A Completeness Theorem for Unrestricted First-Order Languages
Agustín Rayo, Timothy L. Williamson · 2003
Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents and the syntactic differences between their non-logical constituents, treated as variables. A constituent of a sentence is logical just if it is formal in meaning, in the sense roughly that its application is invariant under permutations of individuals. Thus ‘=’ is a logical constant because no permutation maps two individuals to one or one to two; ‘∈’ is not a logical constant because some permutations interchange the null set and its singleton. Truth functions, the usual quantifiers and bound variables also count as logical constants. An argument is logically valid if and only if the conclusion is true under every assignment of semantic values to variables (including all non-logical expressions) under which all its premises are true. A sentence is logically true if and only if the argument with no premises of which it is the conclusion is logically valid, that is, if and only if the sentence is true under every assignment of semantic values to variables. An interpretation assigns values to all variables. For the case of first-order languages, interpretations are standardly cashed out in terms of what might be called model-theoretic interpretations (or MT-interpretations). An MT-interpretation for a first-order language L is an ordered pair 〈D, F 〉. The domain D is a non-empty set, and is intended to specify the range of the variables in L. The interpretation function F is intended to specify semantic values for the variables of L (including non-logical expressions). The semantic value of an n-place predicate-letter is a set of n-tuples of individuals in D, and the semantic value of a first-order variable is an individual in D. Truth on an MT-interpretation can then be characterized as follows: