tag:blogger.com,1999:blog-65945705292217299332018-03-05T13:25:41.015-08:00Reading WittgensteinThis 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)Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.comBlogger69125tag:blogger.com,1999:blog-6594570529221729933.post-16701512950413747532008-04-07T08:00:00.000-07:002008-04-07T08:18:55.156-07:00The world does not depend on me.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 <span style="font-style: italic;">logical</span> necessity exists, so too only <span style="font-style: italic;">logical</span> 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.<br /><br />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.<br /><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-30685321055422179552008-04-07T07:21:00.000-07:002008-04-07T08:00:34.007-07:00The 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 <span style="font-style: italic;">particular way</span>, does tell us something indeed. That one method of theoretical description is simpler than another also tells us something about the world.<br /><br />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 <span style="font-style: italic;">particular</span> point-masses, but only about <span style="font-style: italic;">any whatsoever.</span><br /><br />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.<br /><br />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."<br /><br /> Something entirely analogous applies to space. Wherever one says that neither of two exclusive events can occur because there is <span style="font-style: italic;">no reason</span> 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 <span style="font-style: italic;">cause</span> that made one occur and not the other.<br /><br />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.<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_LuNvpt8PIs8/R_oxW265r4I/AAAAAAAAAA0/-pUIINLW3mQ/s1600-h/6-36111.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://1.bp.blogspot.com/_LuNvpt8PIs8/R_oxW265r4I/AAAAAAAAAA0/-pUIINLW3mQ/s320/6-36111.gif" alt="" id="BLOGGER_PHOTO_ID_5186512189639995266" border="0" /></a><br />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.<br /><br /> 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 <span style="font-style: italic;">logical</span> necessity.<br /><br />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 <span style="font-style: italic;">everything</span> were explained.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-41428500889851396512008-04-06T20:57:00.000-07:002008-04-07T06:35:55.585-07:00Mathematics 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:<br /><div style="text-align: center;">(Ω^ν)^μ'x=Ω^(ν×μ')x Def.,<br /></div><br />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<br />=(Ω'Ω)'(Ω'Ω)'x=Ω'Ω'Ω'Ω'x=Ω^(1+1+1+1')x=Ω4'x.<br /><br />Exploring logic exploring everything that is <span style="font-style: italic;">subject to regularity,</span> 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.)<br /><br />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.<br /><br /> 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.<br /><br />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.)Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-40644813279122706182008-04-06T20:45:00.000-07:002008-04-07T07:17:28.755-07:00Logic 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.<br /><br />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.)<br /><br />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.<br /><br />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)'.<br /><br />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.<br /><br />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.<br /><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-52218453568957252972008-04-06T20:23:00.000-07:002008-04-06T20:45:41.271-07:00One 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 <span style="font-style: italic;">never</span> be surprises in logic.<br /><br />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 <span style="font-style: italic;">rules that deal with signs</span>. 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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.)Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-52448828142893063362008-04-06T13:39:00.000-07:002008-04-06T20:23:37.887-07:00We 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.<br /><br /> 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.<br /><br />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.)<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-3475230841600713752008-04-06T09:13:00.000-07:002008-04-06T13:39:25.825-07:00Recognizing a Tautology.In order to recognize a tautology in cases where no generality-sign occurs in it, one can employ the following method.<br /><br />Instead of 'p', 'q', 'r', etc. one writes elemental propositions as 'WpF', 'WqF', 'WrF', etc. ('Wahr' is 'True' in German.)<br /><br />One expresses combinations using brackets, e.g.<br /><br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_LuNvpt8PIs8/R_j5sm65rzI/AAAAAAAAAAM/1Vy2u538AeI/s1600-h/6-1203-1.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://2.bp.blogspot.com/_LuNvpt8PIs8/R_j5sm65rzI/AAAAAAAAAAM/1Vy2u538AeI/s320/6-1203-1.gif" alt="" id="BLOGGER_PHOTO_ID_5186169515674283826" border="0" /></a>Lines connect the truth or falsity of the compound proposition with the truth value combinations that are its arguments.<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_LuNvpt8PIs8/R_j-OW65r0I/AAAAAAAAAAU/6zBAUSYYOe8/s1600-h/6-1203-2.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://1.bp.blogspot.com/_LuNvpt8PIs8/R_j-OW65r0I/AAAAAAAAAAU/6zBAUSYYOe8/s320/6-1203-2.gif" alt="" id="BLOGGER_PHOTO_ID_5186174493541379906" border="0" /></a><br />This sign, for instance, represents the proposition (p⊃q) 'p implies q'.<br /><br />Now, one can examine the proposition ~(p .~p) (the law of contradiction) in order to determine whether it is a tautology.<br /><br />In our notation the form '~ξ' is written as<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/_LuNvpt8PIs8/R_j-uG65r1I/AAAAAAAAAAc/M2BlKFtdtDg/s1600-h/6-1203-3.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://4.bp.blogspot.com/_LuNvpt8PIs8/R_j-uG65r1I/AAAAAAAAAAc/M2BlKFtdtDg/s320/6-1203-3.gif" alt="" id="BLOGGER_PHOTO_ID_5186175039002226514" border="0" /></a><br />and the form 'ξ.η' as:<br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/_LuNvpt8PIs8/R_j-_m65r2I/AAAAAAAAAAk/Yb17LK-rbEY/s1600-h/6-1203-4.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://2.bp.blogspot.com/_LuNvpt8PIs8/R_j-_m65r2I/AAAAAAAAAAk/Yb17LK-rbEY/s320/6-1203-4.gif" alt="" id="BLOGGER_PHOTO_ID_5186175339649937250" border="0" /></a>Hence, one writes the form ~(~p .q) as<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/_LuNvpt8PIs8/R_j-_265r3I/AAAAAAAAAAs/ciF-LcTmpG0/s1600-h/6-1203-5.gif"><img style="margin: 0px auto 10px; display: block; text-align: center; cursor: pointer;" src="http://3.bp.blogspot.com/_LuNvpt8PIs8/R_j-_265r3I/AAAAAAAAAAs/ciF-LcTmpG0/s320/6-1203-5.gif" alt="" id="BLOGGER_PHOTO_ID_5186175343944904562" border="0" /></a><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-27468775421493964572008-04-05T20:25:00.000-07:002008-04-06T09:13:27.990-07:00The 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.<br /><br />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.<br /><br /> That the propositions of logic are tautologies, is <span style="font-style: italic;">shown</span> 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.<br /><br />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. <br /><br />It is clear that one would reach the same conclusions using contradictions instead of tautologies.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-16796974069005725282008-04-05T18:33:00.000-07:002008-04-06T09:01:35.680-07:00The General Form of a Truth Function.Wittgenstein did not fully explain his symbolism in the Tractatus. The following uses text from Russell's introduction.<br /><br />The general form of a truth function is: [p-, ξ-, N(ξ-)] where<br />p- stands for all elemental propositions,<br />ξ- stands for any set of propositions, and<br />N(ξ-) stands for the negation of all members of ξ-.<br /><br />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:<br /> • select a set of any of the atomic propositions,<br /> • negate them all,<br /> • then take any selection of the set of propositions now obtained,<br /> • together with any of the originals<br /> • -- and so on indefinitely.<br /><br />This describes a procedure which, when performed on given elemental propositions, can generate all other propositions that are not elemental. The process depends upon:<br />(a) Sheffer's proof that all truth-functions can be obtained out of simultaneous negation, i.e. out of ``not-p and not-q'';<br />(b) Wittgenstein's theory of the derivation of general propositions from conjunctions and disjunctions; and<br />(c) The assertion that a proposition can only occur in another proposition as argument to a truth-function.<br /><br />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.<br /><br />To apply this method to arrive at numbers one defines:<br /><div style="text-align: center;">x=Ω^(0)'x Def.<br /></div>and<br /><div style="text-align: center;">Ω'Ω^(ν)'x=Ω^(ν+1)'x Def.<br /><div style="text-align: left;"><br />Now, according to the rules of our notation, one write the series<br /></div></div><div style="text-align: center;">x, Ω'x, Ω'Ω'x, Ω'Ω'Ω'x,...,<br /></div>so that<br /><div style="text-align: center;">Ω^(0)'x, Ω^(0+1)'x, Ω^(0+1+1)'x, Ω^(0+1+1+1)'x, .....<br /><div style="text-align: left;"><br />Therfore, one can rewrite the general form [x, ξ, Ω'ξ]) as<br /></div></div><div style="text-align: center;">[Ω0'x, Ων'x, Ων+1'x]<br /></div>and define:<br /><div style="text-align: center;"><div style="text-align: center;">0+1=1 Def.,<br /></div>0+1+1=2 Def.,<br />0+1+1+1=3 Def.,<br />etc.<br /></div>One sees from this that a number is the exponent of an operation.<br /><br /> 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.<br /><br />So the general form of an integer is: [0, ξ, ξ+1].<br /><br />The theory of classes is completely superfluous in mathematics due to the fact that the generality required in mathematics is not <span style="font-style: italic;">accidental</span> generality.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com1tag:blogger.com,1999:blog-6594570529221729933.post-63972962154902018442008-04-05T14:04:00.000-07:002008-04-05T14:47:28.908-07:00The microcosm.I am my universe of discourse; there is no thinking, imagining subject.<br /><br />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 <span style="font-style: italic;">not</span> be discussed in that book. The subject is not a part of the world, but part of its boundary.<br /><br />Where <span style="font-style: italic;">in</span> 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.<br /><br />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.<br /><br />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'.<br /><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-7600540877698475512008-04-05T13:27:00.000-07:002008-04-05T14:04:44.020-07:00The 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 <span style="font-style: italic;">say</span> what we cannot think.<br /><br />This remark provides the key to deciding how much truth there is to solipsism. What the solipsist <span style="font-style: italic;">means</span> to say is quite correct; only it cannot be <span style="font-style: italic;">said</span>, 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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-27048088946819306412008-04-05T06:11:00.000-07:002008-04-05T13:27:18.846-07:00Elemental 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.<br /><br />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.)<br /><br />What one needs to understand the logic of something is not <span style="font-style: italic;">how</span> that something is, but rather, <span style="font-style: italic;">that</span> it is. But that is not an experience. Logic is <span style="font-style: italic;">prior</span> to every experience, the experience that something <span style="font-style: italic;">is so</span>. 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?<br /><br />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.)<br /><br />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 <span style="font-style: italic;">be</span> for something to be the case?<br /><br />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.<br /><br /> 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.<br /><br />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.)<br /><br />Logic is <span style="font-style: italic;">applied</span> 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 <span style="font-style: italic;">a priori</span>, then trying to state them anyway makes no sense.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-86952290209425825032008-04-05T05:06:00.000-07:002008-04-05T06:09:38.097-07:00Propositions 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.<br /><br />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.<br /><br /> 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.<br /><br />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.<br /><br />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 <span style="font-style: italic;">all else</span> has remained the same.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-30529655058131396762008-04-04T19:32:00.000-07:002008-04-05T05:06:27.856-07:00Expressions.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.<br /><br />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?Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-9408517053121538552008-04-04T18:52:00.000-07:002008-04-05T04:59:38.154-07:00Identity.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."<br /><div style="text-align: center;"> ((x):fx.⊃.x = a).<br /></div>This proposition simply says: "<span style="font-style: italic;">Only</span> '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 <span style="font-style: italic;">only</span> '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 <span style="font-style: italic;">makes sense</span>.)<br /><br />By the way: to say <span style="font-style: italic;">two</span> things are identical is absurd, and to say that <span style="font-style: italic;">one</span> thing is identical with itself is to say nothing at all.<br /><br />In Wittgenstein's notation, one does not write (f(a, b).a = b) but (f(a, a)) or (f(b, b)).<br />Not (f(a,b).~a = b) but (f(a, b)).<br />Analogously, one does not write ((∃x, y). f(x, y). x = y), but ((∃x).f(x, x));<br />and not ((∃x, y).f(x, y)).~x = y), but ((∃x, y).f(x, y)).<br /><br />So instead of Russels ((∃x,y).f(x,y)) one writes ((∃x,y).f(x,y).∨.(∃x).f(x,x)).<br />Instead of ((x):fx⊃x = a) one writes ((∃x).fx.⊃:(∃x, y).fx.fy).<br />And the proposition, 'Only one x satisfies f( )', will read ((∃x).fx:~(∃x, y).fx.fy).<br /><br /> 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.<br /><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-34066593789223805092008-04-04T14:07:00.000-07:002008-04-04T18:15:53.858-07:00Truth 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.<br /><br /><div style="text-align: center;">N(ξ‾) = ~(∃x).fx<br /></div><br /> 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.<br /><br />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 <span style="font-style: italic;">all</span> objects. If elementary propositions are given, then <span style="font-style: italic;">all</span> elementary propositions are thereby given.<br /><br />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.<br /><br /> 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".<br /><br />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 <span style="font-style: italic;">other</span> symbols.<br /><br />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 <span style="font-style: italic;">more</span> elementary proposition i true.)Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-70584239861857972842008-04-04T13:28:00.000-07:002008-04-04T14:07:21.718-07:00How 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.<br /><br /> (~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 <span style="font-style: italic;">rule</span> that mirrors negation.<br /><br /> 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.<br /><br /> 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.<br /><br />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).<br /> <br /> 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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-85600402239827941332008-04-03T12:44:00.000-07:002008-04-04T13:37:12.404-07:00Every 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.<br /><br />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).<br /><br />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:<br /><ol><li>Direct enumeration, in which case we can simply substitute constant values for the variable.</li><li>Stating a function fx whose values for all values of x are the propositions to be described.</li><li>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.<br /></li></ol>So instead of '(-----T)(ξ,....) we can write 'N(ξ‾)' where N(ξ‾) is the negation of all the values of ξ. If ξ‾ has only one value, then N(ξ‾) = ~p (not p); if it has two values, then N(ξ‾) = ~p . ~q. (neither p nor g).<br /><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-81943702562389510542008-04-03T12:24:00.000-07:002008-04-03T12:44:38.061-07:00Occam'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. <br /><br />Occam's maxim is, of course, not an arbitrary rule, nor is it one justified by practice: It points out that <span style="font-style: italic;">unnecessary</span> signs mean nothing. So signs that serve <span style="font-style: italic;">one</span> purpose are logically equivalent, and signs that serve <span style="font-style: italic;">none</span> are logically meaningless.<br /><br />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 <span style="font-style: italic;">meaning</span>. (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.<br /><br />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 <span style="font-style: italic;">some primitive ideas</span> that need a sign, but rather of expressing a rule.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-46721759485617415032008-04-03T09:11:00.000-07:002008-04-03T12:22:35.240-07:00Signs 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.<br /><br />Clearly, whatever we can say <span style="font-style: italic;">in advance</span> about the form of all propositions, we must be able to say <span style="font-style: italic;">all at once</span>. In the event, an elemental proposition already contains all logical operations in itself. For (fa) says the same thing as<br /><div style="text-align: center;">((∃x) . fx . x = a).<br /></div><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-13131022491357118822008-04-03T07:00:00.000-07:002008-04-03T09:10:47.559-07:00Logic 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.<br /><br />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.<br /><br />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.<br /><br />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 <span style="font-style: italic;">must</span> it be used?<br /><br />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.<br /><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-27398702090384417082008-04-03T06:29:00.000-07:002008-04-03T07:00:17.749-07:00Logical 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.<br /><br />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.<br /><br />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.<br /><br />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?<br /><br />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.<br /><br />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).<br /><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-15959063094159654662008-04-03T05:48:00.000-07:002008-04-03T06:29:11.730-07:00All 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.<br /><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-68619851527416046122008-04-02T19:45:00.000-07:002008-04-03T05:48:22.880-07:00The 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.<br /><br />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.<br /><br />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.<br /><br />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.<br /><br /> 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.<br /><br /> 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.<br /><br /> 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).Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0tag:blogger.com,1999:blog-6594570529221729933.post-60548054303776339152008-04-02T17:33:00.000-07:002008-04-07T19:40:00.815-07:00Propositions 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.<br /><br />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.<br /><br />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.<br /><br />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.Bohemianhttp://www.blogger.com/profile/08198013460863446098noreply@blogger.com0