PHIL 4160: Symbolic Logic
Dr. Robert Lane
Lecture Notes: Thursday January 23, 2014

 

 

[3.] Truth Tables

 

[3.1.] Computing Truth-Values.

 

In this section, we will learn how to compute the truth value of longer compound sentences, given the truth values of each atomic component sentence.

 

Remember that we can use the truth table definitions of the five operators plus the truth values of atomic sentences to figure out the truth value of less complex compound sentences. For example, we can figure out the truth value of A B if we know the values of A and B.

 

Similarly, if we are given the truth-value of every atomic sentence in a compound sentence, then we can figure out the truth-value of the entire compound sentence, even if it is relatively long and complex, e.g.,

 

~[(A B) ~C] (~C B)

 

There are two methods you can use to figure out the truth value of such a lengthy sentence:

  the loop method -- more graphical, so easier for beginners to grasp; but it will work for only one assignment of truth values to atomic sentences at a time;

  the tabular method -- preferred, because it works for more than one assignment of truth values to atomic sentences at a time. This is important, since well soon be analyzing compound sentences without assuming a specific set of truth values for their component atomic sentences.

 

The steps are the same in each method... the only difference is the notation used.

 

Step 1: Place a truth value beneath each sentence constant.

 

Assume the following values:

A = true

B = false

C = true

 

LOOP METHOD TABULAR METHOD

~[(A  B)  ~C]  (~C  B)
   T    F     T      T   F
 


~[(A B) ~C] (~C B)

T F T T F

 

 

 

 

 

 

 

Step 2: Place a truth value beneath each tilde that negates a SINGLE sentence constant (if there are any such tildes).

~[(A  B)  ~C]  (~C  B)
    T   F    FT     FT   F
 


~[(A B) ~C] (~C B)

T F T T F

 

 

 

 

 

 

 

 

Step 3: Continue to place truth values beneath operators, beginning with the operator(s) with the smallest scope and working through them in order of increasing scope.

~[(A  B)  ~C]  (~C  B)
T  T F F   F FT  F  FT T F
,
 

 


~[(A B) ~C] (~C B)

T F T T F

 

 

 

 

 

 

 

 

 

 

 

[*I am not able to duplicate the loop method in the online lectures notes-- if you print this day's lecture out, you should draw in the loops as we did in class.]

 

EXAMPLES: pp.56-57

 

 

 

 

EXERCISE 3-1 (p.57):

  for next time, complete all of the problems in exercise 3-1, AT LEAST USING THE TABULAR METHOD; we will go over some of the odd-numbered problems in class next time.

 

 

 

***This is the end of the material that will be covered on your first test. I will revise the Study Guide to reflect this and email you all when the revised guide is online.

 

Stopping point for Thursday January 23. For next time:

         complete exercise 3-1

         we will review for your first exam.

 

 

 

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