This log was inspired by "How to Read Wittgenstein" and "Ludwig Wittgenstein: the duty of genius" by Ray Monk. It is based on reading Tractatus Logico-Philosophicus by Ludwig Wittgenstein translated by D. F. Pears & B. F. McGuinness (Routledge and Kegan Paul:1963)
Monday, April 7, 2008
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.
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
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.)
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 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.)
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.
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
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 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:
Now, according to the rules of our notation, one write the series
Therfore, one can rewrite the general form [x, ξ, Ω'ξ]) as
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.
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.
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.
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.
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
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?
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.
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.)
(~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
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 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.
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.
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.
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.
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
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 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.
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.
A proposition affirms every proposition that follows from it. (p.q) is a proposition that affirms (p) as well as (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.
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)
(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.
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
(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
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.
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.
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.
F T T
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.)
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.
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.
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
and so forth.
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.
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.
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.
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.
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.
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.)
- Tractatus Logico-Philosophicus 1
- Tractatus Logico-Philosophicus 2
- Tractatus Logico-Philosophicus 2.01
- Tractatus Logico-Philosophicus 2.02
- Tractatus Logico-Philosophicus 2.03 to 2.063
- Tractatus Logico-Philosophicus 2.1
- Tractatus Logico-Philosophicus 2.2
- Tractatus Logico-Philosophicus 3
- Tractatus Logico-Philosophicus 3.0
- Tractatus Logico-Philosophicus 3.1
- Tractatus Logico-Philosophicus 3.2
- Tractatus Logico-Philosophicus 3.3
- Tractatus Logico-Philosophicus 3.32
- Tractatus Logico-Philosophicus 3.33
- Tractatus Logico-Philosophicus 3.34
- Tractatus Logico-Philosophicus 3.4 to 3.5
- Tractatus Logico-Philosophicus 4
- Tractatus Logico-Philosophicus 4.00
- Tractatus Logico-Philosophicus 4.01 to 4.022
- Tractatus Logico-Philosophicus 4.023 to 4.027
- Tractatus Logico-Philosophicus 4.03
- Tractatus Logico-Philosophicus 4.04
- Tractatus Logico-Philosophicus 4.05 to 4.0621
- Tractatus Logico-Philosophicus 4.1
- Tractatus Logico-Philosophicus 4.12 to 4.1213
- Tractatus Logico-Philosophicus 4.122 to 4.1252
- Tractatus Logico-Philosophicus 4.126 to 4.128
- Tractatus Logico-Philosophicus 4.2 to 4.28
- Tractatus Logico-Philosophicus 4.3 to 4.442
- Tractatus Logico-Philosophicus 4.45 TO 4.4661
- Tractatus Logico-Philosophicus 4.5 to 4.53
- Tractatus Logico-Philosophicus 5
- Tractatus Logico-Philosophicus 5 to 5.101
- Tractatus Logico-Philosophicus 5.05 to 5.156
- Tractatus Logico-Philosophicus 5.11 to 5.132
- Tractatus Logico-Philosophicus 5.133 to 5.143
- Tractatus Logico-Philosophicus 5.2 to 5.254
- Tractatus Logico-Philosophicus 5.3
- Tractatus Logico-Philosophicus 5.4 to 5.44
- Tractatus Logico-Philosophicus 5.45
- Tractatus Logico-Philosophicus 5.46 to 5.472
- Tractatus Logico-Philosophicus 5.473 to5.476
- Tractatus Logico-Philosophicus 5.5 to 5.503
- Tractatus Logico-Philosophicus 5.51
- Tractatus Logico-Philosophicus 5.52
- Tractatus Logico-Philosophicus 5.53 to 5.535
- Tractatus Logico-Philosophicus 5.5351 to 5.5352
- Tractatus Logico-Philosophicus 5.55 to 5.5571
- Tractatus Logico-Philosophicus 5.6 to 5.621
- Tractatus Logico-Philosophicus 5.63 to 5.641
- Tractatus Logico-Philosophicus 6
- Tractatus Logico-Philosophicus 6 to 6.01
- Tractatus Logico-Philosophicus 6.1 to 6.1202
- Tractatus Logico-Philosophicus 6.1203
- Tractatus Logico-Philosophicus 6.121 to 6.124
- Tractatus Logico-Philosophicus 6.125 to 6.1271
- Tractatus Logico-Philosophicus 6.13 to 6.2331
- Tractatus Logico-Philosophicus 6.234 to 6.3432
- Tractatus Logico-Philosophicus 6.342 to 6.372
- Tractatus Logico-Philosophicus 6.373 to 6.3751
- Tractatus Logico-Philosophicus 6.5
- Tractatus Logico-Philosophicus 7
- The world does not depend on me.
- The relative position of logic and science.
- Mathematics is a method of logic.
- Logic is transcendental.
- One can describe all true logical propositions in ...
- We can do without logical propositions.
- Recognizing a Tautology.
- The propositions of logic are tautologies.
- The General Form of a Truth Function.
- The microcosm.
- The boundary of my language represents the boundar...
- Elemental Propositions.
- Propositions occur in each other only as bases of ...
- Truth Functions do not Include the Concept All.
- How is this useful?
- Every truth-function an be obtained by successivel...
- Occam's rule points out that unnecessary signs mea...
- Signs for logical operations are punctuation marks...
- Logic must be clearly constructed from its primiti...
- Logical objects or logical constants in Frege's an...
- All propositions are the result of truth operators...
- The structures of propositions relate internally t...
- Propositions of probability.
- All deduction is a priori.
- Logical Inference.
- A proposition is a truth function of elemental pro...
- The general propositional form is the variable: 'I...
- Tautology and Contradiction.
- Truth Value Tables and Propositions.
- The sense of a proposition.
- A variable is the sign of a formal concept.
- Formal Properties and Relations.
- Logical Form - What can be shown, cannot be said.
- A proposition exhibits matters of fact which both ...
- Every proposition must make sense on its own.
- A proposition can be true or false only by being a...
- A proposition must have just as many degrees of fr...
- A sentence must use old expressions to tell us som...
- ▼ April (40)