Logical connective in the context of "Binary relation"

Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.

Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like ${\displaystyle \land }$ (and) or ${\displaystyle \to }$ (if...then). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts.

In logic, mathematics and linguistics, and (${\displaystyle \wedge }$) is the truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as ${\displaystyle \wedge }$ or ${\displaystyle \&}$ or ${\displaystyle K}$ (prefix) or ${\displaystyle \times }$ or ${\displaystyle \cdot }$ in which ${\displaystyle \wedge }$ is the most modern and widely used.

The and of a set of operands is true if and only if all of its operands are true, i.e., ${\displaystyle A\land B}$ is true if and only if ${\displaystyle A}$ is true and ${\displaystyle B}$ is true.

In predicate logic, an existential quantification is a type of quantifier which asserts the existence of an object with a given property. It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)"), read as "there exists", "there is at least one", or "for some". Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain. Some sources use the term existentialization to refer to existential quantification.

Quantification in general is covered in the article on quantification (logic). The existential quantifier is encoded as U+2203 ∃ THERE EXISTS in Unicode, and as \exists in LaTeX and related formula editors.

In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, P if and only if Q means that P is true whenever Q is true, and the only case in which P is true is if Q is also true, whereas in the case of P if Q, there could be other scenarios where P is true and Q is false.

In writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary and sufficient for P, for P it is necessary and sufficient that Q, P is equivalent (or materially equivalent) to Q (compare with material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. Some authors regard "iff" as unsuitable in formal writing; others consider it a "borderline case" and tolerate its use. In logical formulae, logical symbols, such as ${\displaystyle \leftrightarrow }$ and ${\displaystyle \Leftrightarrow }$, are used instead of these phrases; see § Notation below.

In logic, false (Its noun form is falsity) or untrue is the state of possessing negative truth value and is a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol ${\displaystyle \bot }$.

Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), ${\displaystyle \bot }$, is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition, and is often called absurdity.