Philosophers who accept the inference rules of intuitionistic logic as unproblematically valid, while taking issue with specifically classical principles of deduction such as the law of excluded middle or double-negation elimination, will typically justify their stance by citing some variant of the BHK (Brouwer-Heyting-Kolmogorov) interpretation of the logical connectives. Whereas standard semantics for classical logic takes as its departure point an epistemically untethered conception of *truth*, the BHK interpretation assigns meaning to a sentence by specifying what it takes for something to qualify as a *proof* of it; and in so doing (so the intuitionist argues) it manages to tie linguistic meaning in a straightforward way to norms of correct language use, thus gaining a philosophical advantage over the classicist.

However, in order to be able claim an edge over the truth-conditional classicist, a proponent of the BHK interpretation must take some care in the formulation of its details. For conditionals, for instance, it will not do to simply say that a proof of "If A then B" consists in an operation which, when applied to a proof of A, will unfailingly yield a proof of B; if such an operation is to serve any epistemic purpose, we must also require that the operation be *recognizable* as having this property of taking proofs of A to proofs of B. This requirement of recognizability leads to a difficulty: if the notion of recognizability is explicated in terms of the existence of a proof, an infinite regress looms; if, on the other hand, it is assumed as primitive, the semantic theory is left too vague to be of much meta-mathematical use.

In my talk I shall be proposing that, for some philosophical purposes at least, the vagueness just mentioned is perhaps not a fatal flaw. For the purpose of displaying the soundness of intuitionistic logic, and indicating where a similar justification of classical logic would founder, it is sufficient that the property of recognizability be assumed to satisfy certain formal requirements; the details of its nature can be left unspecified. But once this is accepted, it becomes possible to discard the proof-transforming operations from the semantic theory altogether, leaving us with just a set of simple "demonstration-conditional" clauses reminiscent of the truth-conditional clauses of standard classical semantics (and also, incidentally, allowing us to dispense with the distinction between canonical proofs and indirect demonstrations). No less partial towards intuitionistic logic than the original BKH interpretation, this simplified semantic theory thus offers an alternative answer to the question in the title.

If time allows, I will also discuss how the same approach can be applied to universal quantification; it seems to me that the approach opens up the possibility of a representational, as opposed to substitutional, account of quantification. But I am not sure about this -- maybe the audience will be able to help me sort out my thoughts on the matter.

# CLLAM Seminar: Tor Sandqvist

## EVENEMANG

**Datum:**08 november 2019 10:00 - 08 november 2019 12:00

**Plats:**D700

What has intuitionistic logic that classical logic hasn't?

**Senast uppdaterad:**
4 november 2019

**Webbredaktör: **
Peter Pagin

**Sidansvarig:** Filosofiska institutionen

## Kontakt

**Prefekt**

Björn Eriksson

**Studierektor på forskarnivå **

*Praktisk filosofi*

Gunnar Björnsson

*Teoretisk filosofi*

Peter Pagin