SYMBOLIC TRAILS AND FORMAL PROOFS OF VALIDITY
Rules With Two Premises
MODUS PONENS (MP) Has, two premises, one which has a  as a major connective.
The other premise matches exactly the antecedent of the first premise. The conclusion
matches the consequent of the first premise exactly.
Premise Connective: 
Example: p  q
Conclusion Connective: None
p

q
MODUS TOLLENS (MT) Has two premises, one of which has a  as a major connective.
The other premise negates the consequent of the  premise. The conclusion is always a
negation of the antecedent of the  premise.
Premise Connectives: , ~
Example: p  q
Conclusion Connective: ~
~q 
~p
DISJUNCTIVE SYLLOGISM (DS) One premise must be a disjunction and the other
premise must negate the left disjunct. The conclusion matches the right disjunct exactly.
Premise Connectives: , ~
Example: p  q
Conclusion Connective: None
~p 
q
HYPOTHETICAL SYLLOGISM: (HS) The major connective for both of the premises
and the conclusion is a . The consequent of one premise matches the antecedent of the
other premise. The conclusion then links the antecedent of one premise with the consequent
of the other.
Premise Connectives: , 
Example: p  q
Conclusion Connective: 
qr  pr
CONSTRUCTIVE DILEMMA: (CD) Has, two premises. One of which is a conjunction
of two conditionals. The other must be a disjunction of the two antecedents of the conditionals. The conclusion must be a disjunction of the two consequents of the conditionals.
Premise Connectives: , , 
Example: (p  q)  (r  s)
Conclusion Connective: 
pr 
qs
CONJUNCTION: (CJ) No matter what the premises are, the conclusion is simply a
conjunction of the two premises.
Premise Connective: None
Example: p
Conclusion Connective: 
q

pq
Rules With One Premise
ABSORPTION: (AB) Only, one premise. Both the premise and the conclusion must have a
 as a major connective. The antecedent of the conclusion is the same as the premise and the
consequent of the conclusion must be a conjunction of the antecedent and the consequent of the
premise.
Premise Connective: 
Example: p  q 
p  (p  q)
Conclusion Connectives:  
SIMPLIFICATION: (SP) Only one premise and the major connective of the premise must be
a conjunction. The Conclusion must be the left hand conjunct.
Premise Connective: 
Example: p  q 
p
Conclusion Connective: None
ADDITION: (AD) Let’s you to add any statement to any premise by using disjunction. The
major connective of the conclusion must be a disjunction, with the premise as the left disjunct.
What is added to the premise by the disjunction can be anything needed to help solve the
problem.
Premise Connective: None
Example: p

pq
Conclusion Connective: 
Proof Strategies
STRATEGY I: If the conclusion is part of a premise, match with MP, DS, or SP.
STRATEGY II: Does the major connective of the conclusion match any of the rules.
Logical Fallacies
Confirming the Consequent: Closely resembles Modes Ponens.
Premise Connective: 
Example: p  q
Conclusion Connective: None
q

p
Denying the Antecedent: Closely resembles Modus Tollens.
Premise Connectives: , ~
Example:
Conclusion Connective: ~
~q
Truth Table for Logical Connectives
p q ~p ~q p  q p  q p  q p  q
TT
TF
FT
FF
F
F
T
T
F
T
F
T
T
F
F
F
T
T
T
F
T
F
T
T
T
F
F
T
pq
~p 