Any formula whose truth table contains only F's [FALSE]
is truth functionally inconsistant or a contradiction.
Disjunctive and Hypothetical Syllogisms
1. Whales are mammals. = (p) A categorical proposition.
2. Either whales are mammals, or they are very big fish. =
(p or q) A DISJUNCTIVE proposition
3. If whales are mammals, then they cannot breathe underwater. =
(if p then q) A HYPOTHETICAL proposition.
Disjunctive Syllogisms
The components of a disjunctive proposition ( p and q ) are called disjuncts.
Such a statement does not actually assert that p is true, or that q is; but it does say that one or the other of them is true. Thus if we knew that one of the disjuncts was not true, we could infer that the other must be true.
Truth Table for p∨q:
(we can see that if one of the disjuncts is true, the whole disjunction is true)
p 
q 
T/F 
T 
T 
T 
T 
F 
T 
F 
T 
T 
F 
F 
F 
Therefore, a disjunctive syllogism has the structure:
either p or q,
not p
therefore q.
This is the INFERENCE RULE called "Disjunctive Syllogism"
[DS]:
pq
~p
q 
It also works in the opposite direction:
pq
~q
p 
(p or q, not q, therefore p)
Chapter 3 The Propositional Calculus
3.3 NONHYPOTHETICAL INFERENCE RULES
The propositional calculus provides a system of inference rules
which are capable of generating all the valid argument forms expressible
in the language of the propositional calculus and only valid forms.
There are ten basic inference rules: two –
an introduction rule and an elimination rule –
for each of the five logical operators.
The Five Logical Operators:
Name 
Symbol

Alternate




Conjunction 
&

·

and 
p · q

"p and q" 
Negation 
~


not 
~p

"not
p; it is not the case that p" 
Disjunction 

v 
either . . . or 
p v q

"either p or
q" 
Conditional (also called "material conditional") 
É

®

if . . . then 
p É
q

"if p, then q" 
Biconditional 
º

«

if and only if . . . then 
p º q

"if and only if p, then q" 
Equivalence Rule: Double negation [DN]
p º
~~p
also known as
Negation elimination [~E]
definition: From a wff of the form ~ ~ Φ, infer Φ.
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Conditionals; Elimination
Inference Rule: Modus Ponens [MP]
the elimination rule for conditionals
A É
B If A, then B; AH
B