Wednesday, October 31, 2007

5.234 The truth-functions of elementary propositions are results of operations that have the elementary propositions as bases. (I call these operations, truth-operations.)


5.233 An operation can first occur in the [a?] place where a proposition is generated from another in a logically meaningful [bedeutungsvolle] way. Thus in the place where the logical construction of the proposition begins.

So propositions are constructed? By us? Arbitrarily? Or by what rules?

Black (p. 260): “The idea is rather cryptically expressed. W. wishes simply to deny that there can be other than truth-functional operations.”

5.232 An internal relation that orders a series is equivalent to an operation by which one term is generated from another.

So internal relations are not so mysterious. Very like operations, which are surely rather arbitrary.

Black points out (p. 260) that 4.1252 tells how a formal series arises.

5.231 And that of course will depend on their formal properties, on the internal similarity of their forms.


Tuesday, October 30, 2007

5.23 An operation is that which must happen to a proposition in order to make another out of it.

Black (p. 259): “W. wishes to make a distinction between an operation and a function (5.251), yet the difference between the two seems at first nothing more substantial than a difference in point of view (and consequently in terminology). Mathematicians commonly use the terms ‘function’ and ‘operation’ interchangeably.”

Black (p. 260): “The important point is that W. restricts ‘operation’ to the case where the ‘bases’ and the ‘result’ (5.22) of the operation are internally related.”

5.22 An operation is the expression of a relation between the structures of its result and its bases.

Black (p. 259) says that “relation” here means internal relation.

5.21 We can bring out these internal relations in our means of expression by presenting a proposition as the result of an operation that produces one proposition out of others (the bases of the operation).


5.2 The structures of propositions stand in internal relations to one another.

Cf. 5.13 and 5.131 on internal relations and deductive connections, Black notes (p. 259).

Monday, October 29, 2007

5.156 Hence probability is a generalization.

It involves a general description of a propositional form.

Only in the absence of certainty do we need probability. – If we are indeed not completely acquainted with a fact but do know something about its form.

(A proposition can indeed be an incomplete picture of a certain state of things, but it is always a complete picture [i.e. a complete picture of something, or a complete picture in some sense].)

A proposition of probability is, as it were, an extract from other propositions.

Why talk about needing probability? And how can a proposition be an incomplete picture of something other than that of which it is a complete picture? And why “as it were”? Otherwise this seems straightforward.

Black’s conclusion (p. 256): “Wittgenstein’s definition [of probability] may tell us how probability might be computed in a language that does not exist; applied to any language actually available, it would compel us to treat all probability measures as unknown.”

5.155 The unit of the proposition of probability is: The circumstances – with which I am otherwise no further acquainted – give to the occurrence of a particular event such and such a degree of probability.

Banal. Suspiciously so?

5.154 In an urn there are equal numbers of white and black balls (and no others). I draw one ball after another and lay them back again in the urn. Then by this experiment I can determine that the number of black balls drawn and the number of white balls drawn get nearer to one another as the drawing goes on.

That is therefore not a mathematical fact.

If I now say: It is as probable that I will draw a white ball as a black, then that means: All circumstances known to me (including laws of nature assumed as hypotheses) give to the occurrence of the one event no more probability than to the occurrence of the other. That is, they give – as can be easily gathered from the explanations above – to each the probability ½.

What I confirm by the experiment is that the occurrence of both events is independent of the circumstances with which I am no closer acquainted.

Since the first paragraph is entirely a priori, why is this not a mathematical fact? Is it logical? Empirical? And what circumstances is he talking about at the end? Obscure to me.

5.153 A proposition is in itself neither probable nor improbable. An event occurs, or it does not occur: there is no middle ground.


Thursday, October 25, 2007

5.152 We call propositions that have no truth-arguments in common with one another independent of one another.

Propositions independent of one another (e.g. any two elementary propositions) give one another the probability ½.

If p follows from q, then the proposition “q” gives the proposition “p” the probability 1. The certainty of a logical conclusion is a limiting case of probability.

(Application to tautology and contradiction.)

Does probability thus conceived make any sense? Independent propositions cannot give each other any probability they do not already have, i.e. they cannot give them any probability at all. So does talk of probability belong in logic?

5.1511 There is no special object peculiar to propositions of probability.

Modals don’t stand for something, for instance, I suppose, in any metaphysical way. “There is a high probability” is not like “There is a planet.”

5.151 In a schema like the one above in no. 5.101, let Tr be the number of Ts in proposition “r”; Trs the number of those Ts in proposition “s” that stand in the same column as Ts of the proposition “r”. The proposition “r” then gives the proposition “s” the probability Trs: Tr.

OK, follows from 5.15, no?

5.15 If Tr is the number of truth-grounds of the proposition “r”, and Trs the number of those truth-grounds of the proposition “s” that are simultaneously truth-grounds of “r”, then we call the ratio Trs: Tr the measure of the probability that the proposition “r” gives to the proposition “s”.

OK, definition.

Wednesday, October 24, 2007

5.143 Contradiction is the common property of propositions, what no proposition has in common with another. Tautology is the common property of all propositions that have nothing in common with one another.

Contradiction vanishes, so to speak, outside, tautology inside, all propositions.

Contradiction is the outer limit of propositions, tautology their substanceless center.

Very strange. It is common to all propositions that none of them have contradictions in common, and that is what contradiction is. Tautologies are equally strangely ‘defined’ here. Both vanish (verschwinden: vanish, disappear, be lost, pass away), so to speak. One is outside all propositions, the other without substance. How close we are to saying that the notions are quite without meaning. I suppose you could tag the denial of a contradiction onto any proposition, or the affirmation of a tautology.

5.142 Tautologies follow from all propositions: they say nothing.

We already know they say nothing, but by 5.14 they must say less than any other proposition.

5.141 If p follows from q and q from p, then they are one and the same proposition.

OK, so the worry about equivalent propositions is dealt with here.

5.14 If a proposition follows from another, then this latter one says more than the former, the former less than the latter.

This is oddly repetitive, and is it true? Couldn’t they each say the same, being equivalent?

Tuesday, October 23, 2007

5.1363 If it does not follow that it is true from a proposition’s being obvious to us, then obviousness is also no justification for our belief in its truth.

True, but we will believe what is obvious to us, won’t we? Justified or not. Cf. On Certainty.

See Marie McGinn pp. 64-68 on the importance of self-evidence for Frege and Russell. “Wittgenstein believes that the problem with any account of logic that treats the propositions of logic as substantial truths, in the way that Frege and Russell do, is that it is forced to rely on a notion of self-evidence to explain our a priori knowledge of their truth. And the problem with any appeal to a notion of self-evidence as a justification for acknowledging a proposition as true is that the truth of a proposition does not follow from its seeming to us to be self-evident.” (pp. 66-67)

5.1362 Freedom of the will consists in the fact that it impossible now to know future actions. We could only know them if causality were an inner 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” is senseless [sinnlos] if p is a tautology.)

Marie McGinn (pp. 218-9) says that Black and Anscombe read the last sentence of the first paragraph here as noting the logical connection between “A knows that p” and “p”. However, in the context McGinn reads it instead as follows: “The point of the final sentence of the first paragraph is that our knowledge extends only so far as what is logically entailed by what we know, and no further.” (p. 219)

My comment: So the connection between knowledge and what is known is logical. And freedom of the will consists in the lack of such a connection between our possible knowledge and future events. So it is logical, not metaphysical. Presumably Wittgenstein says this odd thing because he thinks metaphysical statements are all senseless or nonsensical. So the only meaning a proposition about the freedom of the will could have would be logical, and not metaphysical. But then it is/becomes a tautology to say that we cannot know future events. So to say the will is free is to speak tautologically, and so senselessly. Is this right?

5.1361 We cannot infer the events of the future from those of the present.

Belief in the causal nexus is superstition.

Wittgenstein explains the second sentence thus: “I didn’t mean to say that the belief in the causal nexus was one amongst superstitions but rather that superstition is nothing else than the belief in the causal nexus.” (Letters to Ogden, p. 31) The prime superstition is belief in a causal nexus that allows us to infer future from present, state of affairs from state of affairs. The belief that we can know the world a priori. Metaphysics is superstition. Since logic is the weapon to be used against metaphysics, logic is the means to fight idolatry. The way to salvation is logic. Or is it this: superstition is the belief that everything is connected causally to everything else?

Stokhof (pp. 99-100): “Wittgenstein’s remarks on causality should not be interpreted as claiming there is no such thing as a causal relationship in the first place. What is denied is that causality is an internal relation between situations.”

This is tricky though. Schopenhauer and Kant seem to find the idea of causality without a causal nexus to be inconceivable or incomprehensible. It is all very well to say that Wittgenstein is not denying causality. Indeed he is not explicitly doing so. But what sense of causal regularity is left? On p. 77 Schopenhauer calls the law of causality the sole form of the understanding. It might be relevant that on p. 116 he criticizes Kant for identifying perception with sensation and thus for holding that “perception is something quite direct and brought about entirely without the assistance of the causal nexus” (his italics). As a result, on p. 117, Schopenhauer says that for Kant the origin of empirical intuitive perception is left wholly unexplained and so is “given as it were by a miracle.” Is this Wittgenstein’s view?

Compare Nietzsche Beyond Good and Evil “On the Prejudices of Philosophers” §21 (p. 219 Everyman, 1992, Basic Writings of Nietzsche, edited by W. Kaufmann): “It is we alone who have devised cause, sequence, for-each-other, relativity, constraint, number, law, freedom, motive, and purpose; and when we project and mix this symbol world into things as if it existed “in itself,” we act once more as we have always acted—mythologically.”

On p. 133 Frascolla interprets Wittgenstein’s remarks here about causality as a rejection of some of Schopenhauer’s views. For instance, Schopenhauer’s idea that “motivation is causality seen from the inside” (Frascolla p. 133), and that the relation cause-effect can be assimilated to the relation premise-consequence. On Frascolla’s reading of Schopenhauer, one could know one’s own future actions (with certainty) if one knew one’s own motivational dispositions or character. Hence, one might say, one’s will would not be free. But this leaves out the fact that Schopenhauer believes that two people of the same character will do different things in different circumstances, and we cannot know in what circumstances we will find ourselves in the future, surely. Also, Schopenhauer identifies the causal nexus with the world as representation, which he regards as little more real than a dream. Rejecting belief in it as superstition is something he might endorse.

Monday, October 22, 2007

5.136 There is no causal nexus that justifies such an inference.

How could a causal nexus (metaphysical) justify a logical inference? It couldn’t, but this underlines the importance of not confusing metaphysics and logic. Schopenhauer and Frege emphasize this point, in different ways.

Stenius (p. 60) says “By ‘causal nexus’ he obviously means the aprioristic certainty of causal connections.”

Black (p. 244) says “W. does not mean to deny the existence of causal regularities: he does deny that they are a priori.”

Frascolla (pp. 130-131) connects this remark and the next with 6.37. Wittgenstein is denying “that there is any necessity in the so-called causal nexus between the events of one type, identified as causes, and the events of another type, identified as effects of those cause.” (p. 130)

Schopenhauer refers to the causal nexus in The Fourfold Root of the Principle of Sufficient Reason (translated by E. F. J. Payne, Open Court, La Salle, Ill., 1974), p. 56. On pp. 55-56 he writes: “it is most important for us clearly to recognize first and foremost that the law of causality relates solely and exclusively to changes of material states, and to nothing else whatever. Consequently, it must not be introduced when these are not mentioned. Thus the law of causality is the regulator of the changes undergone in time by objects of external experience; but all these are material. Every change can take place only through another having preceded it, which is determined according to a rule, but by which it then takes place as having been necessarily brought about. This necessity is the causal nexus.” (Italics in the original translation.) He begins this book with an emphatic distinction between the idea that there must be a reason for all things and the idea that the existence of each thing must have a cause. He also emphatically rejects all forms of the cosmological argument, and mocks the Hegelian “Absolute” as a ruse to get around Kant’s demolition of all attempts to prove the existence of God. His points against Hegel might be used against Heidegger. Or at least, the attempt to do so would appeal to many analytic philosophers, possibly including Wittgenstein. Schopenhauer might accept 5.135 and 5.136. He surely would reject 5.1361. For Schopenhauer, the relation between cause and effect is necessary, and the law of causality is a form of the principle of sufficient reason, on which all necessity is based. See p. 63.

As for the law of causality being a priori, see Schopenhauer again, p. 110: “No one who himself has any intelligence will doubt its existence in the higher animals. But it is at times quite evident that their knowledge of causality is actually a priori, and has not resulted from the habit of seeing one thing follow another. A young puppy does not jump down from a table because he anticipates the effect.” He goes on to tell of his pet poodle’s being astonished by curtains that are moved by means of a cord pulled at the side. The dog naturally looked for the cause of the movement. Schopenhauer calls such a natural reaction a priori knowledge of the law of causality. If Wittgenstein denies that it is a priori, then what would he say instead? Has he an explanation?

5.135 By no means can an inference be made from the existence of one state of things to the existence of another quite different state of things.

5.134 repeated emphatically. Why? And isn’t this a “law of inference”?

5.134 No other proposition can be inferred from an elementary proposition.

Because they are independent. See 4.27.

5.133 All inferences are made a priori.

Laws of inference would perhaps make it look like an a posteriori matter, depending on what the laws are. So stating them is misleading as well as pointless.

Friday, October 19, 2007

5.132 If p follows from q then I can infer from q to p; deduce p from q.

The nature of the inference is to be gathered only from the two propositions.

Only they themselves can justify the inference.

“Laws of inference” that – as in Frege and Russell – are supposed to justify inferences, are senseless [sinnlos], and would be superfluous.

Some clarification of what ‘sinnlos’ means for Wittgenstein here. The laws of logic cannot be put into words [cf. On Certainty?]. The attempt to do so is useless and senseless, if this is a different thing.

Ostrow (p. 111) suggests that Wittgenstein’s concern here and elsewhere in the TLP is primarily “to shift our perspective so that we no longer feel any urge to account for why, for example, “q” follows from “p” and “p → q” in the first place.”

Proops points out (pp. 80-86) that Frege and Russell seem to use “laws of inference” to mean laws of logic, including axioms, not just the rules for inference in a particular axiom system. As evidence he cites Frege’s “Foundations of Geometry: First Series” in his Collected Papers p. 319, and Russell’s (1905) “Necessity and Possibility” in his Collected Papers Volume IV, p. 515. Proops takes Wittgenstein to think that (p. 90): “far from expressing truths which lie at the bottom of all valid inferences, the laws of logic are to be viewed as expressing no facts of any kind. Secondly, even if Russell’s conception of logic were correct, even if, that is to say, the laws of logic were not sinnlos, the appeal to logical laws in the explanation of valid inference would in any case be superfluous.” The concept of entailment cannot be explained in other terms, including those of derivability in a sound system.

5.1311 When we infer q from p v q and ~p, then the way of symbolizing here veils the relation of the propositional forms of “p v q” and “~p”. But if instead of “p v q”, e.g., we write “p|q .|. p|q” and instead of “~p” “p|p” (p|q = neither p nor q), then the internal connection becomes clear.

(The fact that one can infer fa from (x) . fx shows that generality is present in the symbol “(x) . fx” itself.)

But of course generality is not a metaphysical property of the symbol.

5.131 If the truth of a proposition follows from the truth of another, this is expressed in the relations in which the forms of these propositions stand to one another; and we certainly do not need to put them in these relations first by combining them with one another in a proposition, but rather these relations are internal and exist as soon as, and by the fact that, these propositions exist.

So these internal relations express the fact that one proposition follows from another. But they are internal and do not need to be put into an explicit sentence or thought.

5.13 We see from the structure of the propositions that the truth of one proposition follows from the truth of another.

OK, so is this what “structure of a proposition” means?

Thursday, October 18, 2007

5.1241 “p. q” is one of the propositions that assert “p” and also one of the propositions that assert “q”.

Two propositions are opposed to one another if asserting them both makes no meaningful proposition [wenn es keinen sinnvollen Satz gibt].

Every proposition that contradicts another denies it.

Black (p. 242) says of the last sentence that “It is hard to see the point of this remark.” Presumably the point has to do with sentences referring to or dealing with each other. A red light and a green light just as such do not deny each other, but when used as traffic signals they do. Incompatible sentences are not merely incompatible, so that you cannot have both, as it were. They actually rule each other out. The reason why you cannot have both is internal to the sentences.

5.124 A proposition affirms every proposition that follows from it.

Ogden has ‘asserts’ for bejaht which is also correct, but lacks some of the positive endorsement implied by ja here.

5.123 If a god creates a world wherein certain propositions are true then he thereby also creates a world in which all the propositions that follow from them are already true. And similarly he could create no world in which the proposition “p” is true without creating all its objects.

Logic here sounds like a kind of metaphysical limit on God.

Wednesday, October 17, 2007

5.122 If p follows from q, then the sense of “p” is contained in the sense of “q”.

So sense (Sinn) is a matter of truth-grounds, as it would not be (I think) for Frege.

5.121 The truth-grounds of the one are contained in those of the other; p follows from q.


5.12 In particular, the truth of a proposition “p” follows from the truth of another “q” if all truth-grounds of the second are truth-conditions of the first.


Tuesday, October 16, 2007

5.11 If the truth-grounds that are common to a number of propositions are the same for a particular proposition, then we say that the truth of this proposition follows from the truth of those propositions.


5.101 The truth-functions of every number of elementary propositions can be written out in a schema in the following way:

(T T T T)(p, q) Tautology (if p then p, and if q then q) [p  HOOK p . q  HOOK q]
(F T T T)(p, q) in words: Not both p and q. [~(p . q)]
(T F T T)(p, q) '' '' If q then p. [q  HOOK p]
(T T F T)(p, q) '' '' If p then q. [p  HOOK q]
(T T T F)(p, q) '' '' p or q. [p v q]
(F F T T )(p, q) '' '' Not q. [~q]
(F T F T)(p, q) '' '' Not p. [~p]
(F T T F)(p, q) '' '' p or q, but not both. [p . ~q :v: q . ~p]
(T F F T)(p, q) '' '' If p, then q; and if q, then p. [p  == q]
(T F T F)(p, q) '' '' p
(T T F F)(p, q) '' '' q
(F F F T)(p, q) '' '' Neither p nor q. [p . ~q or p | q]
(F F T F)(p, q) '' '' p and not q. [p . ~q]
(F T F F)(p, q) '' '' q and not p. [q . ~p]
(T F F F)(p, q) '' '' p and q. [p . q]
(F F F F)(p, q) Contradiction (p and not p; and q and not q.) [p . ~p . q . ~q]

Those truth-possibilities of its truth-arguments that verify a proposition, I will call its truth-grounds.

So, this is what those series would be. And here we have the in-between of tautology and contradiction that I doubted earlier (i.e. from the second to the penultimate line of the schema). Does this show my doubt to have been misplaced? Well, all we have here is a kind of chart. What is going to be done with it? How is it to be interpreted?

5.1 Truth-functions can be ordered in series.

That is the foundation of the theory of probability.

See comment on 4.464. What would these series be? Well, see below for an example.

Monday, October 15, 2007

5.02 It is natural to confuse the argument of functions with the affixes of names. This is because I recognize just as well from the argument as from the affix the meaning of the sign containing it.

In Russell’s “+c”, e.g., “c” is an affix that indicates [hinweist] that the whole sign is the addition sign for cardinal numbers. But this signifying depends on arbitrary agreement and one could choose a simple sign instead of “+c”; in “~p” however, “p” is not an affix, but an argument: the sense of “~p” cannot be understood without the sense of “p” having first been understood. (In the name Julius Caesar “Julius” is an affix. An affix is always a part of a description of an object, to whose name we attach it. E.g. the Caesar of the genus Julius.)

The confusion of argument and affix, if I am not mistaken, is at the bottom of Frege’s theory of the meaning [Bedeutung] of propositions and functions. For Frege the propositions of logic were names, and their arguments the affixes of these names.

Black (p. 239) says Wittgenstein gets Frege wrong here. “Wittgenstein’s allegation is incorrect. Had Frege really thought of the names composing a proposition as ‘indices’ [what I call affixes] in Wittgenstein’s sense, he must have conceded that the meaning of any proposition could just as well have been conveyed by a simple symbol—say T for a true proposition, and F for a false one. Now, Frege would have agreed that the reference (Bedeutung) of a proposition could be identified by a name; but he also held that the sense of a proposition was a function of the senses of its components (as Wittgenstein himself seems to recognize at 3.318 in his allusion to Frege).”

Russell uses “+c” in Principia vol. II, p. 73.

5.01 The elementary propositions are the truth-arguments of the proposition.

Arguments in the mathematical sense, presumably. Other translations have “propositions” at the end, but surely “des Satzes” is singular? Presumably it is like “the zebra” meaning something like “zebras in general.”

5 A proposition is a truth-function of elementary propositions.

(An elementary proposition is a truth-function of itself.)

Frascolla (p. 118): “The principle which is enunciated at 5 is universally known as the Thesis of Extensionality.”

The parenthetical remark here seems eyebrow-raising to me.

4.53 The general propositional form is a variable.


Friday, October 12, 2007

4.52 Sentences are everything that follows from the totality of all elementary sentences (and of course from the fact that this is the totality). (Thus one could in a certain sense say that all sentences are generalizations of elementary sentences.)

One could say this, in some sense. Why so tentative?

4.51 Suppose that all elementary sentences were given to me: then it can be asked simply: which sentences can I build from them? And those are all [the] sentences [that there are] and thus are they limited.

The last sentence here is translated by Ogden and Pears & McGuinness as if it meant what I have implied by the addition of the words in brackets , but these are absent from what Wittgenstein actually wrote. Perhaps this is significant. I don't know.

4.5 Now it seems possible for the most general propositional form to be given: that is to say, to give a description of the propositions of any sign-language whatsoever, so that every possible sense can be expressed by a symbol that fits the description, and that every symbol that fits the description can express a sense, if the meanings [Bedeutungen] of the names are suitably chosen.

It is clear that in the description of the most general propositional form only what is essential to it may be described, -- otherwise it would precisely [nämlich] not be the most general form.

That there is a general propositional form is indicated [bewiesen] by the fact that there may be no proposition whose form one could not have foreseen (i.e. constructed). The general form of the proposition is: Things are thus and so.

Bathos! Surely. All seems right, although flags are raised by the “seems” of the first sentence, until the final “such and such” [German: “so und so.”]

Ostrow (p. 114): “in the transparent vacuity of this culminating statement we are meant to see the vacuity of the Frege/Russell logic, of any attempt to specify a priori the limits of thought and language.” Black (p. 236) calls Wittgenstein’s answer to the question of what the general form of the proposition is “cryptic and unsatisfactory” and adds on the next page that “indeed, the form of words offered is cryptic to the point of unintelligibility.”

White (p. 82): “…in any natural translation, the final sentence of 4.5 looks astonishingly banal, even silly, as a statement of the goal Wittgenstein is struggling to arrive at. Perhaps we should take the overtones of sich verhälten that are lost in the English and render it as: ‘This is how things are arranged’, with the idea, that if we have the general form of proposition, then it will show for any proposition how things must be arranged in the world for it to be true.”

See also PI § 136.

Thursday, October 11, 2007

4.4661 To be sure, even in tautologies and contradictions signs are still combined with one another, i.e. they stand in relations to one another, but these relations are meaningless [bedeutungslos], inessential to the symbol.

There is a question whether signs and symbols are the same things. Here it certainly seems that they could be.

4.466 To a definite logical combination of symbols corresponds a definite logical combination of their meanings [Bedeutungen]; every arbitrary combination corresponds only to unconnected symbols.

That is to say, propositions that are true for every state of things cannot after all be combinations of symbols, because otherwise only definite combinations of objects could correspond to them.

(And there is no logical combination to which no combination of objects corresponds.)

Tautology and contradiction are the limiting cases of the combination of symbols, namely their dissolution.

So tautologies and contradictions are not, after all (überhaupt), combinations of signs. Everything in between is a combination, but at the limits, these bindings become undone. So we need to re-think, if not take back, 4.4611. But is that enough? What sense can we now make of these “limiting cases” as limits of anything? And then what do we mean by “everything in between”?

I have changed “signs” to ‘symbols” here in line with p. 60 of Letters to Ogden.

4.465 The logical product of a tautology and a proposition says the same as the proposition [on its own]. Thus that product is identical with the proposition. Because one cannot change the essence of the symbol without changing its sense.

See 3.24 for the first (I think) reference to symbols.

Wednesday, October 10, 2007

4.464 The truth of tautology is certain, of propositions possible, of contradiction impossible.

(Certain, possible, impossible: here we have an indication of the gradation we need in the theory of probability.)

It’s dodgy though, isn’t it? The certain must belong to the class of the possible, surely, not be a separate class from it.

No article before “tautology” or “contradiction” here, or in 4.463, as per Wittgenstein’s wishes in Letters to Ogden, p. 30. No plural because “there are in fact no contradictions but there is only contradiction, for they all mean the same, i.e. nothing. And the same applies to tautology.”

4.463 Truth-conditions define the room to move [Spielraum – literally “play space”] left to the facts by a proposition.

(A proposition, a picture, a model, are in a negative sense like a solid body that restricts the free movement of others; in a positive sense [they are] like a space limited by solid substance wherein there is room for a body.)

Tautology leaves to reality the whole – endless – logical space; contradiction fills the whole of logical space and leaves reality not a point. Neither of them, therefore, can determine reality in any way.

Odd to say that a contradiction fills logical space, taking up every point, as if there were room for a contradiction in logical space. To say it belongs to the symbolism is, I suppose, to say just this. But in that sense tautology too is within logical space, yet Wittgenstein denies this here. So what is he saying?

Cf. Notebooks November 14th 1914.

4.462 Tautology and contradiction are not pictures of reality. They represent no possible states of things. Because one lets every possible state of things be, and the other none.

In tautology the conditions of agreement with the world – the representing relations – cancel each other out, so that it stands in no representing relation to reality.

So representation is not necessary for meaning (as it is for sense or representation).

Tuesday, October 09, 2007

4.4611 But tautology and contradiction are not nonsensical [unsinnig]; they belong to the symbolism, in a way similar to that in which “0” belongs to the symbolism of arithmetic.

They are bedeutungslos, presumably, referring to nothing, and they are sinnlos (by 4.461). But not, supposedly, meaningless. My translation is based, partly, on Wittgenstein’s remarks on p. 49 of his Letters to Ogden. (Or it was--I just changed "meaningless" to "nonsensical" without re-checking what Wittgenstein wrote to Ogden.)

Mounce (p. 43) says that Wittgenstein means that tautologies and contradictions say nothing (about the weather, e.g.) but nevertheless are not gibberish. They are not gibberish because there are rules for constructing truth tables that yield tautologies and contradictions, but not for gibberish. Also, tautologies and contradictions “show something about the nature of logical structure. Thus ‘p . ~p’ says nothing, but it shows something about logic that this cannot be said, or rather, that these signs when put together say nothing.” (What does it show?!) Mounce again: “In ‘p . ~p’, one might say, is revealed a disintegration of sense, but the value of ‘p . ~p’ is that the disintegration is revealed by means of it not to be arbitrary. One is aware, by means of it, of rules which reflect logical form and that enable one to construct out of the symbols which constitute it propositions that do say something.” (Is such a thing thereby revealed? Was it not already apparent? Could we understand ‘p . ~p’ unless we already grasped the rules of logic of which it supposedly makes us aware? Maybe it makes us more aware, or reminds us of what we already knew, but I’m not sure. And don’t the rules of logic, by which we construct truth tables, for instance, also mean that anything else is gibberish? So there are rules for constructing gibberish, namely (violating) the rules we must follow if we are to make sense.

4.461 A sentence shows what it says, a tautology and a contradiction [show] that they say nothing.

A tautology has no truth-conditions because it is unconditionally true; and a contradiction is true under no condition.

Tautology and contradiction are senseless [sinnlos].

(Like a point from which two arrows go out in opposite directions to one another.)

(I know nothing, e.g., about the weather if I know that it is raining or not raining.)

So a tautology tells us nothing and thus has no sense. Like a contradiction. But a tautology is always true. So somehow it corresponds to the world. How? What does it picture?

4.46 Of all the possible groups of truth-conditions there are two extreme cases.

In one case the proposition is true for all truth-possibilities of the elementary propositions. We say that the truth-conditions are tautological.

In the second case the proposition is false for all truth-possibilities: the truth-conditions are contradictory.

In the first case we call the proposition a tautology, in the second case a contradiction.


4.45 For n elementary propositions there are Ln possible groups of truth-conditions.

The groups of truth-conditions that belong to a number of truth-possibilities can be ordered in a series.


Monday, October 08, 2007

4.442 This, e.g., is a propositional sign:

“p q




F F T"

(Frege’s “assertion sign” “‌‌–“ is logically quite meaningless [ganz bedeutungslos]; for Frege (and Russell) it shows only that these authors hold the propositions thus marked as true. Therefore “‌‌–“ belongs as little to the proposition as does the number of a proposition. A proposition cannot possibly assert of itself that it is true.)

If the order of the truth-possibilities in a schema is fixed once and for all by a rule of combination, then the last column by itself is already an expression of the truth-conditions. If we write this column as a row, then the propositional sign will be:

“(TT—T) (p, q)” Or more distinctly “(TTFT) (p, q)”.

(The number of places inside the brackets on the left is determined by the number of terms in the brackets on the right.)

cf. 4.022. Propositions, I suppose, implicitly say that they are true, in the sense that adding “It is true that” before a proposition (often) makes no difference, adds nothing to it. This is the sense in which they cannot possibly assert [any more than they already do] that they are true. Frege confusedly imports psychology into his logical work. Otherwise this all seems fairly straightforward.

Proops (see pp. 29-57) treats this, along with 4.064 and 4.063 as containing the core of Wittgenstein’s critique of Frege’s assertion sign. He argues that Wittgenstein misunderstands Frege’s position, noting that these remarks are virtually identical to ones Wittgenstein wrote in 1913 in his Notes on Logic (see Proops, p. 31, note 86). Proops (p. 38): “what Wittgenstein means by a “proposition” at 4.442 is, in effect, what Frege calls a “Proposition of Begriffsschrift”—i.e. a sentence of Begriffsschrift immediately preceded by the judgement stroke. Wittgenstein understands such signs … as translatable into English by expressions of the form ‘S is true’ and ‘S is false’, where ‘S’ is replaceable by a sentence of English. So ‘is true’ and ‘is false’ can also be said to be “verbs” in a derivative sense.”

Black (pp. 226-227) says that the assertion sign is unnecessary in Russell’s Principia because it is introduced explicitly (see vol. I, p. 8) to distinguish complete propositions from subordinate propositions contained within them. But, according to Black at least, this is unnecessary, as the difference is already clear, and so the effect is that the sign indicates simply that the authors are putting the proposition forward as true. Frege’s case is different. “Frege is not required to say, pace Wittgenstein, that the presence of his sign-post in his text means that Frege judges the proposition in question to be true. But on the other hand, it is hard to make sense of this supposedly ‘objective’ judgement, which cannot, on Frege’s principles, be a thought. Frege’s sign-post serves no purpose. Frege’s introduction of the assertion-sign may be viewed as an unsuccessful attempt to restore to the propositional sign, which he had degraded to a mere designation, its truth-claiming aspect. Wittgenstein’s account of the proposition does justice to this aspect from the start.”

Philosophical Investigations § 22: “Frege’s idea that every assertion contains an assumption, which is the thing that is asserted, really rests on the possibility found in our language of writing every statement in the form: “It is asserted that such-and-such is the case.”—But “that such-and-such is the case” is not a sentence in our language—so far it is not a move in the language-game. And if I write, not “It is asserted that….”, but “It is asserted: such-and-such is the case”, the words “It is asserted” simply become superfluous.”

Friday, October 05, 2007

4.441 It is clear that no object (or complex of objects) corresponds to the complex of the signs “F” and “T;” just as little as do horizontal and vertical lines or brackets. – There are no “logical objects.”

Of course the same goes for all signs that express the same as the schemata of “T” and “F.”

Black (p. 224): “Wittgenstein’s important conclusion that there are no ‘logical objects’ follows immediately from his principle that objects are represented by genuine names, while the logical signs are ‘interdefinable’ (cf. 5.42b).”

4.44 The sign that is produced by the coordination of each sign “T” with the truth-possibilities is a propositional sign.

OK. More stipulation.

Thursday, October 04, 2007

4.431 The expression of agreement and disagreement with the truth-possibilities of elementary propositions expresses the truth-conditions of a proposition.

A proposition is the expression of its truth-conditions.

(Frege quite rightly therefore put them first [vorausgeschickt] as an explanation of the signs of his concept-script. Only the explanation of the concept of truth that we get from Frege is false: if “the True” and “the False” were really objects and the arguments in ~p etc. then Frege’s determination of the sense of “~p” would in no way determine it.)

Anscombe (p. 107): “As a criticism of Frege the point can be summarized by saying: ‘If truth-values are the references of propositions, then you do not specify a sense by specifying a truth-value.’” Because, according to Frege, determining or specifying sense is not the same as determining or specifying reference. Proops (p. 41, note 123) has ‘determination’ here too, noting that in Prototractatus 4.4221 “Wittgenstein makes it plain that in the envisaged circumstances Frege’s “determination” would be unworthy of the name.”

4.43 We can express agreement with truth-possibilities by coordinating the sign “T” (for true) with them in the schema.

The absence of this sign means disagreement.


Wednesday, October 03, 2007

4.42 As to the agreement and disagreement of a proposition with the truth-possibilities of n elementary propositions, there are SUMMATION(kappa=0 to Kn, binom-coeff(Kn over kappa) = Ln possibilities.

The last part of the symbolism, Black says (p. 222), means 22n.

4.411 It is plausible from the beginning that the introduction of elementary propositions is foundational for the understanding of all other kinds of propositions. Indeed the understanding of all propositions depends palpably [fühlbar – ‘feel-ably’ or sensibly, the Notes on Logic say ‘obviously’ at the corresponding point (p. 100)] on the understanding of the elementary propositions.

‘Plausible’ is suggested by Black (p. 220). Otherwise we have “It is probable from the beginning…” which does not seem to make much sense. Ogden ends up with “It seems probable even at first sight,” while P&McG have “It immediately strikes one as probable.”

He meant, perhaps, that it feels as though this is the case. Because the previous sentence seems to imply that all this is not true in fact. And indeed it seems not to be true, since we do understand propositions but cannot identify a single elementary proposition. Although perhaps we do not have a philosophical understanding of the nature of any propositions.

Tuesday, October 02, 2007

4.41 The truth-possibilities of elementary propositions are the conditions of the truth and falsehood of the propositions.

Sounds tautological, but OK.

4.4 A proposition is the expression of agreement and disagreement with the truth-possibilities of elementary propositions.


Monday, October 01, 2007

4.31 We can represent the truth-possibilities by schemata in the following way (“T” means “true,” “F” means “false.” The rows of “T”s and “F”s under the row of elementary propositions indicate [bedeuten] their truth-possibilities in a readily comprehensible symbolism):

p q r p q p









These should be three truth tables. Must try harder to get them to show up properly.