# In how many and which ways can a logic be non-classical? Are there systems for organizing them?

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.

• Edited to avoid the "duplicate objection".
– J D
Nov 14 '20 at 17:12
• 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. Nov 15 '20 at 3:21

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.

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.

• Aren't provability logics and doxastic logics the same thing just with a slightly different semantic interpretation? 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. Nov 14 '20 at 21:42