By Schmidt H.A., Schutte K., Thiele H.-J. (eds.)

ISBN-10: 0444534148

ISBN-13: 9780444534149

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**Example text**

We will discuss these systems in greater detail in the context of evaluating Goodman and Kreisel’s response to the paradox in Sect. 5. 5 Goodman’s Kreisel’s Theory of Constructions, the Kreisel-Goodman Paradox … 35 and predicates (which are formalized as boolean-valued terms). e. what the theory seeks to axiomatize is a notion of “self-applicable” proof. The distinctive feature of all versions of the Theory of Constructions is the inclusion of a proof operator π whose intended role can be most readily described as that of axiomatically mimicking certain properties of a traditional proof predicate ProofT (x, y) for an arithmetical theory T (such as Peano or Heyting arithmetic).

But by this very fact they do violate the principle, which I stated before, that the word “any” can be applied only to those totalities for which we have a finite procedure for generating all their elements. For the totality of all possible proofs certainly does not possess this character, and nevertheless the word “any” is applied to this totality in Heyting’s axioms, as you can see from the example which I mentioned before, which reads: “Given any proof for a proposition p, you can construct a reductio ad absurdum for the proposition ¬ p”.

An argument structure that is valid with respect to a justification that assigns such operations to occurrences of inferences would in itself have an epistemic force. Perhaps one could say that the function of the justifications would then be to verify that they have such a force, whereas valid arguments as they have been defined here often get their entire epistemic force from the justifications. A notion of valid argument based on justifications of this kind would be a quite different concept from the variants of valid argument that have been dealt with in this paper.

### Contributions to Mathematical Logic (Logic Colloquium '66) by Schmidt H.A., Schutte K., Thiele H.-J. (eds.)

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