Classical vs relevance and intuitionistic logics

If you compare classical logic (CL) and intuitionistic logic (IL) at the syntactic level, then the set of theorems of IL is a proper subset of those of CL. So in that sense, classical logic is 'stronger' when ordered by the derivability relation ⊢ or by the logical consequence relation ⊨.

Relevance logics are trickier to compare because there is a whole family of them, and some of them introduce additional connectives, such as fission and fusion. It remains the case that some theorems of CL are not theorems of any relevance logic.

However, it is misleading to compare logics just by talking about the set of their theorems, since their connectives have different meanings. The differences are particularly sharp when comparing the conditional connective →. In classical logic, A → B is a truth function that is logically equivalent to ¬A ∨ B. In intuitionistic logic, A → B can be understood under the BHK interpretation to mean: I can manipulate a proof of A into a proof of B, or I can manipulate a proof of A into falsum. In relevance logics, A → B means B follows from A in some relevant and non-trivial way.

As a result, ¬¬A → A is not a theorem of IL and A → (B → A) is not a theorem of relevance logics. This does not really mean that CL is stronger than IL, since understsood intuitionistically, ¬¬A → A is not a tautology. Likewise, understood from the perspective of relevance logic, A → (B → A) is not a tautology. IL and relevance logics would be unsound if they proved those theorems.

There are translations between logics and ways of embedding one logic into another. For example, intuitionistic logic can be translated into classical S4 using the Gödel-McKinsey-Tarski translation and classical theorems can be mapped into intuitionistic theorems using the double-negation translation.

There are some ways of translating relevance logics into classical logic, but these are rather more complex, because relevance logics are substructural: they lack the rule of weakening that CL and IL have.

Regarding the pair Classical vs Intuitionistic, the existing translations: Glivenko, Godel-Gentzen, extended to arithmetic (Peano vs Heyting one) show that their connection is very interseting.

Under suitable translation, the two are equiconsistent and they are "equivalent" in terms of entailment.

Many resources available, with suitable details.

SEP's entry on Intuitionistic Logic, obviously, as well as here.

The translations above may mean that the issue about the "right" logic is not a "issue of logic".

It is a philosopical issue, and thus can be seen as a relevant recent example of philosophical result.

See also Intermediate Logics.

Yes, in the simplest sense (i.e. not applying any more complicated translation) classical logic is stronger (= has a larger consequence relation) than relevant and intuitionistic logic. Interestingly, classical logic is in fact maximal in a precise sense; see here.

It's also worth noting that when we do (as in Mauro's above-linked answer) take into account interesting translations between logics, we run into lots of subtleties. There are several different ways of making sense of comparisons/embeddings between logics and they disagree in general; see e.g. the first part of the discussion here.

1. The system of propositions of propositional logic follows a Boolean algebra, with the rule of tertium non datur.

   While propositions of the intuitionistic logic follows a Heyting algebra where the tertium non datur is no longer true.

2. A model of the proposition from propositional logic are the subsets of a given set, with negation the set-complement.

   While a model for the propositions of intuitionistic logic is the system of open sets of a topological space, with negation the open kernel of the set-complement. In the latter case, a “proposition OR its negation” is not necessarily true.

3. For an introduction see Sheaves in Geometry and Logic, Chapter I, Sect. 7.