I'm familiar with the translation of intuitionistic propositional logic into modal logic S4, summarized here and I've looked at one of the original sources for this:

  • J. C. C. McKinsey and Alfred Tarski, "Some Theorems About the Sentential Calculi of Lewis and Heyting", Journal of Symbolic Logic 13 (1948), pp. 1-15

I hadn't before seen any discussion of intuitionistic logic being translatable into weaker modal logics, and indeed that Wikipedia link says that:

S4 itself is the smallest modal companion of the intuitionistic logic

using the translation scheme described there and in the McKinsey-Tarski paper:

$T(p)=\Box p\text{ for any propositional variable }p$,
$T(\neg A)=\Box\neg T(A)$,
$T(A\land B)=T(A)\land T(B)$,
$T(A\lor B)=T(A)\lor T(B)$,
$T(A\supset B)=\Box(T(A)\supset T(B))$.

Recently though I came across some references to a Hacking paper:

  • Ian Hacking, "What is Strict Implication?", Journal of Symbolic Logic 28 (1963), 51-71

where he writes (p. 52):

I take the opportunity to remark that under [McKinsey and Tarski's] translations, [Heyting's version of the intuitionistic propositional calculus] is contained in S3.

Hacking describes a proof of this, but doesn't display the details, in the final section of his paper. That is, if I'm understanding things rightly. I definitely am not tracking the details of these papers.

As is well known, modal logic S3 is weaker than S4. (S3 isn't even a normal modal logic.)

So what is going on? Am I misunderstanding what Hacking is claiming? Is Hacking right? If so, is that statement quoted from the Wikipedia page (and sourced from other texts) wrong?

I expect that I'm the one who's made a mistake in seeming to perceive a conflict here, but if so I'd appreciate learning what my mistake is.


It seems that there is no mistake ...

See :

  • Alexander Chagrov & Michael Zakharyaschev, Modal Logic (1997), page 108 :

That Int can be embedded into S4 [...] was noticed by Orlov (1928 : russian) and Gödel (1933). [...] It is of interest that the first Lewis system S3 turned out to be a "modal companion" of Int too, as was shown by Hacking (1963) and strengthened by Chagrov (1981 : Superintuitionistic fragments of non-normal modal logics - russian).


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