Modal logic: One of the types of formal logic developed in the 1960s, modal logic extends classical propositional and predicate logic to include operators that express modality. Modals are words that express modalities (an attribute or circumstance that denotes mode or manner) to qualify a statement and include possibility. For instance using the phrase, Mary is “typically” happy, the word typically is the modality.
Temporal modalities are modalities of time –It is “always” the situation. Deontic modalities are obligatory or “permissible” while epistemic modalities, are of knowledge –it is “known” that. While doxastic modalities are those of belief –it is “believed” that.
Temporal logic: This is a system of rules and symbolism for representing and/or reasoning about, propositions qualified in terms of time. One temporal logic example is –I will “eventually” come home. Temporal logic can also be referred to as “tense logic,” a particular modal logic-based system of temporal logic introduced by Arthur Prior in the late 1950s.
Deontic logic: This is concerned with an obligation, permission, or related concepts. A deontic logic is a formal system attempting to capture the essential logical features of these concepts. “Deontic” comes from from the ancient Greek word déon – δέον which means, “roughly.”
Relevance logic: A kind of non-classical logic, relevance logic requires the antecedent and consequent of implications to be relevantly related. Relevance logic tries to capture aspects of implication that are
ignored by the “material implication” operator in classical truth-functional logic.
“logic … [is] … the name of a discipline which analyzes the meaning of the concepts common to all the sciences, and establishes the general laws governing the concepts.” – Alfred Tarski (1901-1983); Introduction to logic and to the methodology of deductive sciences, Dover, page xi.