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Validity of Arguments

We can now define what it is for an argument to be valid by adopting three principles analogous to the ones stated for valid deductions:

I. A closed canonical argument (A, J ) is valid, if for each immediate sub-argument structure A∗ of A, it holds that (A, J ) is valid.

II. A closed non-canonical argument (A, J ) is valid, if A reduces relative to J to an argument structure Asuch that (A, J ) is valid.

III. An open argument (A, J ) depending on the assumptions A1, A2,..., An is valid, if all its substitution instances (A, J) are valid, where Ais obtained by first substituting any closed terms for free variables in sentences of A, resulting in an argument structure Adepending on the assumptions A, A,..., A◦ , and then

1 2 n

for any valid closed argument structures (A i , Ji ) for A◦, in, substituting A ifor A◦ in A◦, and where J ∗ = J
JiJ .

Because of the assumed condition on the relative complexity of the ingredients of an introduction inference, the principles I-III can again be taken as clauses of a generalized inductive definition of the notion of valid argument relative to a base B, which is to consist of a set of closed argument structures containing only atomic sentences. If A is an argument structure of B, the argument (A,), where ∅ is the empty justification, is counted as canonical and outright as valid relative to B. A base is seen as determining the meanings of the atomic sentences. An argument that is valid relative to any base can be said to be logically valid.

If A is an argument structure representing mathematical induction as exhibited in Sect. 5.1, J is the justification associated with A as described in Sect. 5.2, and B is a

base for arithmetic, say corresponding to Peano's first four axioms and the recursion schemata for addition and multiplication, then the argument (A, J ) is valid relative to B (as was in effect first noted in a different conceptual framework by Martin-Löf (1971) [14]. This is an example of a valid argument that is not logically valid but whose validity depends on the chosen base. However, I shall often leave implicit the relativization of validity to a base.

Instead of saying that the argument (A, J ) is valid it is sometimes convenient to say that the argument structure A is valid with respect to the justification J . But it is argument structures paired with justifications that correspond to proofs and that will be compared to BHK-proofs.

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