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# What is modern logic?

What is modern logic?

Modern logic crystallized gradually during the period 1850–1950. It was not created by any one person or school, but evolved from the work of several. We now recognize that the following features were needed for its mature development:

(a) There is a formal language whose symbols separate it from ordinary language.

(b) In the formal language, L, the syntax is clearly separated from the semantics.

The syntax is like the grammar of a natural language, specifying which sequences of symbols (called ‘formulas’) are permissible. Meaning does not occur in the syntax but in the semantics.

Thus logical concepts such as sentence, axiom, formal proof and consistency are part of the syntax. Concepts such as truth, satisfiability and definability are part of the semantics. Formal proofs can be checked effectively and do not depend on the meaning of the terms in them.

(c) Within the syntax of L, the logical symbols are clearly separated from the non-logical symbols. The logical symbols include connectives such as ‘and’, ‘not’, ‘if…then…’.

The meaning of the non-logical symbols changes with different interpretations, while that of the logical symbols does not.

The non-logical symbols may include individual constants, which, in the semantics, are interpreted as specific individuals. There may also be function constants and relation constants, which are interpreted as specific functions and relations. Besides the logical and non-logical symbols, there are variables, which may be free or bound.

(d) The syntax gives a recursive definition for the formulas of L.

(e) There is a clear distinction between the object language L, in which our formulas occur, and the metalanguage, in which we speak about L. Within the metalanguage, we must define what truth means in L. Theorems in the metalanguage are called metatheorems.

(f) Usually there will be levels within our symbolic logic. The lowest level (called propositional logic) is that of the connectives ‘and’, ‘not’, and so on.

The next level will have individual variables x and quantifiers (‘for all x‘, ‘there exists an x‘) over those variables. If that is the highest level, then we have first-order logic.

But there may also be function variables (or relation variables) and quantifiers over them, giving second-order logic (see Second- and higher-order logics).

In the simple theory of types, there are individual variables, function variables, variables for functions of functions, and so on, and this is sometimes called ω-order logic. We must decide which order our logic will have.

(g) We must also specify other matters. Will the semantics of our logic have only two truth-values, ‘true’ and ‘false’, as in traditional logic, or will it have three or more truth-values (see Many-valued logics)?

Will our logic be ‘classical’ (in which the law of the excluded middle holds) or ‘intuitionistic’ (in which that law fails; see Intuitionism)? Will our logic be finitary (like all the logics discussed above) or will it be infinitary (that is, have infinitely long formulas, or have variable-binding operators that give the same effect, or have rules of inference with infinitely many premises; see Infinitary logics)?

(h) Finally, we must study the metatheory of L, that is, what general results can be proved, in the metalanguage, to be true of L. This includes the consistency and independence of the logical axioms for L as well as whether L is ‘complete’ (that is, whether every true sentence of L is provable in L).