Lecture 8

Propositional Logic

      

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

 

Definition: a Complex Statement 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. (conjunction) - "and", "but", "however", "moreover", "nevertheless", "still", "also", "although", "both"

      

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

 

Symbolized Statement:   I S

 

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

 

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

 

Symbolized Statement:   I S

 

4. (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 S

 

5. (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 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 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 [S (L 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 [L (J 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

 

                                   p    q     p    q

                             

                            T    T       T

                            T    F       F

                            F    T       F

                            F    F       F

                               

 

3. Disjunction

 

                                 p     q     p    q

 


                       T      T         T

                       T      F         T

                       F      T         T

                       F      F         F     

 

 

4. Implication

 

                                 p     q     p    q

 


                       T      T         T

                       T      F         F

                       F      T         T

                       F      F         T     

 

5. Equivalence

 

                                 p     q     p    q

 


                       T      T         T

                       T      F         F

                       F      T         F

                       F      F         T     

             

 

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    q

                      q   r

                 \ p    r

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

 

                      p   q

                      p_____ 

                  \q

 

 

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

 

                      p    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   q)   (r    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   q)   (r    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    q

                      q_____  

                   \p

 

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

 

                      p   q

                      ~ p___ 

                   \~ q