Analyticity - Semantic and Syntactic

Salman Panahy (University of Melbourne), Salman Panahy

Logicians use analyticity in different meanings, some of them think of it as a semantic property of true sentences; they either think of analytic truths as ‘truths in virtue of meaning’ or as ‘truth in virtue of meaning of logical words’. Let us call the first understanding of analyticity the wider notion of analyticity and the second one the narrower notion of it since restricts analyticity to be true in virtue of a specific kind of words.

Some other logicians understand analyticity syntactically, that is literally in terms of syntax and structure of expressions. They prefer to call logical systems with sabformula property as ‘analytic’. One known system with this property is Gentzen’s Sequent Calculus. Since the semantic account of analyticity is silent about the literal composition of an expression out of its elements, sabformula property is not necessary for a logical system to be analytic. Natural Deduction is a well-known system without Sab-Formula property.

I will study the possible relation between the semantic/syntactic notions of analyticity by way of a conjecture: if we think of the meaning of logical vocabulary as their role in an argument, then syntactic issues are significant in characterizing the meaning of them. This conjecture, I think, has a support.  By the normalization theorem, Prawitz has shown that any proof in Natural Deduction system can be transformed into a normal proof. This happens because the Conditional Elimination rule in a Natural Deduction system offers a different meaning for the conditional, when compared with the left-rule for the conditional in the Sequent Calculus.