Deductive Arguments: Truth-Functional Operators
Written by Levi Clancy
First published
January 14, 2010
Last modified
October 10, 2011
| Truth-Functional Operators: What Are They? | ||
| Operator | Construct | Overview |
|---|---|---|
| Conditional | If P, then Q. | P is the antecedent and Q is the consequent. |
| Disjunction | Either P or Q. | |
| Negation | It is not P. | |
| Conjunction | Both P and Q. | |
| Necessary Conditions & Sufficient Conditions | ||
| Operator | Construct | |
|---|---|---|
| Necessary Condition | If not P, then not Q. | |
| Sufficient Condition | If P, then Q. | |
| Neccesary & Sufficient | Q if and only if P. | |
To get an A in math, you must ace the midterm and the final. Acing your midterm and acing your final are each necessary for an A in the class, yet neither is sufficient on its own.
To get an A in literature, the only requirement is to ace the one and only paper. Acing the paper is necessary and sufficient for an A in the class. You ace the class if and only if you ace the paper.
Truth-Functional Operators: Truth-Fuctionally Valid Forms
| Argument | Construct | Overview |
|---|---|---|
| Modus Ponens | If P, then Q. It is P. ∴ It is Q. |
Modus Ponens (Affirming Mode) is perhaps the most prevalent argument form. It contains one premise that is a conditional, which is affirmed by the other premise. |
| Modus Tollens | If P, then Q. It is not Q. ∴ It is P. |
Modus Tollens (Denying Mode) contains one premise that is a conditional statement, which is negated by the other premise. |
| Hypothetical Syllogism | If P, then Q. If Q, then R. ∴ if P then R. |
Hypothetical Syllogism contains premises and conclusions that are all conditionals. The consequent of one premise is identical with the antecedent of the other premise. The antecedent of the former and consequent of the latter are identical to the antecedent and consequent of the conclusion. |
| Dilemma | Either P or Q. If P, then R. If Q, then S. ∴ Either R or S. |
Dilemma can be viewed as a beast with two horns — seize the P horn, you get R; seize te Q horn, you get S. Therefore, either Q or S is begotten. |
| Simplified Dilemma | EIther P or Q. If P, then R. If Q, then R. ∴ R. |
Simplified Dilemma lists all possible conditionals; since these conditional share the same consequent, the conclusion is the consequent. |
| Disjunctive Syllogism | Either P or Q. Not P. ∴ Q. |
Truth-Functional Operators: Fallacies
| Argument | Construct | Overview |
|---|---|---|
| Affirming the Consequent | If P, then Q. Q. ∴ P. |
Affirming the Consequent is an invalid argument, which is regardless of the premises’ truthfulness. The conclusion is the antecedent of the conditional, as opposed to the consequent or the negation of the antecedent. |
| Denying the Antecedent | If P, then Q. Not P. ∴ Not Q. |
Denying the Consequent is an invalid argument, which is regardless of the premises’ truthfulness. The conclusion is the negation of the consequent of the conditional premise. |
| Begging the Question | P ∴ P. |
Begging the Question is a valid fallacy whereby the conclusion is included among the premises. Begging the Question is oft misunderstood; it actually refers to stealing (begging) the conclusion and smuggling it into the premises. |
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