Logical system in the context of Consistency

In logic, a rule of replacement is a transformation rule that may be applied to only a particular segment of an expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for logical expressions in the system. Whereas a rule of inference is always applied to a whole logical expression, a rule of replacement may be applied to only a particular segment. Within the context of a logical proof, logically equivalent expressions may replace each other. Rules of replacement are used in propositional logic to manipulate propositions.

Common rules of replacement include de Morgan's laws, commutation, association, distribution, double negation, transposition, material implication, logical equivalence, exportation, and tautology.

Metalogic is the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.

The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and the study of deductive systems is the branch that is known as proof theory.

A logical possibility is a logical proposition that cannot be disproved, using the axioms and rules of a given system of logic. The logical possibility of a proposition will depend upon the system of logic being considered, rather than on the violation of any single rule. Some systems of logic restrict inferences from inconsistent propositions or even allow for true contradictions. Other logical systems have more than two truth-values instead of a binary of such values. Some assume the system in question is classical propositional logic. Similarly, the criterion for logical possibility is often based on whether or not a proposition is contradictory and as such, is often thought of as the broadest type of possibility.

In modal logic, a logical proposition is possible if it is true in some possible world. The universe of "possible worlds" depends upon the axioms and rules of the logical system in which one is working, but given some logical system, any logically consistent collection of statements is a possible world. The modal diamond operator ${\displaystyle \lozenge }$ is used to express possibility: ${\displaystyle \lozenge P}$ denotes "proposition ${\displaystyle P}$ is possible".