## Tuesday, 19 April 2011

### On the Interpretation of the Propositional Calculus

I've just posted another (more recent) longer article on my homepage, On the Interpretation of the Propositional Calculus. The next post will be a short article, I promise.

Comments are welcome.  Here is the abstract:

The question considered is 'How can formulae of the propositional calculus be brought into a representational relation with the world?'. Four approaches are discussed: (1) the denotational approach, on which formulae are taken to denote objects, (2) the abbreviational approach, on which formulae and connectives are taken to abbreviate natural-language expressions, (3) the truth-conditional approach, on which truth-conditions are stipulated for formulae, and (4) the modelling approach, on which formulae, together with either valuation- or proof-theory, are regarded as an abstract structure capable of bearing (via stipulation) a representational relation to the world.

The modelling approach is developed here for the first time. The simple technical apparatus used for this is then applied to two issues in the philosophy of logic. (1) I demonstrate a corollary or converse to Carnap's result that certain 'non-normal' valuation-functions can be added to the set of admissible valuations of formulae without destroying the soundness and completeness of standard proof-theories. This sheds considerable light on a recent thread of the inferentialism debate which involves dialectical use of Carnap's result. (2) I show how the approach can be extended to quantification theory, by defining a model-theoretic notion of validity equivalent to the usual one, but making use of a proof-theoretic apparatus in place of the device of assigning values to formulae. This sheds light on the close relationship between proof- and valuation-theory.

1. I don't understand the question. Why should we think that a formula, say

(E) P only if (Q only if P)

ought to represent anything in the world?

On the other hand, if someone asks, "How do you express this without symbols?", I might try to put it thus:

(E') Something is true only if given the truth of something else, its truth follows.

Then apparently I am talking about things which are true.

2. The above, (E), is not really a formula of the propositional calculus, since it contains English words.

Anyway, I'm talking about particular interpretations of formulae. The idea that you can fix things so that the atomic formula 'p', for example, "says" something in particular - e.g. that snow is white. Or so that 'p & q', to take another example, "says" something like 'snow is white and grass is green'. (I put 'says' in inverted commas because it is not the best word to use for some of the approaches I consider.)