The Method of Deduction Chapter 9 The Method of Deduction

Method of Deduction Though truth tables are adequate ways of testing very simple types of arguments, they quickly grow unwieldy as the number of component statements increases. However, by translating arguments into symbolism and using the method of deduction, even these arguments can easily be tested for validity.

Rules of Inference There are just nine rules of inference that logicians have deduced which can be used in constructing formal proofs: 1) Modus ponens 2) Modus tollens 3) Hypothetical syllogism 4) Disjunctive syllogism 5) Constructive dilemma 6) Absorption 7) Simplification 8) Conjunction 9) Addition Each of these rules of inference corresponds to valid elementary argument forms. By identifying these rules in steps of a proof, we can label the arguments as valid.

Rule of replacement Some obviously valid arguments cannot be proved valid using just the nine rules of inference. For many of these, the rule of replacement—which permits us to infer from any statement the result of replacing any component of the statement with any other logically equivalent component—can be helpful.

Logical Equivalences There are ten such logical equivalences: 1)De Morgan’s theorems 2) Commutation 3)Association 4) Distribution 5) Double negation 6)Transposition 7)Material implication 8) Material equivalence 9) Exportation 10) Tautology Whenever the formula for one is found in an argument, it can be replaced with its logically equivalent form to facilitate analysis.

Proving Invalidity Proving invalidity is more difficult and complex than proving validity, for simply failing to discover a formal proof of validity does not constitute a proof of invalidity. However, there is an effective method for proving invalidity, related to the truth table method— but shorter.

Proving Invalidity In essence, this method requires the construction of just one line of the truth table; and if in that line all the premises are true and the conclusion is false, then the argument is invalid. One can also construct an indirect proof of validity by stating an additional assumed premise that is the negation of the conclusion. If one can derive from this assumed premise and the original set of premises an explicit contradiction then the original argument must be invalid.

Paradox One strange and potentially confusing situation arises when inconsistent premises occur. If a set of premises is inconsistent, they will validly yield any conclusion at all—no matter how irrelevant. Of course, arguments that are valid because of inconsistent premises are unsound. This situation is related to the “paradox” of material implication.