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.