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The No-Assumptions View

The most radical alternative to the placeholder view of assumptions is the claim that there are no assumptions at all. This view is much older than the placeholder view and was strongly advocated by Frege (see for example Frege [22]). Frege argues that the aim of deduction is to establish truth, and, in order to achieve that goal, deductions proceed from true assertions to true assertions. They start with assertions that are evident or for other reasons true. This view of deduction can be traced back to Bolzano's Wissenschaftslehre [1] and its notion of 'Abfolge'. This notion means the relationship between true propositions A and B, which obtains if B holds because A holds.

However, in view of the fact that hypothetical claims abound in everyday life, science, legal reasoning etc., it is not very productive simply to deny the idea of hypotheses. Frege was of course aware that 'if …, then …' statements play a central role wherever we apply logic. His logical notation (the Begriffsschrift) uses implication as one of the primitive connectives. The fact that he can still oppose the idea of reasoning from assumptions is that he denies a two-layer concept of hypotheses. As we have the connective of implication at hand, there is no need for a second kind of hypothetical entity that consists of a hypothetical proof from assumptions. Instead of maintaining that B can be proved from the hypothesis A, we should just be able to prove A implies B non-hypothetically, which has the same effect. There is no need to consider a second structural layer at which hypotheses reside.

For Frege this means, of course, that implication is not justified by some sort of introduction rule, which was a much later invention of Jas´kowski and Gentzen. The laws of implication are justified by truth-theoretic considerations and codified by certain axioms. That A implies itself, is, for example, one of these axioms (in Frege's Grundgesetze, [21]), from which a proof can start, as it is true.

The philosophical burden here lies in the justification of the primitive axioms. If, like Frege, we have a truth-theoretic semantics at hand, this is no fundamental problem, as Frege demonstrates in detail by a truth-valuational procedure. It becomes a problem when the meaning of implication is to be explained in terms of proofs. This is actually where the need for the second structural layer arises. It was the ingenious idea of Jas´kowski and Gentzen to devise a two-layer method that reduces the meaning of implication to something categorically different. Even though this interpretation was not, or was only partly, intended as a meaning theory for implication (in Gentzen [25] as an explication of actual reasoning in mathematics, in Jas´kowski [34] as an explication of suppositional reasoning) it has become crucial in that respect in later proof-theoretic semantics. The single-layer alternative of starting from axioms in reasoning presupposes an external semantics that is not framed in terms of proofs.

Such an external semantics need not be a classical truth-condition semantics, or an intuitionistic Kripke-style semantics. It could, for example, be a constructive BHK-style semantics, perhaps along the lines of Goodman-Kreisel or a variant of realisability (see, for example, Dean and Kurokawa [4]). We would then have a justification of a formal system by means of a soundness proof. As soon as the axioms or rules of our system are sound with respect to this external semantics, they are justified. In proof-theoretic semantics, understood in the strict sense of the term, such an external semantics is not available. This means that a single-layer concept of implication based only on axioms and rules, but without assumptions, is not a viable option.1

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