Deductions Involving Complex Propositions

Some deductions involve complex propositions. A complex proposition is a proposition that is made from simpler propositions combined with connectives like "and," "or," and "if... then."

When discussing the form of such propositions, we use lowercase letters to represent the simple propositions.

In this article we will cover arguments involving conditionals (complex propositions joined by "if... then..." and disjunctions (complex propositions joined by "or").

Deductions Involving Conditionals

In a conditional, the proposition following the "if" is called the "antecedent" and the one following the "then" is called the consequent.

Modus Ponens

Modus Ponens (or "affirming the antecedent") is a two-premise deduction of which one premise is a conditional, the other is the antecedent of the conditional, and the conclusion is the consequent of the same conditional.

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Modus ponens should not be confused with the fallacy of affirming the consequent.

Modus Tollens

Modus Tollens (or "denying the consequent") is a two-premise deduction of which one premise is a conditional, the other is the denial of the consequent, and the conclusion is the denial of the antecedent.

Modus Tollens should not be confused with the fallacy of denying the antecedent.

Hypothetical Syllogism

A hypothetical syllogism is a two-premise deductive argument in which each premise is a conditional, the consequent of one these conditionals is the antecedent of the other, and the conclusion is a third conditional with the remaining antecedent as its antecedent and the remaining consequent as its consequent.

Disjunctive Syllogism

A disjunction is a complex proposition in which two other propositions (called the "disjuncts") are connected by an "or". In logic "or" is usually interpreted inclusively, meaning that at least one of the disjuncts must be true, but that both might be. The main type of argument involving disjunctions is called disjunctive syllogism. It's a two-premise argument in which one premise is a disjunction, the second premise denies one of the disjuncts, and the conclusion affirms the remaining disjunct.