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

Question

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.

Answer 1

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:

• no LEM (intuitionistic logic)
• no explosion (paraconsistent logics)
• no (or restricted) disjunction introduction (relevance logics)
• no (or restricted) LNC (dialethic logics)
• restricted distributivity (quantum logic)
• no (or restricted) structural rules (substructural logics)
• no idempotency of entailment (linear logic)
• restricted law of identity (non-reflexive logic)
• no monotonicity of entailment (non-monotonic logics, default logics)

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

• alethic (necessary truth)
• doxastic, provability
• epistemic
• deontic
• interrogatory
• imperative
• action
• justification
• etc.

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.