Derivations can prove but not test validity. They cannot tell you whether or not an argument is valid. They cannot prove that invalid arguments are invalid, and they can only prove that valid arguments are valid if you (the logician) are clever enough. They are not "effective methods" (dumb and infallible) in the mathematical sense; on the contrary, they require ingenuity and insight. This has the disadvantage that we must now be smart, not just patient and legible, in order to succeed. For all these reasons, they are inferior to truth tables.
But derivations are superior to truth tables on several other fronts. Because they require ingenuity and insight, they are more like chess than tic-tac-toe; they are much less boring than truth-tables. They come close to how we actually reason --closer than truth tables at any rate. In addition to proving validity, they can generate valid conclusions from a given set of premises, to see where they lead, or to "unfold" them. Hence they can model exploratory reasoning. Moreover, they can handle any number of propositional variables without clumsiness, unlike truth tables. Finally, they are adequate for predicate logic (our next unit) with only a few additions, while truth tables are useless in predicate logic.
In derivations we use two kinds of rule: rules of inference and rules of replacement. (The term "rule of inference" is sometimes used to cover both types.) The former are tautologous conditionals, and the latter are tautologous biconditionals. The former apply only to entire lines of proof, or whole propositions, while the latter apply to components within compounds as well as to the whole compounds themselves.
Any tautologous conditionals and biconditionals could serve as valid rules of inference; but these are infinite in number. If we let ourselves use any or all of these, then every valid argument could be proved in one step, without further proof. This would be unilluminating. It would also be impractical (actually, impossible) to name each of the rules we were to use. By contrast, then, we use only 19 rules, with two supplementary techniques, that suffice to prove every valid argument valid. And we can call each by name.
Another perspective on the same point. Copi's 19 rules of inference are like the multiplication table. If you memorize a few basic multiplications, then you can use them as tools to compute any complex multiplication. However, if you were superhuman, you could memorize every one of the infinity of correct multiplications so that you would never have to compute any. Similarly, by using a finite set of rules of inference, you prove every complex argument to be valid. However, if you were superhuman, you could memorize every one of the infinity of tautologous conditionals and biconditionals and never have to perform any derivations (of more than one step).
A proof is successful if each step is justified by one of our rules of inference, and if the last step is the proposition to be proven. Everything else (except spelling and neatness) is irrelevant, including the length and niftiness of the proof, and repeated or unnecessary steps. Elegance is desirable, but only validity is essential.
Come back to these after you have learned Copi's rules of inference for derivations.
1. Derive any statement you need from a contradiction
A · ~A given A simplification ~A simplification A B addition; B is any statement we happen to need B disjunctive syllogism
2. Move from any B to A B
B given B ~A logical addition ~A B commutation A B material implication
Along the same lines, do you see how to move from any given ~A to A B?
3. Move from A(B · C) to AB
A(B · C) given ~A(B · C) material implication (~AB) · (~AC) distribution ~AB simplification AB material implication
4. Move from (AB)C to AC
(AB)C given ~(AB)C material implication (~A · ~B)C DeMorgan's theorem C(~A · ~B) commutation (C~A) · (C~B) distribution C~A simplification ~AC commutation AC material implication
5. Move from A ~A to ~A
A ~A given ~A ~A material implication ~A tautology
6. Move from ~A A to A
7. Add any tautology to any proof
~A A given A A material implication A tautology
You know that the negation of a tautology is a contradiction. Hence, assume the negation of the tautology desired. This will be a contradiction, and it will be your fault if you cannot, eventually, derive an explicit contradiction from it (the conjunction of some statement with its negation). When you do, discharge your assumption with the (unnegated) tautology, by Indirect Proof.
Can you see how to move from [A (B C)] to [B (A C)] using only exportation and commutation?
This file is an electronic hand-out for the course, Symbolic Logic.
Most of the logic symbols in this file are GIFs. See my Notes on Logic Notation on the Web.
Peter Suber, Department of Philosophy, Earlham College, Richmond, Indiana, 47374, U.S.A.
firstname.lastname@example.org. Copyright © 1997, Peter Suber.