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Symbolic Argument Forms

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Modus Tollens

If P, then Q. Q is not true. Therefore, P is not true. Symbolic form:

(P \rightarrow Q), \neg Q \vdash \neg P

Modus Ponens

If P, then Q. P is true. Therefore, Q must be true. Symbolic form:

(P \rightarrow Q), P \vdash Q

Hypothetical Syllogism

If P, then Q. If Q, then R. Therefore, if P, then R. Symbolic form:

((P \rightarrow Q) \land (Q \rightarrow R)) \vdash (P \rightarrow R)

Destructive Dilemma

If P, then Q. If R, then S. Neither Q nor S is true. Therefore, neither P nor R is true. Symbolic form:

((P \rightarrow Q) \land (R \rightarrow S)) \land (\neg Q \land \neg S) \vdash (\neg P \land \neg R)

Affirming the Consequent

If P, then Q. Q is true. Therefore, P might be true. Symbolic form:

(P \rightarrow Q), Q

*Note: This is a logical fallacy, and does not constitute a valid argument form.

Disjunctive Syllogism

Either P or Q is true. P is not true. Therefore, Q must be true. Symbolic form:

(P \lor Q), \neg P \vdash Q

Constructive Dilemma

If P, then Q. If R, then S. Either P or R is true. Therefore, either Q or S is true. Symbolic form:

((P \rightarrow Q) \land (R \rightarrow S)) \land (P \lor R) \vdash (Q \lor S)

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