The Proposition in Wittgenstein's Tractatus


An attempt to "make sense" of some of the Tractatus by comparing it with the standard account of the semantics of first order logic.
The Semantics of FOL
From the standard account of the semantics of First Order Logic it is possible to obtain an account of what is a proposition.
Wittgenstein's Propositions


This is part of my "positioning" exercise, trying to work out and articulate my attitude to the main features of analytic philosophy in the 20th Century.
Making Sense of The Tractatus
This is my best shot at extracting something worthwhile out of the Tractatus. This probably falls squarely in those aspects of the Tractatus which Wittgenstein was later to repudiate. In particular the sense I can make of it is in the account of the proposition (which was the original title of the work) which still hangs together as an element of the semantics of first order logic. (admittedly not what it was intended for)

The Semantics of FOL:

From the standard account of the semantics of First Order Logic it is possible to obtain an account of what is a proposition (even though that term is not used).
For our present purposes the simplest account of first order logic will suffice, i.e. one without function symbols and without any special treatment for equality. See, for example, my semi-formal description of first order logic. The account in any text on first order logic is likely to suffice for present purposes.
The notion of interpretation is key. An interpretation is an assignment to each relation symbol of a relation over the domain of discourse. An interpretation may be thought of as a possible world. The semantics is to be given by showing how the truth value of a sentence depends upon the structure of the particular world in which it is evaluated.
It is the definition of the relationship of satisfaction which performs this role. This is defined as a relationship between a sentence and an interpretation (in the context of an assignment of values to the free variables in the sentence, which we can ignore for present purposes).
Lack of Denotation
Because the semantics is presented as a relationship it does not appear to be denotational, there is no talk of "the meaning" of a sentence as an entity, no talk of propositions. However, this can be done without changing the pragmatics of the semantics in any way. A logically equivalent account can be presented in which each sentence denotes a proposition.
In this alternative rendition of the semantics of first order logic we simply talk of the denotation of a sentence as the set of interpretations in which it is true, or as a "propositional function" which takes interpretations as arguments and yields truth values. (still ignoring free variables)

Wittgenstein's Propositions:

5. A proposition is a truth function of elementary propositions.
A function must usually have some fixed number of arguments, so to make this "work" we need to think of the elementary propositions as bundled together, say into a set. A proposition is then a function from sets of elementary propositions to truth values. To make the connection with first order logic I just need to persuade you that a "set of elementary" propositions is essentially the same thing as a first order interpretation.


Present Day Validity
If we construe the Tractatus as (among other things) offering an answer to the question:
What must the world be like if it is to be possible to talk about it using a logical language like Frege's Begriffschrift?
Then the answer it supplies is not so bad, even viewed from the end of the century. Since scientific theories amount to mathematical models and modern mathematics can be (and often has been) construed in the context of first order set theory, this way of looking at the world is good enough for modern science.
This viewpoint does not really support inference to the structure of reality, since what we are discussing here is one way of structuring models of reality. All we can conclude about reality from the adequacy of this kind of scientific model is that it behaves as if it had the same structure as the model, and we expect there to be many alternative models with different structures which predict the same behaviour.
Logical Independence of Atomic Propositions
Apparently a decisive factor in Wittgenstein's repudiation of the Tractactus was his concluding that atomic propositions could not be logically independent. First order logic makes a distinction between logical and non-logical constants and axioms. From the usual semantics for first order logic (which we have argued corresponds closely to the ideas in the Tractatus) there derives a notion of what is a logical truth. In relation to this notion of logical truth the "atomic propositions" are indeed logically independent, but logical truth is a much narrower concept than analyticity (in which a truth may be analytic as a result of the relationship between the meaning of two non-logical concepts, e.g. a brother is a male sibling).

The weakness of the theory of the Tractatus for an adequate account of analyticity is connected with the weakness of first order logic as a basis for mathematics. The gap is filled by set theory, which not only provides a foundation for mathematics but which also can provide a broader account of logical truth corresponding to the notion of analyticity.


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