This 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)

Wednesday, April 2, 2008

Logical Inference.

If all the verifiers common to a number of propositions are also verifiers of a particular proposition, then the truth of that proposition follows from the truth of the others. In particular, the truth of one proposition (p) follows from the truth of another proposition (q) if all the verifiers of (q) are also verifiers of (p). In that case, the sense of (p) is contained in the sense of (q). If a god creates a world in which certain propositions are true, then by that very act he also creates a world in which all the propositions that follow from them come true. And similarly he could not create a world in which the proposition (p) was true without creating all its objects.

A proposition affirms every proposition that follows from it. (p.q) is a proposition that affirms (p) as well as (q).
(TFTF)(p,q) (p)
(TTFF)(p,q) (q)
(TFFF)(p,q) (p.q)

Two propositions are opposed to one another if no proposition that makes sense affirms them both. Every proposition that contradicts another negates it.

One can see that the truth of one proposition follows from the truth of others from the structure of the propositions. When that is the case, the relations between the forms of the propositions express that. Furthermore, we need not associate them in a compound proposition. Rather, these relations are internal, an immediate consequence of the existence of the propositions.

When we infer q from (p∨q and ~p), the relation between the propositional forms of (p∨q) and (~p) is masked by our notation.

(TTTF)(p,q) (p∨q)
(FTFT)(p,q) (~p)
(TTFF)(p,q) (q)

But if instead of (p∨q) we write, for example, (p|q . | . p|q), and instead of (~p), (p|p) where (p|q = ~p.~q), then the internal relation becomes apparent.
(FFFT)(p,q) (~p.~q or p|q)
(FTFT)(p,q) (~p)
(FFTT)(p,q) (~q)

(TTTF)(p.q) (~(p|p).~(p|p))
(FTFT)(p,q) (p|p)
(TTFF)(p,q) (q)

(That one can start with (x).fx and conclude fa shows that the symbol (x).fx itself is generalizable.)

If p follows from q, I can infer p from q; deduce p from q. The nature of the conclusion can be gathered only from the two propositions; only they themselves can justify the inference. 'Laws of inference', which are supposed to justify inferences, as in the works of Frege and Russell, have no purpose, and would be superfluous.

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