Dr. Robert Lane

Lecture Notes: Thursday January 23, 2014

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**[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 ****É 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
****É**** B) ****·**** ~C] ****º**** (~C ****Ú**** B)**

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

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**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;

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**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 we’ll 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 ****É**** B)
****·**** ~C] ****º**** (~C ****Ú**** 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 ****É**** B)
****·**** ~C] ****º**** (~C ****Ú**** 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 ****É**** B)
****·**** ~C] ****º**** (~C ****Ú**** 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**

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**EXERCISE 3-1 (p.57):**

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**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.**

*This page last updated 1/23/2014.*

Copyright © 2014 Robert Lane. All rights reserved.