Symbolic Trails and Formal Proof's of Validity

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: p  q Conclusion Connective: ~ ~p  ~q

Truth Table for Logical Connectives p q ~p ~q p  q p  q p  q p  q

T T F F T T T T T F F T F T F F F T T F F T T F F F T T F F T T