Wittgenstein: Why is bipolarity necessary?

Question

I do get that for a certain proposition there may be the possibility of it being either true or false, but why must this possibility exist for every proposition?

Answer 1

This is a deceptively complex question, and very on the nose when it comes to Tractarian interpretation. My line on this is to say that we need to pay attention to the distinction in the semantics of TLP between "Propositions" (the German "Satz") and "Elementary Propositions" ("Elementarsatz"), and to note the theoretical difficulties in explaining how to cash out the more general notion of Proposition in terms of Truth functions over Elementary Propositions, as in section $5, without sacrificing the Picture Theory approach that makes the semantics of Elementary Propositions seem appealing

For Wittgenstein, although propositional (Elementarsatz, at least) meaning is tied to the existence of states of affairs, how do we explain the meaningfulness of contingently false sentences, such as “It is warmer in Britain than in Australia”? If there is a state of affairs of objects joined in the way indicated by the sentence, then it is clearly a non-actual one. On the other hand, if there is no such state of affairs, then how do we resolve the reference of the sentence to some other state of affairs so as to preserve sensibility?

We have a choice to make between accepting that non-actual states of affairs can be the subjects of utterance and fleshing out some mechanism by which we speak of the facts of the matter even in error. Max Black, in his “Companion to Wittgenstein's Tractatus”, denotes these two interpretations as Possibility and Fact Theories, respectively, and different philosophers have developed both interpretations of the Tractatus as in keeping with the themes and approaches of the original text (in as much as any interpretation can be in keeping with the TLP).

If we can take it as read that the Truth functions exist as mathematical functions over a Boolean-valued domain, such that there are only two possible truth values that any given proposition might take, then this is a fairly cheap Fact-Theory way to get the additional abstraction over our base level of elementary propositions that we need to speak "falsely about" the facts of the matter. Elementary propositions describe states of affairs, these states of affairs combine together into "facts" in a way strictly mediated by how complex propositions are structured in boolean functional terms, and the "negative fact" is just a consequence of the negation operation in boolean logic, rather than any kind of complicated alternative possibility. In the example above, Britain is at least as cold as Australia, and so the falsity of Britain being warmer can be accounted for from the state of affairs describing the relevant climates of the two and the boolean function taking truth to falsity and vice versa.

(In effect, the "possibility" of truth or falsehood in a proposition doesn't commit us to an ontology of alternate universes of states of affairs if we can get this kind of abstract truth function construction going. We do need such an account, because the more natural way to interpret negative propositions in the picture theory is to point to an inflated ontology with non-actual states of affairs for them to correspond to. And if we can account for negated propositions in this kind of mathematical way, we get the bivalence of any proposition as part of the package.)

This interpretation is very along lines that Bertrand Russell might have endorsed. So naturally as Wittgenstein scholars we might want to be skeptical that it's the intended one. Indeed, there seems reason to suspect that W himself might be interpreting negation some other way. There is a distinction to be drawn in TLP between "~" negation (the German "Verneinung", or "denial") and the "N(ζ)" "Negation" described in $5.5 in terms of repeated schematic applications of the Sheffer Stroke - the former, if thought of as Boolean negation, is a Semantic conception of negation and the latter Syntactic. It's the Syntactic version "N(ζ)" that is called upon in $6 to explain the logical form of Truth Functions (rather than the more contemporary tactic of evaluating the definable functions set-theoretically), and hence that informs his conclusion about the limits of what can be logically expressed.

Mathematically, if we think the syntactic conception of negation is the intended one, then as Benubird pointed out above, we don't necessarily thereby commit ourselves to a strictly Boolean logical framework. For Wittgenstein in the Tractatus, it is entirely possible that problems around infinity and countability hadn't yet come into view (his Philosophy of Mathematics work would come later and largely miss the massive developments happening in Set Theory at the time) but issues around how exactly to cash out this notion of repeated application of functions and the mathematical disputes around foundations and axioms in maths would be very relevant here. We could entirely well say something very similar about the formal limitations of assertion using Heyting Algebras or some alternative semantic framework as would be favoured by Intuitionists; in which case, we would arrive at a different conclusion of what possible descriptions of the Facts there might be too.

Given what Wittgenstein says throughout about the use of Truth Tables, and his position in the history of mathematics, it seems reasonable to suppose that he was relying on the technology of boolean logic over finite models. Since model theory, proof theory and set theory have come a long way since the 1920's, we might be more prepared from our current standpoint to deal with pluralities that Wittgenstein as a non-mathematician may not have been aware he was skipping over.

Answer 2

First off, I think we need to understand what is meant by a proposition, how it differs from a fact, which differs from a thing and what it means to bear truth, and what is truth.

Wittgenstein in the opening section of the Tractatus says that:

1.1 The world is the totality of facts, and not of things

Following Kant, we cannot access, the thing-in-itself (ding-an-sich), but we have access to facts. Facts simply are - they do not bear truth-values, that is we do not say that they are true or false. So, what is it that we do when we say something is true or not? According to Wittgenstein:

2.1 We picture facts to ourselves.

Notably the mind must be involved - and it is that picture that we make in our minds that have truth or falsity, that is they have logic, which is confirmed in:

3.0 A logical picture of facts is a thought.

Where logic is introduced. But this thought, this picture of the facts, is not a proposition. The picture I have of green grass is not the same as the proposition that 'grass is green', which he points out in:

3.1 In a proposition a thought finds an expression that can be perceived by the senses.

This does not mean that a proposition is something that is written or spoken. When a proposition is written out or spoken out aloud, he calls that the sign of the proposition - the perceptible sign - since we can read or hear it:

3.11 We use the perceptible sign of a proposition (spoken or written, etc.) as a projection of a possible situation. The method of projection is to think of the sense of the proposition.

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.

So a proposition is that which is common to all signs that express it. This is similar to the idea that 'oneness' is common to 1, one, ein, ek etc.

So propositions bear truth - but what is truth for Wittgenstein? He terms it as agreement with the facts of the world or not:

4.3 Truth-possibilities of elementary propositions mean Possibilities of existence and non-existence of states of affairs.

(He excludes tautologies and contradictions as propositions are they are neccessarily true or false, and admit no possibility).

To return to your question:

I do get that for a certain proposition there may be the possibility of it being either true or false, but why must this possibility exist for every proposition?

So, propositions for Wittgenstein must have the possibility of being either true or false; he expressly forbids analytic propositions which by their very form are always true or false. In his view propositions must have possibility. But can one have more than true or false as truth-values? It first appears that the answer to this question must be yes, for in modern logic there are species of logic that are multi-valued, for example intuitionistic logic, but note that this is a formal system and the meta-logic we use to manipulate this formal system is still classical and thus admits only true & false as the only truth-values.

Answer 3

You can see Gregory Landini's paper : "Wittgenstein's Tractarian Apprenticeship" (Russell: the Journal of Bertrand Russell Studies, 2003) or Landini's book : Wittgenstein's Apprenticeship with Russell (2007).

In "Wittgenstein's Tractarian Apprenticeship", Landini writes (106),

In fact, Wittgenstein’s famous self-imposed isolation in Norway “until he has solved all the problems of logic” failed to produce the reduction that Nicod found in 1916. ...

There is no question that Wittgenstein had ideas for improving the philosophical foundation of Principia [...]. For example, Wittgenstein proclaimed that a proper elimination of the identity symbol would obviate Principia’s need to add statements of the infinity of individuals as antecedents of central theorems concerning inductive cardinals. He also advocated an extensionalist position that “a function can appear only through its values” and suggested that this would avoid the need for Principia’s Reducibility Principle.

The truth-functional foundation of logic (extensionality) was one of the building blocks of the "revolution" in logic of Frege and Russell.

Wittgenstein's Tractatus is born in that "environment"; so is natural that we find it at the core of the book, that was intended as a "philosophical foundation" of the new logic.

Answer 4

It is not true to say that it must exist for every proposition. There are plenty of logic systems that do not admit to only true and false values. For example, see here: http://en.wikipedia.org/wiki/Three-valued_logic