Traditional first-order logic

https://web.archive.org/web/20060225052821/http://www.earlham.edu/~peters/courses/logsys/nonstbib.htm

In standard first-order logics:

• Wffs are finite in length (although there may be infinitely many of them)

• Rules of inference take only finitely many premises

• There are only two truth values

• Truth-values of given proposition symbols do not change within a given interpretation, only between or across interpretations

• All propositional operators and connectives are truth-functional

• p ∨ ¬p is provable even if we do not have |- p or |- ¬p separately; that is, the principle of excluded middle holds

• Contradictions are always false (as opposed to being both true and false)

• Contradictions imply everything (ex falsum quidlibet). If the axioms contain an inconsistency, then all wffs are theorems

• |- is monotonic - if the set of premises is enlarged, the set of derivable conclusions doesn't shrink

• Inferences are from wffs to wffs, or from truth-values to truth-values (by means of rules), not from meanings to meanings. Rules of inference refer to syntactic features of wffs or to semantic truth-values, but not to other semantic features beyond truth-value such as meaning or intension

• There are individual variables (as opposed to none)

• There are quantifiers (as opposed to none)

• Predicates take only individuals as arguments (as opposed to other predicates)

• Quantifiers bind only individual variables (as opposed to binding predicates)

• Domains are non-empty by default, or at least one individual exists in every interpretation

• Universal quantifiers lack existential import (hence, Aristotle is non-standard)

• all structure inside predicates, in a natural language, is ignored (e.g. tense, adverbs, adjectives, etc.), except the order of args and quantifiers, which can help us distinguish the subject from the objects of the verb