Antecedent (logic) in the context of "Vacuous truth"

A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if P implies Q, then P is called the antecedent and Q is called the consequent. In some contexts, the consequent is called the apodosis.

In propositional logic, affirming the consequent (also known as converse error, fallacy of the converse, or confusion of necessity and sufficiency) is a formal fallacy (or an invalid form of argument) that is committed when, in the context of an indicative conditional statement, it is stated that because the consequent is true, therefore the antecedent is true. It takes on the following form:

Denying the antecedent (also known as denial of the antecedent, inverse error, or fallacy of the inverse) is a formal fallacy of inferring the inverse from an original statement. Phrased another way, denying the antecedent occurs in the context of an indicative conditional statement and assumes that the negation of the antecedent implies the negation of the consequent. It is a type of mixed hypothetical syllogism that takes on the following form:

The paradoxes of material implication are a group of classically true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated with English words such as "implies" or "if ... then ...". They are sometimes phrased as arguments, since they are easily turned into arguments with modus ponens: if it is true that "if ${\displaystyle P}$ then ${\displaystyle Q}$" (${\displaystyle P\rightarrow Q}$), then from that together with ${\displaystyle P}$, one may argue for ${\displaystyle Q}$. Among them are the following:

A material conditional formula ${\displaystyle P\rightarrow Q}$ is true unless ${\displaystyle P}$ is true and ${\displaystyle Q}$ is false; it is synonymous with "either P is false, or Q is true, or both". This gives rise to vacuous truths such as, "if 2+2=5, then this Wikipedia article is accurate", which is true regardless of the contents of this article, because the antecedent is false. Given that such problematic consequences follow from an extremely popular and widely accepted model of reasoning, namely the material implication in classical logic, they are called paradoxes. They demonstrate a mismatch between classical logic and robust intuitions about meaning and reasoning.

In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive. The contrapositive of a statement has its antecedent and consequent negated and swapped.

Conditional statement ${\displaystyle P\rightarrow Q}$. In formulas: the contrapositive of ${\displaystyle P\rightarrow Q}$ is ${\displaystyle \neg Q\rightarrow \neg P}$.

In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or bidirectional implication or biimplication or bientailment, is the logical connective used to conjoin two statements ${\displaystyle P}$ and ${\displaystyle Q}$ to form the statement "${\displaystyle P}$ if and only if ${\displaystyle Q}$" (often abbreviated as "${\displaystyle P}$ iff ${\displaystyle Q}$"), where ${\displaystyle P}$ is known as the antecedent, and ${\displaystyle Q}$ the consequent.

Nowadays, notations to represent equivalence include ${\displaystyle \leftrightarrow ,\Leftrightarrow ,\equiv }$.

Destructive dilemma is the name of a valid rule of inference of propositional logic. It is the inference that, if P implies Q and R implies S and either Q is false or S is false, then either P or R must be false. In sum, if two conditionals are true, but one of their consequents is false, then one of their antecedents has to be false. Destructive dilemma is the disjunctive version of modus tollens. The disjunctive version of modus ponens is the constructive dilemma. The destructive dilemma rule can be stated:

where the rule is that wherever instances of "${\displaystyle P\to Q}$", "${\displaystyle R\to S}$", and "${\displaystyle \neg Q\lor \neg S}$" appear on lines of a proof, "${\displaystyle \neg P\lor \neg R}$" can be placed on a subsequent line.

where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a proof with", P and Q are any given logical statements, and ${\displaystyle \neg P\lor Q}$ can be read as "(not P) or Q". To illustrate this, consider the following statements: