Logical Systems

Overview

A logical system is a formal system endowed with semantics. The syntax of WFFs found in these systems are interpretable. In many cases, such as propositional and predicate logic, WFFs are defined recursively.

Paradigms

There exist a number of different paradigms that dictate what rules exist in the underlying proof system and how semantics should be interpreted.

Classical

Classical logic restricts truth values to one of truth $T$ or falsity $F$. It is the most prevalent form of logic; other paradigms are often categorized as simply non-classical.

Certain laws hold in every classical logic by virtue of propositions only being able to take one of two mutually exclusive options. Let $P$ and $Q$ be propositions.

• Law of Double Negation. $\mathrm{\neg }(\mathrm{\neg }P)⟺P$.
• Law of Excluded Middle. $(P\vee \mathrm{\neg }P)⟺T$.
• Principle of Explosion. $(P\wedge \mathrm{\neg }P)⊢Q$.

Intuitionistic

Intuitionistic logic, also known as constructive logic, refers to a logic that does not assume the law of excluded middle or double negation elimination.

Paraconsistent

A paraconsistent logic refers to a logic that is not explosive, i.e. it does not follow the principle of explosion.

Dialetheism

A dialetheia is a proposition $A$ such that both it and its negation ($\mathrm{\neg }A$) are true. Dialetheism is the view that there are dialetheias. In other words, dialetheism admits the existence of true contradictions.

Semantic Consequence

TODO