Main Page Philosophical Terms Reconstructing an Argument Short List of Definitions Glances Ahead Rotating Validity Exercises

I. Introduction to Symbolic Logic- the Use of the Truth Table for Determining Validity
-Truth tables are useful formal tools for determining validity of arguments because they specify the truth value of every premise in every possible case
-Truth tables are constructed of logical symbols used to represent the validity- determining aspects of an argument
-Symbols:
the dot (.) is used to represent any word that joins two conjuncts (ex. 'and', 'moreover', 'furthermore', 'but', 'yet', 'still', 'however', 'also', 'nevertheless', 'although')
the wedge (v) is used to represent any word that joins two disjuncts, most frequently representing the word “or” in an inclusive sense (that is, the inclusive “or” asserts that at least one disjunct is true, while the exclusive “or” asserts that at least one disjunct is true, but not both are true)        the tilde (~) is used the represent the negation of any simple statement (ex. p= lead is heavy; ~p= lead is not heavy)
the horseshoe (>) is used to represent the equivalent of ~(p . ~q); it is used for any conditional statement; for any  conditional “if, then” statement to be true, p>q, the negation of the conjunction of its antecedent with the negation of its consequent, must be true also

Examples:
Truth Table for: p . q
p          q          p . q
T          T          T
T          F          F
F          T          F
F          F          F

Truth Table for: p v q
p          q          p v q
T          T          T
T          F          T
F          T          T
F          F          F

Truth Table** for:  A > (B v C)
B
Therefore, A > ~C
A         B         C         B v C   A > (B v C)      ~C       A > ~C
T          T          T          T          T                      F          F
T          T          F          T          T                      T          T
T          F          T          T          T                      F          T
T          F          F          F          F                      T          T
F          T          T          T          T                      F          T
F          T          F          T          T                      T          T
F          F          T          T          T                      F          T
F          F          F          F          T                      T          T

**This argument is shown to be INVALID by the above truth table because, in the first row, both premises are true, but the conclusion is false.

Either Atlanta wins their conference championship and Baltimore wins their conference championship  or Chicago wins the superbowl.
Translation: (A . B) v C
Truth table:
A         B         C         A . B    (A . B) v C
T          T          T          T          T
T          T          F          T          T
T          F          T          F          T
T          F          F          F          F
F          T          T          F          T
F          T          F          F          F
F          F          T          F          T
F          F          F          F          F

The truth of the statement can be assessed, but, because this is not an argument, validity cannot be assessed.

Exercises
Determine the validity of the following arguments by first translating them into symbols and then constructing truth tables.
* Mary has brown hair and James has black hair.  Therefore, Mary has brown hair.
* It is not the case that humans have ten toes and humans do not have either blond hair or brown hair.  Humans have blond hair. Therefore, It is not the case that humans have ten toes and have brown hair.
* Mars is uninhabited by human life or is made of solidified soda.  Mars is uninhabited by human life.  Therefore, mars is not made of soda.

Solutions
* first argument is valid and the second two are invalid

II. Philosophical Analysis

III. Logic and Natural Language

IV. The Law of the Excluded Middle