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I asked on MathSE What are the various respects under which a logic can deviate from classical logic, thus being “ non-classical”? and received one short answer. So, I'm interested in responses from Philosophy SE, too.

In what ways can a logic deviate from classical logic? I think one can find rather easily a list (though maybe incomplete) of non-classical logics. But it seems more difficult to find a presentation of the field that exhibits in a systematic fashion under which respects a logic can be non-classical.

The aspects I can think of are the following:

  1. Type of objects over which quantifiers range --> first-order/ second-order logic
  2. Validity of "ex falso" or not --> paraconsistent logics
  3. Use of modal operators, or not --> modal logics
  4. Finite or infinite number of premises --> compactness maybe?

There is an attempt at such a presentation in Theodore Sider's book Logic For Philosophy, but I'd be much interested in other references.

Note: I'm not asking for an absolutely complete list of points of departure from classical logic; I suppose it would be too long. Rather, what interests me is the systematicity of the presentation.

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  • Edited to avoid the "duplicate objection". – J D Nov 14 '20 at 17:12
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    Your list strangely omits the tremendously important intuitionistic logics, which are characterized by the omission of DNE (double negation elimination, ¬¬p→p) and related matters. – Mark Dominus Nov 15 '20 at 3:21
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Some of these might plausibly be called extensions of classical logic rather than non-classical in the strict sense, but I'll take your question as a broad one about logics that progress beyond elementary classical first order predicate logic. It is not exhaustive.

Valency: Bivalent. Multivalent. Not n-valent for any n. Fuzzy. Probabilistic.

Order: First-order. Second-order. Higher-order.

Domain: Free logics that permit quantification over non-existent objects, or allow empty domains.

Variables: Typeless vs. typed/many-sorted. Dependence logic.

Quantification: Generalised quantifiers. Branching quantifiers. Plural quantification. Objectual vs. substitutional.

Rules of implication:

Modal: De dicto and de re. Quantified modal logic. Intensional logics. Hyperintensional logics. Possible world semantics.

Temporal logic.

Conditional logics. Counterfactuals. Causal calculus.

Multiple conclusion logic.

The group under 'rules of implication' are strictly non-classical, as would be any non-bivalent logic. Modal and temporal logics are usually extensions of classical logic, though one could have a modal intuitionistic logic, for example. Others might be non-classical or not depending on how they are set up.

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  • Aren't provability logics and doxastic logics the same thing just with a slightly different semantic interpretation? – Sriotchilism O'Zaic Nov 14 '20 at 19:37
  • This answer is good, but I think it could be improved if you split into things that are extensions of classical logic, v.s. things that do have all the laws of classical logic. I know you mentioned the distinction, but structuring the answer based on that could be helpful. – PyRulez Nov 14 '20 at 21:42

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