Methods of Deduction
Our set of nine elementary valid argument forms is now complete.
All nine of these argument forms are very plainly valid. Any one of them whose
validity we may doubt can be readily proved to be valid using a truth table. Each of
them is simple and intuitively clear; as a set we will find them powerful as we go on
to construct formal proofs for the validity of more extended arguments.
overview
Rules of Inference:
Elementary Valid Argument Forms
Name Abbreviation Form
1.Modus Ponens M.P. p)q
p
‹ q
2.Modus Tollens M.T. p)q
'q
‹ 'p
3.Hypothetical Syllogism H.S. p)q
q)r
‹ p)r
4.Disjunctive Syllogism D.S. p q
'p
‹¡q
#
5.Constructive Dilemma C.D. (p)q) (r)s)
p r
‹ q s
¡
6.Absorption Abs. p)q
¡ #
‹ p)(p q)
#
7.Simplification Simp. p q
‹ p
8.Conjunction Conj. p
q
#
‹ p q
9.Addition Add. p
‹ p q
¡
Two features of these elementary argument forms must be emphasized. First,
they must be applied with exactitude. An argument that one proves valid using
Modus Ponens must have that exact form: p)q, p, therefore q. Each statement
variable must be replaced by some statement (simple or compound) consistently
and accurately. Thus, for example, if we are given (C D))(J K) and
(C D), we may infer (J K)by Modus Ponens. But we may not infer (K J)by
¡ ¡
Modus Ponens, even though it may be true. The elementary argument form must
¡ ¡ ¡
be fitted preciselyto the argument with which we are working. No shortcut—no
(cid:22)(cid:26)(cid:20)
Methods of Deduction
De Morgan also advanced the field called the logic of relatives. Identity and
difference are relations to which logicians have given great attention, but there
are other relations, such as equality, affinity, and especially equivalence, that
also deserve the logician’s attention, as De Morgan showed.
Two logical equivalences, widely useful and intuitively clear, received
from De Morgan their time-honored formulation and carry his name:
De Morgan’s Theorems, which remain a permanent and prominent instrument
in deductive reasoning. !
overview
The Rules of Replacement:
Logically Equivalent Expressions
Any of the following logically equivalent expressions may replace each other
wherever they occur.
Name Abbreviation Form
#
10.De Morgan’s theorems De M. '(p q)!T ('p 'q)
#
'(p q)!T ('p¡ 'q)
11.Commutation Com. (p ¡q)!T (q p)
# #
(p ¡q)!T (q p¡)
12.Association Assoc. [p (q r)]!T [(p q) r]
# # # #
[p ¡(q r)¡]!T [(p q) ¡r] ¡
# # #
13.Distribution Dist. [p (q r)]!T [(p q) (p r)]
# #
[p (q¡ r)]!T [(p q)¡ (p r)]
14.Double Negation D.N. p!T¡''p ¡ ¡
15.Transposition Trans. (p)q)!T ('q) 'p)
16.Material Implication Impl. (p)q)!T ('p q)
#
17.Material Equivalence Equiv. (p K q)!T [(p)¡q) (q)p)]
# #
(p K q)!T [(p q) ('p 'q)]
#
18.Exportation Exp. [(p q))r]!T [p)¡(q)r)]
19.Tautology Taut. p!T (p p)
#
p!T (p ¡p)
(cid:22)(cid:27)(cid:26)
