Adequacy of a logic

[Tennant, Neil]

In the following, "the set X of sentences has a model" will mean that there is an interpretation of the non-logical vocabulary of the sentences in X that makes all of them true. We can also express this by saying "X is satisfiable".

Assume you have a logical system S of proof for which the following propositions hold:

0. S-proofs are finite. So the set of premises of any S-proof is finite. The relation "Y is an S-proof of the sentence P whose premises form the set X" is effectively decidable.

1. If X is a set of sentences that has no model, then there is an S-proof of absurdity from premises in X.

2. If there is an S-proof of absurdity whose premises form the set X, then X has no model.

3. If X has a model, and every model of X makes P true, then there is an S-proof of P from premises in X.

4. If X has a model, and there is an S-proof of P from premises in X, then every model of X makes P true.

The combination of (1) and (2) can be thought of as "soundness and completeness of S with respect to unsatisfiable sets of sentences".

The combination of (3) and (4) can be thought of as "soundness and completeness of S with respect to satisfiable sets of sentences".

Note that (1) entails that every logically false sentence Q can be refuted in S; that is, there is an S-proof of absurdity from (the singleton of) Q.

Note that (3) entails that every logically true sentence Q can be proved outright in S; that is, Q is the conclusion of an S-proof from the empty set of premises.

My question for the list is this:

What extra condition, if any, not already entailed by (0), (1), (2), (3), and (4) can possibly be required of any logic that is to be adequate for the formal regimentation of the deductive reasoning that is involved in mathematics and science?

Here, being adequate is to be understood as being able to furnish whatever proofs and disproofs might be required as fully formal regimentations of the deductive reasoning that is involved in these areas of intellectual endeavor. We include, of course, theorem-proving in mathematics from decidable sets of mathematical axioms; discovery of inconsistencies in proposed axiom-sets; making scientific predictions from scientific hypotheses combined with statements of initial and boundary conditions for experiments; and discovery of any conflicts that could arise between such predictions and the observations/measurements that might result from experimental testing.

Neil Tennant
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