PART II • SECTION B MODERN LOGIC

METHODS OF DEDUCTION

10.1 Formal Proof of Validity10.2 The Rule of Replacement10.3 Proof of Invalidity10.4 Inconsistency10.5 Indirect Proof of Validity10.6 Shorter Truth-Table Technique

10.1 Formal Proof of Validity

In theory, truth tables are adequate to test the validity of any argument of the gen­eral type here considered. But in practice they grow unwieldy as the number ofcomponent statements increases. A more efficient method of establishing the va­lidity of an extended argument is to deduce its conclusion from its premises by asequence of elementary arguments each of which is known to be valid. This tech­nique accords fairly well "vith ordinary methods of argumentation.

Consider, for example, the following argument:

If Anderson was nominated, then she went to Boston.If she went to Boston, then she campaigned there.If she campaigned there, she met Douglas.Anderson did not meet Douglas.Either Anderson was nominated or someone more eligible was selected.Therefore someone more eligible was selected.

It cannot be

demanded that

we should prove

everything,because that is

impossible; but

we can require

that all proposi-tions used with-

out proof be

expresslydeclared to be so

... Furthermore

I demand-and

in this I go

beyond Euclid­that all of

the methods of

inference used

must be specifiedin advance.

-Gottlob Frege

359

360 Chapter IO Methods of Deduction

Its validity may be intuitively obvious, but let us consider the matter ofproof. The discussion will be facilitated by translating the argument intosymbolism as:

A~BB~ CC~D-DAvE:. E

Rules of Inference

The rules that permitvalid inferences fromstatements assumedas premises.

NaturalDeductionA method of prov­ing the validity of adeductive argumentby using the rules ofinference.

Formal Proof of

ValidityA sequence ofstatements, each ofwhich is either apremise of a givenargument or isdeduced, using therules of inference,from precedingstatements in thatsequence, such thatthe last statement inthe sequence is theconclusion of theargument whosevalidity is being ­proved.

To establish the validity of this argument by means of a truth table would re­quire a table with 32 rows, since there are five different simple statements in­volved. But we can prove the given argument valid by deducing its conclusionfrom its premises by a sequence of just four elementary valid arguments. Fromthe first two premises, A ~ Band B ~ C, we validly infer A ~ C by aHypothetical Syllogism. From A ~ C and the third premise C ~ D, we validlyinfer A ~ D by another Hypothetical Syllogism. From A ~ D and the fourth

premise - D, we validly infer ~ A by modus tollens. And from -A and the fifthpremise A v E, by a Disjunctive Syllogism we validly infer E, the conclusion ofthe original argument. That the conclusion can be deduced from the fivepremises of the original argument by four elementary valid arguments provesthe original argument to be valid. Here the elementary valid argument formsHypothetical Syllogism (H.S.), modus tollens (M.T.), and Disjunctive Syllogism(D.S.) are used as rules of inference in accordance with which conclusions are

validly inferred or deduced from premises.This method of deriving the conclusion of a deductive argument-using

rules of inference successively to prove the validity of the argument-is as reli­able as the truth table method exhibited in the preceding chapter, if the rulesare used with meticulous care. But it improves upon the truth table method in

two ways. It is vastly more efficient, as has just been shown. And it enables us tofollow the flow of the reasoning process from the premises to the conclusion andis therefore much more illuminating, more intuitive. The method is often called

that of natural deduction. Using natural deduction we can provide a formal Proof

of the validity of an argument that is valid.A formal proof of validity is given by writing the premises and the state­

ments that we deduce from them in a single column, and setting off in an­

other column, to the right of each such statement, its 'Justification," or thereason we can give for including it in the proof. It is convenient to list all thepremises first and to write the conclusion either on a separate line, or slight­

ly to one side and separated by a diagonal line from the premises. If all thestatements in the column are numbered, the 'Justification" for each state­ment consists of the numbers of the preceding statements from which it is in­ferred, together with the abbreviation for the rule of inference by which it

follows from them. The formal proof of the argument above is written as:

IS

m

l.A::JB

2. B::J C

3. C::J D

4. ~D5.AvE

:. E

6. A ::J C

7. A ::J D8. ~A

9.E

1,2, H.S.6,3, H.S.7,4, M.T.

5,8, D.S.

Formal Proof of Validity 361

We define a formal proof that a given argument is valid as a sequence of state­ments each of which is either a premise of that argument or follows from pre­ceding statements of the sequence by an elementary valid argument, such thatthe last statement in the sequence is the conclusion of the argument whose va­lidity is being proved.

We define an elementary valid argument as any argument that is a substitu­tion instance of an elementary valid argument form. One matter to be empha­sized is that any substitution instance of an elementary valid argument form isan elementary valid argument. Thus the argument

(A - B) ::J [C== (Dv E)]A-B

:. C== (Dv E)

is an elementary valid argument because it is a substitution instance of the ele­mentary valid argument form modus ponens (M.P.). It results from

p::Jq

P

:. q

by substituting A - B for p and C == (D v E) for q, and it is therefore of that formeven though modus ponens is not the specific form of the given argument.

j\;fodus ponens is a very elementary valid argument form indeed, but whatother valid argument forms are to be included as rules of inference? vVe beginwith a list of just nine rules of inference to be used in constructing formalproofs of validity.

These nine rules of inference correspond to elementary argumentforms whose validity is easily established by truth tables. With their aid, for­mal proofs of validity can be constructed for a wide range of more compli­cated arguments. The names listed are for the most part standard, and theuse of their abbreviations permits formal proofs to be set down with a mini­mum of writing.

Elementary ValidArgumentAnyone of a set ofspecified deductivearguments thatserves as a rule ofinference and canbe used to con­struct a formal

proof of validity.