Bipolarity and Sense in the Tractatus Peter Hanks University of Minnesota [email protected]

SSHAP Conference, May 2012 1

Plan • What is bipolarity? • Bipolarity and sense in Notes on Logic • Bipolarity and sense in the Tractatus

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What is bipolarity? Bipolarity as the capacity for both truth and falsity “Although Wittgenstein’s preconception that ordinary language is in good logical order committed him to the requirement of bivalence and the applicability of the Law of Excluded Middle, he adopted, in the course of his atomism, the much more radical position of Bipolarity for elementary propositions. This of course satisfies bivalency, but it goes much further, for it commits him to the unique position of denying that there are any necessary elementary propositions. Any proposition that has a sense must not just be capable of being true or false, it must be capable of being true and also capable of being false.” - P.M.S. Hacker, “The Rise and Fall of the Picture Theory,” 1981 “The only genuine propositions are pictures of possible states of affairs. bipolar — capable of being true but also capable of being false ...,” - Hans-Johann Glock, “Necessity and Normativity,” 1996

These are

“In the 1913 Notes on Logic the bi-polarity of genuine propositions is characterized in terms of their being “essentially true-false” (Notebooks, p.98). By their very nature they are capable both of truth and falsity.” - Ian Proops, Logic and Language in Wittgenstein’s Tractatus, 2000 3

What is bipolarity? Bipolarity as the representation of positive and negative states of affairs “The picture-theory of the proposition is that the proposition in the positive sense says: ‘This is how things are’ and in the negative sense says: ‘This is how things aren’t‘ — the ‘this’ in both cases being the same. ... This rendering of the picture become proposition would stress the fact that it has acquired two ‘poles’, or senses in which it can be thought, by having the drawn figures correlated with actual men. ... I believe that the most that we can say is that the bi-polarity of the picture, of the occurrence of one picture in two senses, has a very striking analogy in the fact that if we have a proposition, and insert a ‘not’ into it, then what is being denied is exactly what the original proposition said.” - Anscombe, An Introduction To Wittgenstein’s Tractatus, 1959 “W. liked to think of this ‘bi-polarity’ by imagining the proposition to draw a boundary in ‘logical space’, with the verifying conditions on one side and the falsifying conditions on the other. The need for a proposition to specify both falsifying and verifying conditions then appears as the truism that a boundary must have two sides to it.” - Black, A Companion to Wittgenstein’s ‘Tractatus’, 1964

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Bipolarity as the representation of positive and negative states of affairs

‘aRb’

possible positive state of affairs: a’s bearing R to b

possible negative state of affairs: a’s not bearing R to b

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Bipolarity in Notes on Logic

“Every proposition is essentially true-false: to understand it, we must know both what must be the case if it is true, and what must be the case if it is false. Thus a proposition has two poles, corresponding to the case of its truth and the case of its falsehood. We call this the sense of a proposition.” - Notes on Logic, 2nd. ed., pp.98-99

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Bipolarity in Notes on Logic The “definition of sense” in Notes on Logic “The form of a proposition has meaning in the following way. Consider a symbol “xRy”. To symbols of this form correspond couples of things whose names are respectively “x” and “y”. The things xy stand to one another in all sorts of relations, amongst others some stand in the relation R, and some not; just as I single out a particular thing by a particular name I single out all behaviors of the points x and y with respect to the relation R. I say that if an x stands in the relation R to a y the sign “xRy” is to be called true to the fact and otherwise false. This is a definition of sense.” - Notes on Logic, 2nd ed., p.95

“The form of a proposition is like a straight line, which divides all points of a plane into right and left.” - Notes on Logic, 2nd ed., p.102

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Bipolarity in Notes on Logic (cont.)

‘xRy’

pairs that bear R

‘a’,‘b’

pairs that do not bear R



‘aRb’ possible positive state of affairs: a bears R to b

possible negative state of affairs: a does not bear R to b 8

Sense in Notes on Logic

bipolarity

‘aRb’

possible positive state of affairs: a bears R to b

possible negative state of affairs: a does not bear R to b

‘aRb’ sense

possible positive state of affairs: a bears R to b

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Sense in Notes on Logic (cont.) “The sense of a proposition is determined by the two poles true and false. The form of a proposition is like a straight line, which divides all points of a plane into right and left. The line does this automatically, the form of a proposition only by convention.” (2nd ed., p.102) “I now determine the sense of ‘xRy’ by laying down: when the facts behave in regard to ‘xRy’ so that the meaning of ‘x’ stands in the relation R to the meaning of ‘y’, then I say that the [facts] are ‘of like sense’ with the proposition ‘xRy’; otherwise ‘of opposite sense’: I correlate the facts to the symbol ‘xRy’ by thus dividing them into those of like sense and those of opposite sense. To this correlation corresponds the correlation of name and meaning. Both are psychological.” (2nd ed., p.104)

‘xRy’ pairs that bear R

pairs that do not bear R

“of like sense” truth

“of opposite sense” falsity 10

Sense in Notes on Logic (cont.) ‘xRy’

‘a’,‘b’

pairs that bear R

pairs that do not bear R

truth

falsity



‘aRb’ possible positive state of affairs: a bears R to b

possible negative state of affairs: a does not bear R to b

truth pole

falsity pole

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Sense in Notes on Logic (cont.)

“Thus a proposition has two poles, corresponding to the case of its truth and the case of its falsehood. We call this the sense of a proposition.” (NL, 2nd ed., p.98-99) “From this it results that ‘true’ and ‘false’ are not accidental properties of a proposition, such that, when it has meaning, we can say it is also true or false: on the contrary, to have meaning means to be true or false: the being true or false actually constitutes the relation of the proposition to reality, which we mean by saying that it has meaning (Sinn).” (Notes Dictated to G.E. Moore in Norway, 2nd ed., p.113)

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Bipolarity in the Tractatus The terms “bipolar”, “bipolarity”, and “poles” do not appear anywhere in the Tractatus.

2.11: A picture presents a situation in logical space, the existence and non-existence of states of affairs. 2.201: A picture depicts reality by representing a possibility of existence and non-existence of states of affairs. 4.1: Propositions represent the existence and non-existence of states of affairs. 2.06: (We also call the existence of states of affairs a positive fact, and their nonexistence a negative fact.)

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Bipolarity in the Tractatus (cont.) picture

?

positive fact: two men fighting

negative fact: two men not fighting

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Bipolarity in the Tractatus (cont.) On the representation of negative facts in the Notebooks. “A picture can present relations that do not exist! How is that possible?” (2nd ed., p.8) “Could we say: In ‘~φ(x)’ ‘φ(x)’ images how things are not?” (2nd ed., p.21) “That shadow which the picture as it were casts upon the world: How am I to get an exact grasp of it? Here is a deep mystery. It is the mystery of negation: This is not how things are, and yet we can say how things are not.” (2nd. ed., p.30)

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On the representation of negative facts in the Notebooks (cont.) “If a picture presents what-is-not-the-case in the forementioned way, this only happens through its presenting that which is not the case. For the picture says, as it were: “This is how it is not”, and to the question “How is it not?” just the positive proposition is the answer.” (2nd ed., p.25)

“Think of the representation of negative facts by means of models. E.g.: two railway trains must not stand on the rails in such-and-such a way. The proposition, the picture, the model are — in the negative sense —  like a solid body restricting the freedom of movement of others; in the positive sense, like the space bounded by solid substance, in which there is room for a body.

This image is very clear and must lead to a solution.” (2nd. ed., p.30) Cf. 4.463

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On the representation of negative facts in the Notebooks (cont.)

4.0641: One could say that negation must be related to the logical place determined by the negated proposition. The negating proposition determines a logical place different from that of the negated proposition. The negating proposition determines a logical place with the help of the negated proposition. For it describes it as lying outside the latter’s logical place. 5.2341: Negation reverses the sense of a proposition.

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‘~aFb’ ‘aFb’ says

positive fact: two men fighting

says

negative fact: two men not fighting

5.2341: Negation reverses the sense of a proposition.

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Sense in the Tractatus

‘aFb’

bipolarity

positive fact: two men fighting

negative fact: two men not fighting

4.21: The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs.

‘aFb’ sense

positive fact: two men fighting

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Sense in the Tractatus (cont.) On the reduced role for conventions in the Tractatus In Notes on Logic, conventions: 1. associate names with objects 2. associate predicates with relations (divisions in objects) 3. associate truth and falsity with the poles of a proposition In the Tractatus the third role is eliminated. The propositional sign becomes a proposition as soon as its names are associated with objects and its predicate is associated with a relation. 3.12: I call the sign with which we express a thought a propositional sign. — And a proposition is a propositional sign in its projective relation to the world. 5.4733: Frege says that any legitimately constructed proposition must have a sense. And I say that any possible proposition is legitimately constructed, and, if it has no sense, that can only be because we have failed to give a meaning to some of its constituents.

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On the reduced role for conventions in the Tractatus (cont.) Notebooks 4.11.14 “One name is representative of one thing, another of another thing, and they themselves are connected; in this way — like a tableau vivant — the whole images the situation.” (Cf. 4.0311) (2nd ed., p.26) 5.11.14 “In this way the proposition represents the situation — as it were off its own bat. … Then in order for a proposition to present a situation it is only necessary for its component parts to represent those of the situation and for the former to stand in a connexion which is possible for the latter.” (2nd ed., p.26) 26.11.14 “By my correlating the components of the picture with objects, it comes to represent a situation and to be right or wrong.” (2nd ed., p.33)

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Sense in the Tractatus (cont.) Propositions and arrows 3.144: Names are like points; propositions like arrows — they have sense.

bipolarity tail-to-head direction

head-to-tail direction

sense tail-to-head direction

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Sense in the Tractatus (cont.) Propositions and arrows (cont.) Philosophical Investigations 454. “Everything is already there in ….” How does it come about that this arrow → points? Doesn’t it seem to carry in it something besides itself? — “No, not the dead line on paper; only the psychical thing, the meaning, can do that.” —That is both true and false. The arrow points only in the application that a living being makes of it. This pointing is not a hocus-pocus which can be performed only by the soul.

455. We want to say: “When we mean something, it’s like going up to someone, it’s not having a dead picture (of any kind).” We go up to the thing we mean.

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Sense in the Tractatus (cont.) An appeal to use? “Now, confining ourselves to pictures, it is also clear that if we ‘think the sense of the picture’ by correlating its elements with actual objects, we can in fact think it in either of two ways: namely either as depicting what is the case, or as depicting what isn’t the case. That is to say, there are two senses which we can ‘think’ in connection with the picture. For it is the very same picture we hold up if we wish to say that it holds or that it doesn’t hold.” - Anscombe, p.69 4.062: Can we not make ourselves understood with false propositions just as we have done up till now with true ones? — So long as it is known that they are meant to be false. — No! For a proposition is true if we use it to say that things stand in a certain way, and they do; and if by ‘p’ we mean ~p and things stand as we mean that they do, then, construed in the new way, ‘p’ is true and not false. “This is how the picture arises.— In order to designate a logical place with the picture we must attach a way of symbolizing to it (the positive, the negative, etc.). We might, e.g., shew how not to fence by means of fencing puppets.” (Notebooks, 5.11.14, p.27)

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