Lecture 8

## Propositional Logic

Definition:  a ÔSimple StatementÕ is a statement of fact which occurs in a single grammatical unit (i.e., containing one subject and one predicate).

Definition: a ÔComplex StatementÕ is a statement of fact which occurs in a compound grammatical unit (i.e., containing plural subjects or plural predicates).

# I. Symbolic Translation

A. Variables - upper case letters of the Roman alphabet {A, B, C. . . Z} will represent single propositions.

Examples:

Statement:  Iran raised their price for oil.

Symbolized Statement:   I.

Statement:  Iran and Saudi Arabia raised their price for oil.

Symbolized Statement:   I and S.

Statement:  Either we go dancing or go to the movies.

Symbolized Statement:  Either D or M.

B. Symbols - the five operators or connectives for Sentential Logic

1. ~ (negation) - "not", "it is not the case that", "it is false that"

Statement:  Saudi Arabia did not raise their price for oil.

Symbolized Statement:   ~ S

2. á (conjunction) - "and", "but", "however", "moreover", "nevertheless", "still", "also", "although", "both"

Statement:  Iran and Saudi Arabia raised their price for oil.

Symbolized Statement:   I á S

3. v (disjunction) - "or", "unless"

Statement:  Either Iran or Saudi Arabia raised their price for  oil.

Symbolized Statement:   I ò S

4. ƒ (material implication) - "if", "only if", "given that", "provided that", "on the condition that",  "sufficient condition for",  "necessary condition for"

Statement:   If Iran raised the price of oil, then Saudi Arabia did also.

Symbolized Statement:  I ƒ S

5. ¼ (material equivalence) - "if and only if", "sufficient and necessary condition for"

Statement:  Iran raised the price of oil if and only if Saudi Arabia did.

Symbolized Statement:  I ¼ S

C. Punctuation - to demonstrate the order of operators where three or more simple propositions occur within a statement

1. ( ) - first order

Statement:  Not both Iran and Saudi Arabia raised the price of  oil.

Symbolized Statement:   ~ (I á S)

2. [ ] - second order

Statement:  John likes to dance; however, if Sarah does, then either Joe or Laura likes to go to the movies.

Symbolized Statement:  J á [S ƒ (L ò J)]

3. { } - third order

Statement:  That Tom loves Nancy; and if Larry likes Tina, then John and Steve love Laura is not the case.

Symbolized Statement:  ~{T á [L ƒ (J á S)]}

Note:  Just like other languages, SL has a grammatical structure or syntax.  This is the order of the units in the proposition.  If all the elements are in their proper place, we say it is a well-formed-formula or WFF.  Operators, variables, and punctuation must all be in its proper place in order for a proposition to be well formed.

II.  Truth Values for Propositional Logic:

A. Truth Values for Operators:

1. Negation -

p     ~ p

T      F

### F      T

2. Conjunction Ð

p    q     p  á  q

#### TTT

T    F       F

F    T       F

#### FFF

3. Disjunction Ð

p     q     p  ò  q

T      T         T

T      F         T

F      T         T

F      F         F

4. Implication Ð

p     q     p  ƒ  q

T      T         T

T      F         F

F      T         T

##### FFT

5. Equivalence Ð

p     q     p  ¼  q

T      T         T

T      F         F

F      T         F

##### FFT

III. Six Valid Argument Forms:

A. Disjunctive Syllogism (DS) - a deductive argument containing a disjunctive premise and a negation of one of the disjuncts

p v q

~ p__

\ q

B. Hypothetical Syllogism (HS) - a deductive argument composed of two implicative premises and an implicative conclusion

p  ƒ  q

q  ƒ  r

\ p  ƒ  r

C. Modus Ponens (MP) - a deductive argument composed of one implicative premise and the affirmation of the antecedent of the premise

p  ƒ  q

p_____

\q

D. Modus Tollens (MT) - a deductive argument composed of one implicative premise and the denial of the consequent

p  ƒ  q

~ q___

\ ~ p

E. Constructive Dilemma (CD) - a deductive argument composed of a conjunction of two implications and a disjunction of the antecedents of the conjuncts

(p  ƒ  q)  á  (r  ƒ  s)

p  v  r___________

\ q  v  s

F. Destructive Dilemma (DD) - a deductive argument composed of a conjunction of implications and the disjunction of the negations of the consequents of the conjuncts

(p  ƒ  q)  á  (r  ƒ  s)

~ q  v  ~ s

\~ p  v  ~ r

IV.  Two Common Formal Fallacies:

A.  Affirming the Consequent (AC) -  in MP affirming the consequent instead of the antecedent of the first premise.

p  ƒ  q

q_____

\p

B.  Denying the Antecedent (DA) - in MT denying the antecedent instead of the consequent

p  ƒ  q

~ p___

\~ q