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I am currently going through Philip Wadler's "Proposition as Types" and a passage of the introduction has struck me:

Propositions as Types is a notion with breadth. It applies to a range of logics including propositional, predicate, second-order, intuitionistic, classical, modal, and linear.

I am familiar with intuitionistic and classical logic but I could not find any direct, informal explanations of the differences and nuances between each of those on the Internet.

Taroccoesbrocco
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Erwan Aaron
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Proof theoretically intuitionist logic is a sub-theory of classical logic. The Wikipedia article indicates this by indicating that if we have a certain set of axioms for intuitionist logic (those axioms can get found under the section entitled "Hilbert style calculus", then classical logic may get obtained by joining to the system the law of the excluded middle... in Polish notation ApNp, Peirce's law CCCpqpp, or the law of double negation elimination CNNpp, or with the definition Np := Cp0, we might write CCCp00p.

Thus, all formal proofs done in a an intuitionist logic framework work as proofs for any classical logic framework, but not all proofs done in a classical logic framework will work in any intuitionist logic framework.

For example, one might deduce Cpp from a classical logic framework as follows:

thesis 1 CCpqCCqrCpr
thesis 2 CpCNpq
thesis 3 CCNppp
DD123  4 Cpp

This is not a possible proof of Cpp in any intuitionist logic framework, since CCNppp is not a thesis in any intuitionist logic (the term "thesis" refers to a formula which is either an axiom or a theorem).

Semantically speaking intuitionist logic qualifies as much richer than classical logic in that the truth set for intuitionist logic is infinite-valued, while that of classical logic is two-valued. Semantically speaking, intuitionist logic behaves the same way as classical logic when truth-values get confined to "True" and "False". In such a case, each formula whether interpreted from the perspective of classical logic or intuitionist logic yields the same result. But, intuitionist logic can get said to reject the law of bivalence in that no two-valued models of intuitionist logic qualify as adequate for a semantics of it.

Doug Spoonwood
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  • Could you please elaborate on the polish notation thing? How is that related to logic, to LEM or Pierce's Law? Thank you! – Erwan Aaron Jun 05 '15 at 00:00
  • There isn't much relation to LEM or Peirce's law (though identifying the principal connective of those formulas is easier in Polish notation than in infix notation). Polish notation involves putting functions before their arguments uniformly, with "C" standing for implication, and "N" for negation. "C" takes two arguments, and "N" takes one. You might see here http://plato.stanford.edu/entries/lukasiewicz/polish-notation.html or the wikipedia for more details online. It might interest you to know that some of the original papers written in Polish notation got burned. – Doug Spoonwood Jun 05 '15 at 01:25
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Intuitionistic logic could be succintly described as classical logic that violates the law of excluded middle (LEM), i.e. $A\lor \lnot A$ is not always true in intuitionistic logic. This is mainly due to the intuitionistic negation being an intensional operator.

This can be seen if we consider the canonical (Brower-Heyting-Kolmogorov) interpretation: a proof of $\lnot A$ means that there is a proof that there is no proof for $A$. Now take some as yet unproven conjecture---say the twin prime conjecture $TP$.

Now, since there's neither a proof of $TP$ nor is there a proof that there is no proof of $TP$, it follows from the definition of '$\lor$' that there's no proof of $TP \lor \lnot TP$.

Mariusz Popieluch
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  • "Intuitionistic logic could be succintly described as classical logic that violates the law of excluded middle (LEM), i.e. A∨¬A is not always true in intuitionistic logic." From the perspective of classical logic this makes no sense at all. Why? From the perspective of classical logic, in Polish notation, "A" (disjunction) is an abbreviation for "CN" (implication followed by negation). And thus ApNp is just an abbreviated form of CNpNp. CNpNp is not denied by intuitionistic logic. – Doug Spoonwood Jun 01 '15 at 16:42
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    LEM isn't valid in intuitionistic logic --- the statement you're taking issue with says no more than that. Do you then also take issue with the following relevant section of the SEP article? http://plato.stanford.edu/entries/logic-intuitionistic/#RejTerNonDat – Mariusz Popieluch Jun 01 '15 at 18:18
  • Yes, I'd take issue with that also. The definition-free forms of classical logic under Nicod's axiom, for example, only have propositions containing the Sheffer stroke (or "NAND"). Additionally, even with definitions deleting ApNp and it's consequences is simply insufficient. Since the definitions tell us that Np := Dpp and Apq := DNpNq, then ApNp = A p Dpp = D Np NDpp = D Dpp D Dpp Dpp. The problem lies in that deleting DDppDDppDpp is not sufficient. Since we have the rule of substitution, we can't just delete DDppDDppDpp, because it is a special case of the more general DDpqDDpqDqp. – Doug Spoonwood Jun 01 '15 at 19:18
  • Deleting CNNpp has the same problem. Cpq := DpNq. Thus, CNNpp becomes C N Dpp p, then C D Dpp Dpp p, then D D Dpp Dpp Np and finally D D Dpp Dpp Dpp. Now note that Dxy = Dyx (D is the Sheffer Stroke or "NAND"). Thus, since D Dpq DDpqDqp is a tautology, so is D DDpqDqp Dpq. A special case of this is the formula we got above DDDppDppDpp. So, again, just removing CNNpp (and it's consequence in the presence of everything else) won't suffice. We have to remove a more general formula. – Doug Spoonwood Jun 01 '15 at 19:34
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    The general phenomenon that @DougSpoonwood gave several specific instances of is that, in the many different axiomatizations of classical logic, there are different parts that go beyond intuitionistic logic. In some systems, that part is $p\lor\neg p$; in some it is $(\neg\neg p)\to p$; in some it is not an axiom at all but a definition of $p\lor q$ as $(\neg p)\to q$; etc. Furthermore, it is not guaranteed that an arbitrary complete axiomatization of classical logic can be split into a complete axiomatization of intuitionistic logic plus intuitionistically invalid stuff. – Andreas Blass Jun 01 '15 at 22:19
  • @AndreasBlass Correct me if I am wrong but the debate here is about intuitionistic logic being defined by the fact that LEM is not an axiom or double negation? If I am mistaken could you explain me what is the issue with this answer? Thank you! – Erwan Aaron Jun 05 '15 at 00:03
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    @ErwanAaron The difficulty is that "LEM is not an axiom or double negation" doesn't uniquely describe intuitionistic logic. One can axiomatize classical logic in such a way that neither LEM nor double negation is an axiom. Of course, such an axiomatization will contain some other non-intuitionistic principle, which then ought to be removed, but your definition doesn't cover that. Furthermore, I expect that one could axiomatize classical logic in such a way that none of the axioms are intuitionistically acceptable, so they'd all need to be removed. – Andreas Blass Jun 05 '15 at 00:44
  • @AndreasBlass Yes, classical logic can get axiomizated in such a way that none of the axioms are intuitionistically acceptable. Just consider the fact that classical logic has single axioms. There do exist multiple axiom sets such that none of the axioms are intuitionstically acceptable also. An example is {CCCpqrCCrpCsp, CCNpNqCqp}. – Doug Spoonwood Jun 05 '15 at 01:16
  • @AndreasBlass Then what would be an acceptable (and accurate) description of intuitionistic logic? For example would "subset of classical logic in which 'LEM or double negations' are not axioms and where truth is replaced by 'constructive provability'" accurately describe intuitionistic logic? – Erwan Aaron Jun 05 '15 at 02:43
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    @ErwanAaron I hoped you don't mind me responding. Consider the axiom set that the Wikipedia gives for intuitionistic logic under the section for Hilbert-style calculus. Call that system IL. Define a thesis as a theorem or an axiom. An intutionistic logic is any formal system that has the same theses as IL. – Doug Spoonwood Jun 05 '15 at 04:11
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    @ErwanAaron Doug Spoonwood has given a good answer to your first question. I'd add that there are other, equivalent axiomatizations in the literature, and they could be used in place of IL in Doug's answer. For your second question, about "subset of classical ...", your proposed description is true but also applies to some other, weaker systems than intuitionistic logic. Also, it presupposes that "constructive provability" has already been defined, whereas, in fact, defining it seems to be one of the main issues here. – Andreas Blass Jun 05 '15 at 13:44
  • Thanks to both you of being so helpful – Erwan Aaron Jun 07 '15 at 03:43
  • It turns out that there actually exist 16 more general tautologies with the same bracket type as that of the excluded middle "DDppDDppDpp". Also, some of those end up as more general than other tautologies in that list also. I wrote a program which can find all of them which you can see (and potentially use for some other things with some modification) here: http://www.codeskulptor.org/#user40_UkE5jaYJtr_1.py – Doug Spoonwood Jun 16 '15 at 03:42