, only particular combinations of objects could correspond to them. (And what is not a logical combination has no combination of objects corresponding to it.)
Tautology and contradiction are the limiting cases of sign combinations - their dissolution. Admittedly the signs are still combined with one another even in tautologies and contradictions. That is, they relate to one another. But these relations have no meaning, they are not essential to the symbol.
Truth Value Tables and Propositions.
The truth values of elemental propositions denote the possibility that matters of fact may be the case or not. We can exhibit these in tables that can easily be understood. Each column is dedicated to an elemental proposition listed in the heading followed by as many truth values as needed. ('T' means 'true', 'F' means 'false'.) The truth table for a single elemental proposition has two rows and the number of rows in the table is doubled for each additional elemental proposition.
p
=
T
F
p q
===
T T
F T
T F
F F
p q r
====
T T T
F T T
T F T
F F T
T T F
F T F
T F F
F F F
A proposition expresses both agreement and disagreement with the truth values of elemental propositions. This makes those truth values the conditions that determine whether the proposition is true or not.
It immediately strikes one that introducing elemental propositions provided the vasis for understanding all other kinds of sentences. Indeed, elemental propositions are palpably required to understand general propositions.
Given n elemental propositions, we can calculate the L(n) ways a particular proposition can both agree and disagree with their truth values. In an additional column dedicated to that proposition, we can denote that the proposition agrees with the combination of values in that row of the truth table by assigning the mark 'T' to the row. The absence of this mark denotes disagreement.
Now, expressing agreement with the truth values of the elemental propositions expresses the truth values of the proposition itself. Thus, the proposition is the expression of its truth values. The sign that results from adding the correlating marks to the truth table is its propositional sign.
It is clear that the complex of signs 'F' and 'T' in the truth table has no object (or complex of objects) corresponding to it, just as there is none corresponding to the horizontal and vertical lines or to the brackets. There are no 'logical objects'. Of course the same applies to all signs that express what truth tables express.
For example:
(p q
If the order of the truth table is fixed once and for all by a combinatory rule, then the last column by itself will express the truth-conditions. If we now write this column as a row, the propositional sign will become
'(TT-T) (p,q)' or more clearly '(TTFT) (p,q)'.
(Note that the number of places in the left-hand pair of brackets is determined by the number of terms in the right-hand pair.)
p
=
T
F
p q
===
T T
F T
T F
F F
p q r
====
T T T
F T T
T F T
F F T
T T F
F T F
T F F
F F F
A proposition expresses both agreement and disagreement with the truth values of elemental propositions. This makes those truth values the conditions that determine whether the proposition is true or not.
It immediately strikes one that introducing elemental propositions provided the vasis for understanding all other kinds of sentences. Indeed, elemental propositions are palpably required to understand general propositions.
Given n elemental propositions, we can calculate the L(n) ways a particular proposition can both agree and disagree with their truth values. In an additional column dedicated to that proposition, we can denote that the proposition agrees with the combination of values in that row of the truth table by assigning the mark 'T' to the row. The absence of this mark denotes disagreement.
Now, expressing agreement with the truth values of the elemental propositions expresses the truth values of the proposition itself. Thus, the proposition is the expression of its truth values. The sign that results from adding the correlating marks to the truth table is its propositional sign.
It is clear that the complex of signs 'F' and 'T' in the truth table has no object (or complex of objects) corresponding to it, just as there is none corresponding to the horizontal and vertical lines or to the brackets. There are no 'logical objects'. Of course the same applies to all signs that express what truth tables express.
For example:
(p q
T T T
F T T
T F
F F T)
is a propositional sign.F T T
T F
F F T)
If the order of the truth table is fixed once and for all by a combinatory rule, then the last column by itself will express the truth-conditions. If we now write this column as a row, the propositional sign will become
'(TT-T) (p,q)' or more clearly '(TTFT) (p,q)'.
(Note that the number of places in the left-hand pair of brackets is determined by the number of terms in the right-hand pair.)
The sense of a proposition.
A proposition corresponds, or not, to matters of fact. These, in turn, can be the case or not. The sense of a proposition consists of this correspondence. The simplest propositions, the elemental propositions, assert that a matter of fact is the case. An indication that a proposition is elemental is that no elemental proposition can contradict it.
An elemental proposition consists of a list of names liked to each other. It is clear that when we analyze propositions, we must reach elemental propositions which consist of names in direct combination. This raises the question of how to combine them into propositions. Even if the world is infinitely complex, so that every fact consists of infinitely many matters of fact and each of them is composed of infinitely many objects of consideration, there would still have to be objects of consideration and matters of fact.
Names only occur in the context of an elemental proposition. They are simple symbols, indicated by single letters ('x', 'y', 'z'). Wittgenstein writes elemental propositions as functions of names, so that they have the form 'fx', 'O (x,y)', etc. or indicates them by the letters 'p', 'q', 'r'.
When using two signs with one and the same meaning, he expresses this by putting the sign '=' between them. So (a = b) means that the sign 'b' can be substituted for the sign 'a'. If he uses an equation to introduce a new sign 'b', deciding that it shall replace a known sign 'a', then, like Russell, he writes the equation, a definition, in the form 'a = b Def.' A definition is a rule dealing with signs. Expressions of the form 'a = b' are, therefore, mere tools of representation. They say nothing about the meaning of the signs 'a' and 'b'.
Can we understand two names without knowing whether they signify the same thing or two different things? Can we understand a proposition in which two names occur without knowing whether their meaning is the same or not? If one knows both an English word and a German word with the same meaning, then it is impossible for me not to know that. It is not possible that one cannot translate them. Expressions like 'a = a', or those derived from them, are neither elemental propositions nor significant in any other way. (This will become evident later.)
If an elemental proposition is true, the matter of fact it describes is the case; if false, then not. Giving all true elemental propositions describes the world completely. What amounts to the same thing, one can give all elemental propositions and mark them as true or false. Of these matters of fact any combination can be the case and the remainder not be.
An elemental proposition consists of a list of names liked to each other. It is clear that when we analyze propositions, we must reach elemental propositions which consist of names in direct combination. This raises the question of how to combine them into propositions. Even if the world is infinitely complex, so that every fact consists of infinitely many matters of fact and each of them is composed of infinitely many objects of consideration, there would still have to be objects of consideration and matters of fact.
Names only occur in the context of an elemental proposition. They are simple symbols, indicated by single letters ('x', 'y', 'z'). Wittgenstein writes elemental propositions as functions of names, so that they have the form 'fx', 'O (x,y)', etc. or indicates them by the letters 'p', 'q', 'r'.
When using two signs with one and the same meaning, he expresses this by putting the sign '=' between them. So (a = b) means that the sign 'b' can be substituted for the sign 'a'. If he uses an equation to introduce a new sign 'b', deciding that it shall replace a known sign 'a', then, like Russell, he writes the equation, a definition, in the form 'a = b Def.' A definition is a rule dealing with signs. Expressions of the form 'a = b' are, therefore, mere tools of representation. They say nothing about the meaning of the signs 'a' and 'b'.
Can we understand two names without knowing whether they signify the same thing or two different things? Can we understand a proposition in which two names occur without knowing whether their meaning is the same or not? If one knows both an English word and a German word with the same meaning, then it is impossible for me not to know that. It is not possible that one cannot translate them. Expressions like 'a = a', or those derived from them, are neither elemental propositions nor significant in any other way. (This will become evident later.)
If an elemental proposition is true, the matter of fact it describes is the case; if false, then not. Giving all true elemental propositions describes the world completely. What amounts to the same thing, one can give all elemental propositions and mark them as true or false. Of these matters of fact any combination can be the case and the remainder not be.
A variable is the sign of a formal concept.
Wittgenstein now extends the discussion to formal concepts in the same sense as he discussed formal properties. He introduced the expression 'formal concepts' to distinguish between them and concepts proper. (They are confused in the whole of traditional logic.) The sign for the object itself shows us that something falls under a formal concept. A proposition cannot tell us that. (A name obviously signifies an object, a numeral signifies a number, etc.)
Functions can express concepts proper but not formal concepts. This is because functions do not express formal properties. Instead, all symbols whose meaning falls under a formal concept express the formal property that is a characteristic feature of that concept. In this way, a propositional variable in which this characteristic feature alone is constant expresses the formal concept. The propositional variable signifies the formal concept, and its values signify the objects that fall under it.
A variable is the sign of a formal concept, because it displays a constant form that all its values possess, and that form can be regarded as a formal property of those values. Thus the variable name 'x' is actually the sign for the pseudo-concept object of consideration.
Wherever the word 'object' ('thing', 'matter' etc.) is used properly to denote a formal concept, it will become a variable name in conceptual notation. For example, in the proposition, 'There are 2 objects which. . .', 2 objects is expressed by ' (∃x,y) ... ' and not otherwise.
Wherever objects is used as a proper concept, pseudo-propositions result that make no sense. So one cannot say, for example, 'There are objects', as one might say, 'There are books', and it is just as impossible to say, 'There are 100 objects'.
It is absurd to speak of the total number of objects. The same applies to the words 'complex', 'fact', 'function', 'number', etc. They all signify formal concepts, and are represented in conceptual notation by variables, not by functions or classes (as Frege and Russell believed). Expressions such as '1 is a number', 'There is only one zero', and all similar ones make no sense. (It also makes no more sense to say, 'There is only one 1', as to say, '2 + 2 at 3 o'clock equals 4'.)
A formal concept is a given once an object falls under it. One cannot, therefore, introduce both a formal concept as well as objects belonging to it as primitive ideas at the same time. So one cannot introduce both the concept of a function and specific functions as primitive ideas (as Russell does), nor, for that matter, the concept of a number along with particular numbers.
If we want to express the general proposition 'b is a successor of a' in conceptual notation, then we need to express the general term of a series of forms. We can do this by giving its first term and the general form of the operation that produces the next term out of the proposition that precedes it, like this:
We can only express the general term of a series of forms using a variable, because the concept 'term of that series of forms' is a formal concept. (This is what Frege and Russell overlooked: consequently the way in which they want to express general propositions like the one above is incorrect; it contains a vicious circle.)
To ask whether a formal concept exists makes no sense, for no proposition can give an answer. (So, for example, the question, 'Are there unanalysable subject-predicate propositions?' cannot be asked.)
Logical forms are not enumerable. For this reason, there are no preeminent numbers in logic, and no possibility of philosophical monism or dualism, etc.
Functions can express concepts proper but not formal concepts. This is because functions do not express formal properties. Instead, all symbols whose meaning falls under a formal concept express the formal property that is a characteristic feature of that concept. In this way, a propositional variable in which this characteristic feature alone is constant expresses the formal concept. The propositional variable signifies the formal concept, and its values signify the objects that fall under it.
A variable is the sign of a formal concept, because it displays a constant form that all its values possess, and that form can be regarded as a formal property of those values. Thus the variable name 'x' is actually the sign for the pseudo-concept object of consideration.
Wherever the word 'object' ('thing', 'matter' etc.) is used properly to denote a formal concept, it will become a variable name in conceptual notation. For example, in the proposition, 'There are 2 objects which. . .', 2 objects is expressed by ' (∃x,y) ... ' and not otherwise.
Wherever objects is used as a proper concept, pseudo-propositions result that make no sense. So one cannot say, for example, 'There are objects', as one might say, 'There are books', and it is just as impossible to say, 'There are 100 objects'.
It is absurd to speak of the total number of objects. The same applies to the words 'complex', 'fact', 'function', 'number', etc. They all signify formal concepts, and are represented in conceptual notation by variables, not by functions or classes (as Frege and Russell believed). Expressions such as '1 is a number', 'There is only one zero', and all similar ones make no sense. (It also makes no more sense to say, 'There is only one 1', as to say, '2 + 2 at 3 o'clock equals 4'.)
A formal concept is a given once an object falls under it. One cannot, therefore, introduce both a formal concept as well as objects belonging to it as primitive ideas at the same time. So one cannot introduce both the concept of a function and specific functions as primitive ideas (as Russell does), nor, for that matter, the concept of a number along with particular numbers.
If we want to express the general proposition 'b is a successor of a' in conceptual notation, then we need to express the general term of a series of forms. We can do this by giving its first term and the general form of the operation that produces the next term out of the proposition that precedes it, like this:
aRb,
(∃x):aRx.xRb,
(∃x,y):aRx.xRy.yRb,
...
(∃x):aRx.xRb,
(∃x,y):aRx.xRy.yRb,
...
We can only express the general term of a series of forms using a variable, because the concept 'term of that series of forms' is a formal concept. (This is what Frege and Russell overlooked: consequently the way in which they want to express general propositions like the one above is incorrect; it contains a vicious circle.)
To ask whether a formal concept exists makes no sense, for no proposition can give an answer. (So, for example, the question, 'Are there unanalysable subject-predicate propositions?' cannot be asked.)
Logical forms are not enumerable. For this reason, there are no preeminent numbers in logic, and no possibility of philosophical monism or dualism, etc.
Formal Properties and Relations.
In a certain sense, we can assign formal properties to objects and matters of fact as we discuss them. This applies to structural properties of facts as well. In the same sense, we can also discuss formal relations and structural relations.
Also, Wittgenstein prefers to say 'internal property' instead of structural property in case of a fact and say 'internal' instead of structural in case of a relation. He introduces these expressions to clarify the very widespread confusion between internal relations and relations proper (external relations).
A property is internal if it is unthinkable that its object should not possess it. Different shades of blue relate, eo ipso, as lighter to darker. Because it is unthinkable that these two particular objects should not relate in this way, the relation is internal. (Note the shift in the use of 'object of consideration.' It corresponds to both 'property' and 'relation.')
Internal properties and internal relations cannot be declared in assertions. Rather, they can simply be seen in those sentences that describe the relevant circumstances and deal with the relevant objects. An internal property of a fact can also be called a feature of that fact (in the sense in which we speak of facial features, for example).
That a situation has an internal property, becomes apparent in the proposition that describes it as an internal property of the proposition. It is not expressed by means of another proposition. It makes no sense to assert that a proposition has a formal property or not. One cannot distinguish forms from one another by saying that one has this property and another that property because doing that presupposes that it makes sense to ascribe either property to either form.
That situations have an internal relation to each other, expresses itself in language as an internal relation between the sentences describing them. This resolves the controversy of whether all relations are internal or external.
Wittgenstein calls a series that is ordered by an internal relation a series of forms. The series of natural numbers is not ordered by an external relation but by an internal relation. The same is true of the series of propositions
Also, Wittgenstein prefers to say 'internal property' instead of structural property in case of a fact and say 'internal' instead of structural in case of a relation. He introduces these expressions to clarify the very widespread confusion between internal relations and relations proper (external relations).
A property is internal if it is unthinkable that its object should not possess it. Different shades of blue relate, eo ipso, as lighter to darker. Because it is unthinkable that these two particular objects should not relate in this way, the relation is internal. (Note the shift in the use of 'object of consideration.' It corresponds to both 'property' and 'relation.')
Internal properties and internal relations cannot be declared in assertions. Rather, they can simply be seen in those sentences that describe the relevant circumstances and deal with the relevant objects. An internal property of a fact can also be called a feature of that fact (in the sense in which we speak of facial features, for example).
That a situation has an internal property, becomes apparent in the proposition that describes it as an internal property of the proposition. It is not expressed by means of another proposition. It makes no sense to assert that a proposition has a formal property or not. One cannot distinguish forms from one another by saying that one has this property and another that property because doing that presupposes that it makes sense to ascribe either property to either form.
That situations have an internal relation to each other, expresses itself in language as an internal relation between the sentences describing them. This resolves the controversy of whether all relations are internal or external.
Wittgenstein calls a series that is ordered by an internal relation a series of forms. The series of natural numbers is not ordered by an external relation but by an internal relation. The same is true of the series of propositions
»aRb«,
»(∃x):aRx.xRb«,
»(∃x,y):aRx.xRy.yRb«,
and so forth.
(If b stands in one of these relations to a, b is a successor of a.)
»(∃x):aRx.xRb«,
»(∃x,y):aRx.xRy.yRb«,
and so forth.
Logical Form - What can be shown, cannot be said.
Propositions must have logical form in common with reality in order to be able to describe anything real. But one cannot use propositions to describe logical form. In order to do that, one would need to be located somewhere outside of logic, that is to say outside the world. A proposition cannot exhibit a logical form because the form is reflected in it.
Language cannot describe what is reflected in it. We cannot express by means of language what expresses itself in our use of it. Propositions show the logical form of reality: They exhibit it.
Thus, the single proposition (fa) shows that object 'a' occurs in its sense. The two propositions (fa) and (ga) show that the same object is mentioned in both of them. If two propositions contradict one another, then their structure will show it. The same is true if one of them follows from the other, and so on. This explains our conviction that a logical conception is valid once we have the formalism right.
What can be shown, cannot be said.
Language cannot describe what is reflected in it. We cannot express by means of language what expresses itself in our use of it. Propositions show the logical form of reality: They exhibit it.
Thus, the single proposition (fa) shows that object 'a' occurs in its sense. The two propositions (fa) and (ga) show that the same object is mentioned in both of them. If two propositions contradict one another, then their structure will show it. The same is true if one of them follows from the other, and so on. This explains our conviction that a logical conception is valid once we have the formalism right.
What can be shown, cannot be said.
A proposition exhibits matters of fact which both are and are not the case.
The entirety of true propositions is the the entirety of natural science. Philosophy is not a natural science, the aim of philosophy is to clarify thoughts logically. That makes it an activity, not a doctrine. Essentially, a philosophical work explains, so philosophy does not result in 'philosophical propositions', but rather in their clarification. It should, as it were, put thoughts in focus.
Psychology is no more closely related to philosophy than any other natural science. Theory of knowledge is the philosophy of psychology. Does not my study of symbolism correspond to the study of thought processes, which philosophers used to consider so essential to the philosophy of logic? Only in most cases they got entangled in unessential psychological investigations, an analogous risk with my method as well.
Darwin's theory has no more to do with philosophy than any other hypothesis in natural science.
Philosophy delimits what can be discussed in natural science. It delineates what can be thought; and, in doing so, what can not. It marks the border to what cannot be thought by working outwards from what can be thought. It will signify what cannot be said, by presenting clearly what can be said. Everything that can be thought at all can be thought clearly. Everything that can be put into words can be put clearly.
Psychology is no more closely related to philosophy than any other natural science. Theory of knowledge is the philosophy of psychology. Does not my study of symbolism correspond to the study of thought processes, which philosophers used to consider so essential to the philosophy of logic? Only in most cases they got entangled in unessential psychological investigations, an analogous risk with my method as well.
Darwin's theory has no more to do with philosophy than any other hypothesis in natural science.
Philosophy delimits what can be discussed in natural science. It delineates what can be thought; and, in doing so, what can not. It marks the border to what cannot be thought by working outwards from what can be thought. It will signify what cannot be said, by presenting clearly what can be said. Everything that can be thought at all can be thought clearly. Everything that can be put into words can be put clearly.
Every proposition must make sense on its own.
To illustrate the concept of truth, imagine a black spot on white paper. One can describe the shape of the spot by saying, for each point on the sheet, whether it is black or white. To the fact that a point is black there corresponds a positive fact, and to the fact that a point is white (not black), a negative fact. If I designate a point on the sheet (a truth-value according to Frege), then that corresponds to the supposition being evaluated, etc. etc.
But in order to be able to say that a point is black or white, I must first know when a point is called black, and when white. In order to be able to say, " 'p' is true (or false)", I must have determined in what circumstances I call 'p' true, and by that have already determined the sense of the proposition. But now the the simile breaks down.
We can indicate a point on the paper even if we do not know what black or white is. but if a proposition makes no sense, nothing corresponds to it, since it does not designate an object (truth value) which might have properties called 'false' or 'true'. The predicate of a proposition is not 'is true' or 'is false', as Frege thought: rather, that which 'is true' must already include that predicate.
Every proposition must make sense on its own. It cannot be given a sense by affirmation since its sense is just what is affirmed; and the same applies to negation, etc. One could say that negation must be related to the logical space determined by the negated proposition. The negation determines a logical space different from that of the negated proposition. The negation determines a logical space with the help of the negated proposition, for it describes the former as lying outside the latter's logical space. That a negation can be negated shows that what is negated is already a proposition, and not a mere preliminary.
But in order to be able to say that a point is black or white, I must first know when a point is called black, and when white. In order to be able to say, " 'p' is true (or false)", I must have determined in what circumstances I call 'p' true, and by that have already determined the sense of the proposition. But now the the simile breaks down.
We can indicate a point on the paper even if we do not know what black or white is. but if a proposition makes no sense, nothing corresponds to it, since it does not designate an object (truth value) which might have properties called 'false' or 'true'. The predicate of a proposition is not 'is true' or 'is false', as Frege thought: rather, that which 'is true' must already include that predicate.
Every proposition must make sense on its own. It cannot be given a sense by affirmation since its sense is just what is affirmed; and the same applies to negation, etc. One could say that negation must be related to the logical space determined by the negated proposition. The negation determines a logical space different from that of the negated proposition. The negation determines a logical space with the help of the negated proposition, for it describes the former as lying outside the latter's logical space. That a negation can be negated shows that what is negated is already a proposition, and not a mere preliminary.
A proposition can be true or false only by being an image of reality.
When reality and the proposition are compared, one must not overlook that fact that a proposition makes sense independently of the facts. That error would lead one to suppose that true and false are relations between signs and what they signify, and have equal status. One could declare, for example, that (p) signifies in the true way what (~p) signifies in the false way, etc.
Could we not communicate using false propositions just as we have up to now with true ones, as long as one knows that they are meant to be false? No!
A proposition is true if things really are as it says they are. Now let us mean (~p) if we say (p) and let things actually be as we mean them. Then, because we construe reality in the new way, (p) is true and not false.
But that the signs 'p' and '~p' could possibly say the same thing is important. For it shows that nothing in reality corresponds to the sign '~'. The occurrence of negation in a proposition is not enough to characterize its sense (~~p = p). The propositions 'p' and '~p' may have opposite sense, but one and the same reality corresponds to them.
Could we not communicate using false propositions just as we have up to now with true ones, as long as one knows that they are meant to be false? No!
A proposition is true if things really are as it says they are. Now let us mean (~p) if we say (p) and let things actually be as we mean them. Then, because we construe reality in the new way, (p) is true and not false.
But that the signs 'p' and '~p' could possibly say the same thing is important. For it shows that nothing in reality corresponds to the sign '~'. The occurrence of negation in a proposition is not enough to characterize its sense (~~p = p). The propositions 'p' and '~p' may have opposite sense, but one and the same reality corresponds to them.
A proposition must have just as many degrees of freedom as the situation it represents.
They must possess the same logical multiplicity. (See Hertz's Mechanics on dynamical models.) This mathematical multiplicity, of course, cannot itself be depicted and one cannot get out of it while depicting.
If, for example, we wanted to express ((x) . fx) by a prefix such as (Gen. fx), we would not know what was being generalized. If we wanted to signalize it with an index 'α' by writing (f(xα)), we would not know the scope.
If we were to try to do it by introducing a mark into the argument position by writing
All these notations are inadequate because they lack the necessary mathematical multiplicity. For the same reason, the idealist's explanation of seeing of spatial relations with 'spatial spectacles' is inadequate because it cannot explain the multiplicity of these relations.
If, for example, we wanted to express ((x) . fx) by a prefix such as (Gen. fx), we would not know what was being generalized. If we wanted to signalize it with an index 'α' by writing (f(xα)), we would not know the scope.
If we were to try to do it by introducing a mark into the argument position by writing
((G,G) . F(G,G))
we could not establish the identity of the variables. And so on.All these notations are inadequate because they lack the necessary mathematical multiplicity. For the same reason, the idealist's explanation of seeing of spatial relations with 'spatial spectacles' is inadequate because it cannot explain the multiplicity of these relations.
A sentence must use old expressions to tell us something new.
A sentence must use old expressions to communicate a new sense. A sentence must be essentially associated with the situation it tells us about. That association is precisely the fact that the sentence is the situation's logical image. A sentence states something only in so far as it is an image.
In a sentence, a situation is assembled on trial, as it were. Instead of, 'This sentence makes sense in such and such a way', we can simply say, 'This sentence depicts such and such a situation'. A name stands for one thing, another for another, and, they are combined with one another; thus the whole - like a tableau - presents a matter of fact. The principle that signs represent objects makes sentences possible.
Wittgenstein consider it fundamental that the 'logical constants' do not stand for something; that the logic of facts cannot be representational. A sentence is only an image of a situation in so far as it was organized logically. (Even the sentence 'Ambulo' is composite; for its stem with a different ending yields a different sense, and so does its ending with a different stem.)
In a sentence, a situation is assembled on trial, as it were. Instead of, 'This sentence makes sense in such and such a way', we can simply say, 'This sentence depicts such and such a situation'. A name stands for one thing, another for another, and, they are combined with one another; thus the whole - like a tableau - presents a matter of fact. The principle that signs represent objects makes sentences possible.
Wittgenstein consider it fundamental that the 'logical constants' do not stand for something; that the logic of facts cannot be representational. A sentence is only an image of a situation in so far as it was organized logically. (Even the sentence 'Ambulo' is composite; for its stem with a different ending yields a different sense, and so does its ending with a different stem.)
Monday, March 31, 2008
A proposition describes of a matter of fact.
A proposition must restrict reality to two alternatives: yes or no. In order to do that, it must describe reality completely. A proposition describes of a matter of fact. Just as a description of an object describes it by giving its external properties, so a proposition describes reality by its internal properties. A proposition constructs a world with the help of a logical scaffolding, so that one can actually see from the proposition how everything stands logically if it is true. One can draw conclusions from a false proposition.
To understand a proposition means to know what is the case if it is true. (One can understand it, therefore, without knowing whether it is true.) It is understood by anyone who understands its constituents. When translating, we do not translate each sentence of one language into a sentence of the other, but merely translate the constituents. (And the dictionary translates not only substantives, but all the parts of speech and it treats them all in the same way.) The meanings of simple signs (words) must be explained to us if we are to understand them. With sentences, however, we make ourselves understood. It belongs to the nature of a sentence that it should be able to communicate a sense new to us.
To understand a proposition means to know what is the case if it is true. (One can understand it, therefore, without knowing whether it is true.) It is understood by anyone who understands its constituents. When translating, we do not translate each sentence of one language into a sentence of the other, but merely translate the constituents. (And the dictionary translates not only substantives, but all the parts of speech and it treats them all in the same way.) The meanings of simple signs (words) must be explained to us if we are to understand them. With sentences, however, we make ourselves understood. It belongs to the nature of a sentence that it should be able to communicate a sense new to us.
A sentence is an image of reality: it shows its sense.
A sentence is both an image and a model of what we think reality is. At first sight, the (printed) sentence does not seem to be an image of the reality it treats. But neither do written notes seem at first sight to be an image of a piece of music, nor our phonetic notation (the alphabet) to be an image of our speech. And yet these sign languages prove to be images, even in the ordinary sense, of what they represent.
Obviously, we perceive the proposition (aRb) as an image. In this case the sign is a likeness of what it denotes. And if we penetrate to the essence of this representation, we see that it is not impaired by seeming irregularities (such as the use of ♯ and ♭in musical notation). For these exceptions also depict what they are meant to express; only in another way. A gramophone record, the musical idea, the written notes, and the sound-waves, all relate to one another in the same internal relation that holds between language and the world. They are all constructed according to a common logical pattern.
There is a general rule by means of which the musician can obtain the symphony from the score, and which makes it possible to derive the symphony from the groove on the gramophone record, and, using the first rule, to derive the score again. It constitutes the inner similarity between such entirely different constructs. And that rule is the law of projection which projects the symphony into the language of musical notation. It is the rule for translating this language into the language of gramophone records.
The possibility of all imagery, of all vividness of expression, is contained in the logic of depiction. In order to understand the nature of a sentence, consider hieroglyphic script, which depicts the facts that it describes. Alphabetic script developed from it without losing what was essential to depiction. We can see this from the fact that we understand the sense of a propositional sign without its having been explained to us.
A sentence is an image of reality: for if I understand a sentence, I know the situation that it represents and I understand it without having had its sense explained to me. It shows its sense, it shows how things stand if it is true; and it says that they do so stand.
Obviously, we perceive the proposition (aRb) as an image. In this case the sign is a likeness of what it denotes. And if we penetrate to the essence of this representation, we see that it is not impaired by seeming irregularities (such as the use of ♯ and ♭in musical notation). For these exceptions also depict what they are meant to express; only in another way. A gramophone record, the musical idea, the written notes, and the sound-waves, all relate to one another in the same internal relation that holds between language and the world. They are all constructed according to a common logical pattern.
There is a general rule by means of which the musician can obtain the symphony from the score, and which makes it possible to derive the symphony from the groove on the gramophone record, and, using the first rule, to derive the score again. It constitutes the inner similarity between such entirely different constructs. And that rule is the law of projection which projects the symphony into the language of musical notation. It is the rule for translating this language into the language of gramophone records.
The possibility of all imagery, of all vividness of expression, is contained in the logic of depiction. In order to understand the nature of a sentence, consider hieroglyphic script, which depicts the facts that it describes. Alphabetic script developed from it without losing what was essential to depiction. We can see this from the fact that we understand the sense of a propositional sign without its having been explained to us.
A sentence is an image of reality: for if I understand a sentence, I know the situation that it represents and I understand it without having had its sense explained to me. It shows its sense, it shows how things stand if it is true; and it says that they do so stand.
A thought is a sentence that made sense.
Thoughts are sentences that make sense. In their entirety, they constitute language. Mankind constructs languages that allow one to express any sense without needing to have much of an idea what each particular word means or how that is so. This is also just how one speaks, without knowing how individual sounds are made. Natural language is a bodily function and no less complicated than that body. It is not humanly possible to directly obtain the logic of natural language from itself.
Language disguises thought, so that one cannot infer, from the outward form of the clothing, the form of the thought clothed by it. This is so, because the outward form of the clothing is designed for entirely different purposes than to let the form of the body be recognized and the tacit conventions that are part of understanding natural language are so complex.
Most of the statements and questions to be found in philosophical works are not false; they just make no sense. Consequently we cannot give any answer to questions of this kind, but can only point out that they do not make sense. (They belong to the same class as the question whether the good is more or less identical than the beautiful.) Most of the propositions and questions of philosophers arise from our failure to understand the logic of our language and it is not surprising that the deepest problems are in fact not problems at all.
All philosophy is a 'linguistic critique' (though not in Mauthner's sense). Russell's achievement was to show that the apparent logical form of a proposition need not be its real one.
Language disguises thought, so that one cannot infer, from the outward form of the clothing, the form of the thought clothed by it. This is so, because the outward form of the clothing is designed for entirely different purposes than to let the form of the body be recognized and the tacit conventions that are part of understanding natural language are so complex.
Most of the statements and questions to be found in philosophical works are not false; they just make no sense. Consequently we cannot give any answer to questions of this kind, but can only point out that they do not make sense. (They belong to the same class as the question whether the good is more or less identical than the beautiful.) Most of the propositions and questions of philosophers arise from our failure to understand the logic of our language and it is not surprising that the deepest problems are in fact not problems at all.
All philosophy is a 'linguistic critique' (though not in Mauthner's sense). Russell's achievement was to show that the apparent logical form of a proposition need not be its real one.
A propositional sign, applied by thinking it, is a thought.
The sentence determines a location in logical space. The existence of this logical location is established by the mere existence of the components, by the existence of a sentence that makes sense. The sentence and the logical coordinates together constitute a logical location. The geometric and logical location are alike in that both allow something to exist.
Although the sentence is allowed to determine only one location in logical space, nevertheless, it must already establish the whole of logical space. (Otherwise negation, logical sum, logical product, etc., would introduce more and more new elements in co-ordination.) The logical framing of an image determines its logical space, while the sentence extends through the whole of logical space.
Although the sentence is allowed to determine only one location in logical space, nevertheless, it must already establish the whole of logical space. (Otherwise negation, logical sum, logical product, etc., would introduce more and more new elements in co-ordination.) The logical framing of an image determines its logical space, while the sentence extends through the whole of logical space.
Any valid symbolic language must be translatable.
A proposition possesses essential and accidental features. Accidental features are those that result from the particular way in which the propositional sign is created, while essential features enable it to make sense. So what is essential in a particular proposition is what all propositions that can express the same sense have in common. And in the same way, what is generally essential in a symbol is what all symbols that can serve the same purpose have in common.
So one could say: All symbols that signify an object have its actual name in common. Thus, it would follow that no particular composition is essential to a name.
There is indeed something arbitrary in the notations we use, but this is not arbitrary: That once we have fixed on one thing arbitrarily, then something else must necessarily be the case. (This lies in the nature of notation.) A particular notation may be unimportant but that it is possible is always important. And that is generally so in philosophy: again and again the individual case turns out to be unimportant, but the possibility of each individual case discloses something about the nature of the world.
Definitions are rules for translating from one language into another. Any valid symbolic language must be translatable into any other in accordance with such rules: This they all have in common. What denotes in a symbol is what is common to all the substitute symbols that the rules of logical syntax allow.
For instance, what all notations for the truth function have in common can be expressed in this way: They all have in common that the notation '~p' ('not p') and 'p ∨ g' ('p or g') can replace any of them. (This characterizes the way in which a specific notation can disclose something general.)
The sign of a complex is not resolved arbitrarily upon analysis; its resolution would not be different when it is incorporated in a different sentence structure.
So one could say: All symbols that signify an object have its actual name in common. Thus, it would follow that no particular composition is essential to a name.
There is indeed something arbitrary in the notations we use, but this is not arbitrary: That once we have fixed on one thing arbitrarily, then something else must necessarily be the case. (This lies in the nature of notation.) A particular notation may be unimportant but that it is possible is always important. And that is generally so in philosophy: again and again the individual case turns out to be unimportant, but the possibility of each individual case discloses something about the nature of the world.
Definitions are rules for translating from one language into another. Any valid symbolic language must be translatable into any other in accordance with such rules: This they all have in common. What denotes in a symbol is what is common to all the substitute symbols that the rules of logical syntax allow.
For instance, what all notations for the truth function have in common can be expressed in this way: They all have in common that the notation '~p' ('not p') and 'p ∨ g' ('p or g') can replace any of them. (This characterizes the way in which a specific notation can disclose something general.)
The sign of a complex is not resolved arbitrarily upon analysis; its resolution would not be different when it is incorporated in a different sentence structure.
Russell's Paradox
In 'theory of types' Russell had to consider the meaning of signs when establishing the rules for them. This is an error. We can dispose of Russell's paradox as follows: No proposition can make a statement about itself, because a propositional sign cannot contain itself. (That is the whole of the 'theory of types').
The reason why a function cannot be its own argument is that the function sign already contains the prototype of its argument, and it cannot contain itself.
Let us suppose that the function F(fx) could be its own argument. In that case, we would have propositions such as: 'F(F(fx))' where the outer function F and the inner function F must have different meanings. This is because the inner one has the form ϕ(f(x)) and the outer one has the form ψ(ϕ(fx)).
Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(∃φ) : F(φu) . φu = Fu'.
The rules of logical syntax must be self evident, once one knows how each individual sign signifies.
The reason why a function cannot be its own argument is that the function sign already contains the prototype of its argument, and it cannot contain itself.
Let us suppose that the function F(fx) could be its own argument. In that case, we would have propositions such as: 'F(F(fx))' where the outer function F and the inner function F must have different meanings. This is because the inner one has the form ϕ(f(x)) and the outer one has the form ψ(ϕ(fx)).
Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(∃φ) : F(φu) . φu = Fu'.
The rules of logical syntax must be self evident, once one knows how each individual sign signifies.
The sign is that aspect of a symbol perceivable by the senses.
Two different symbols can share a sign (written or spoken etc.) yet still signify in distinctly different ways. However, a common characteristic of two different objects can never be indicated by using a single sign with two different modes of signification. For the sign, of course, is arbitrary. We could just as well choose two different signs instead, and then where would the common significance be?
In everyday language it is common for the same word to have different modes of signification and thus belong to different symbols. By the same token, two words that have different modes of signification are commonly employed in sentences in what is superficially the same manner. Thus the word 'is' figures as the copula, as a sign for identity, and as an expression for existence. 'Exist' figures as an intransitive verb like 'go', and 'identical' as an adjective. We speak of something, but also say something's happening. (In the sentence, 'Green is green' - where the first word is the proper name of a person and the last an adjective - these words do not merely have different meanings: they are different symbols.)
In this way the most fundamental confusions are easily produced (the whole of philosophy is full of them). In order to avoid such errors, we must use an object language that prevents them. It must do so by not using the same sign for different symbols and by not using signs that have different modes of signification in a superficially similar way. A notation, therefore, that observes logical grammar and has logical syntax. (The conceptual notation of Frege and Russell is such an object language, though, it is true, it fails to exclude all mistakes.)
In order to recognize a symbol by its sign we must mind that it be used in a manner that makes sense. The sign only determines a logical form in conjunction with its logical, syntactic use. If a sign is not needed, it has no meaning: That is the point of Occam's maxim. (If everything acts as if it has meaning, then it does.) The meaning of a sign should never play a role in establishing logical syntax; it must be possible to do it by merely describing expressions.
In everyday language it is common for the same word to have different modes of signification and thus belong to different symbols. By the same token, two words that have different modes of signification are commonly employed in sentences in what is superficially the same manner. Thus the word 'is' figures as the copula, as a sign for identity, and as an expression for existence. 'Exist' figures as an intransitive verb like 'go', and 'identical' as an adjective. We speak of something, but also say something's happening. (In the sentence, 'Green is green' - where the first word is the proper name of a person and the last an adjective - these words do not merely have different meanings: they are different symbols.)
In this way the most fundamental confusions are easily produced (the whole of philosophy is full of them). In order to avoid such errors, we must use an object language that prevents them. It must do so by not using the same sign for different symbols and by not using signs that have different modes of signification in a superficially similar way. A notation, therefore, that observes logical grammar and has logical syntax. (The conceptual notation of Frege and Russell is such an object language, though, it is true, it fails to exclude all mistakes.)
In order to recognize a symbol by its sign we must mind that it be used in a manner that makes sense. The sign only determines a logical form in conjunction with its logical, syntactic use. If a sign is not needed, it has no meaning: That is the point of Occam's maxim. (If everything acts as if it has meaning, then it does.) The meaning of a sign should never play a role in establishing logical syntax; it must be possible to do it by merely describing expressions.
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