A deductive system that disagrees with classical logic with regard to the consequence relation (in the language of classical logic). For example, while the principle of excluded middle, i.e.,

• • $\phi \vee \neg \phi$

is a theorem of classical logic, there exist counterexamples to the principle in intuitionistic logic.

As the differences between classical logic and non-classical logics can be characterized in terms of the validity of particular inferences or theorems, the genesis of a non-classical logic is frequently grounded in an assertion that some aspect of the classical theory of deduction is philosophically suspect. For example, the framework that motivates the introduction of logics of nonsense posits the existence of well-formed but senseless formulae that represent grammatical sentences in natural language that are nonsense, e.g.,

• • The mome raths outgrabe.

Most logics of nonsense assume that the negation of a meaningless sentence is likewise meaningless, e.g.:

• • It is not the case that the mome raths outgrabe.

might be regarded as equally meaningless as the first sentence. In such cases, a formula $\phi$ and its negation $\neg \phi$ may be treated as senseless, the consequence being that $\phi \vee \neg \phi$ corresponds to no proposition and can not bear a truth value. In this context, proponents of logics of nonsense assert that classical logic presupposes a flawed theory of meaning, i.e., that sentences necessarily denote propositions, and reject the corresponding inferences of classical logic.

Disagreements with classical logic leading to non-classical systems may not be so broad-sweeping and may be tailored to very particular applications. The three-valued logics formulated by mathematician Stephen Cole Kleene (1909–1994)—which also rejects the principle of excluded middle—are introduced to describe a phenomenon in classical recursion theory. Kleene’s presentations of these systems involve classical valuations of formulae, i.e., recursive truth functions mapping formulae to the values of truth and falsity, that are not necessarily total, that is, there may be occasions in which $v\left(\phi \right)$ is undefined or a truth value gap. The properties of Kleene’s logics follow from a setting in which the calculation of the value of a formula $\phi$ may not converge, that is, a procedure may not halt when calculating an atom $Pa$ where the predicate $P$ corresponds to a property that is not decidable. There is no disagreement in this case with the principle of bivalence, i.e., to say that solving whether $a$ satisfies $Px$ is undecidable is independent of whether $Pa$ is true or false.

Frequently, the ambiguity of the interpretation of the material conditional $\supset$ blurs the line between non-classical logics and deductive systems that merely enrich classical logic by the introduction of new connectives. In, e.g., many formulations of philosopher Clarence Irving Lewis’ (1883–1964) logics of strict implication are axiomatized with a primitive conditional connective $\mathcal{⊰}$ representing entailment between formulae. In the case of, e.g., this type of formulation of the modal logic $\text{S}5$, there are counterexamples to some instances of the scheme:

• • $\phi \mathcal{⊰}\left(\psi \mathcal{⊰}\phi \right)$

This may be contrasted to the classical theorem:

• • $\phi \supset \left(\psi \supset \phi \right)$

Now, if one follows Lewis and interprets $\mathcal{⊰}$ and $\supset$ as representing competing accounts of entailment, it is plausible to assert that the account of entailment in $\text{S}5$ in fact rejects a classical theorem. In this reading, $\text{S}5$ can be considered non-classical.

On the other hand, $\text{S}5$ need not be formulated in terms of strict implication. Indeed, most modern accounts of $\text{S}5$ introduce the system axiomatically by including every classical axiom and adding new axioms and rules corresponding to the behaviour of the modal operator $\square$. So presented, $\text{S}5$ need not be construed as disagreeing with classical logic, but can be instead interpreted as an enrichment or extension of classical logic. This type of ambiguity entails that modal logics are sometimes—but not uniformly—thought of as non-classical.