Ch 4: Entailment

affirming the consequent: a deductively invalid logical form where P is inferred from the premises If P then Q and Q.

antecedent: in a conditional, the clause that expresses the condition—that is, the clause immediately following if.

conditional: a statement composed of two sentential clauses joined by if...then.... (Sometimes then is omitted, as in: If Bob starts dancing, I will leave.)

consequent: in a conditional, the clause that expresses what is said to follow if the antecedent is true. The consequent usually comes right after then.

counterexample: an example that shows that a universal claim is false: for example, if Betty is happy, then Betty is a counterexample to the claim Everyone is unhappy—assuming the speaker is really talking about everyone! As we know, for a logical form to be deductively valid, every instance must be deductively valid. So, a logical form can sometimes be shown to be deductively invalid by providing a counterexample to this universal claim—namely, an instance of it that is not deductively valid. Such a counterexample will be particularly vivid if the premises are actually true and the conclusion false.

deductively valid argument: what makes an argument deductively valid is that its premises entail its conclusion: if the premises were true, the conclusion would have to be true.

deductively valid logical form: what makes a logical form deductively valid is that every argument with that form is deductively valid.

denying the antecedent: a deductively invalid logical form where not Q is inferred from the premises If P then Q and not P.

disjunction: the disjunction of two sentences P and Q is a sentence that is true as long as either P is true or Q is true, and false only when both P and Q are false. It can be formed by combining P and Q with either...or....

disjunctive syllogism: deductively valid logical form where Q is inferred from the premises Either P or Q and not P.

flipping the argument: assuming that the conclusion is false, and asking whether all of the premises could still be true. If so, the premises do not entail the conclusion.

hypothetical syllogism: deductively valid logical form where If P then R is inferred from If P then Q and If Q then R.

logical form: A structure that can be shared by different arguments; it can be illustrated by replacing certain words or sentences with variables until the argument are the same.

modus ponens: deductively valid logical form where Q is inferred from the premises If P then Q and P.

modus tollens: deductively valid logical form where not P is inferred from the premises If P then Q and not Q.

negation: the negation of a sentence is true when the original sentence is false and false when the original sentence is true. Often formed by inserting not into the sentence that is being negated.