[2.12.] “Material Biconditionals.”

 

The final truth-functional connective in sentential logic is the material biconditional (a.k.a. biconditional), symbolized by the tribar: º

 

 

The truth-function expressed by a material biconditional is called material equivalence:

 

material equivalence (df.): two sentences are materially equivalent if they have the same truth value; if they do not have the same truth value, then they are not materially equivalent.

 

Notice that in the truth-table definition for “º”, “p º q” is true when both p and q are true and when both p and q are false. In other words, a material biconditional is true when its component sentences have the same truth-value; it is false exactly when they have different truth values.

 

“p º q” is called a biconditional because it is equivalent to the conjunction of two conditionals:

 

p º q                             is equivalent to           (p É q) · (q É p)

 

 

This explains why we read “p º q” as “p if and only if q”. This is because...

 

 

“p if and only if q” or p º q is equivalent to the conjunction of
“p if q”   which is equivalent to   “if q then p” or q É p	and	“p only if q”   which is equivalent to   “if p then q” or p É q

 

So “Amy is an actress if and only if Bert is a bachelor” is equivalent to “If Amy is an actress, then Bert is a bachelor, and if Bert is a bachelor, then Amy is an actress.”

 

So “A º B” is equivalent to “(A É B) · (B É A)”.

 

Notice that the phrase “just in case” expresses material equivalence. So an accurate translation of “Murphy is happy just in case he has a bone” is “H º B”.

 

--

 

You have now been introduced to all five connectives of sentential logic. You MUST master these connectives. In particular, you should be able to reproduce the truth table definitions for all of them.  In summary, they are:

 

function	negation	conjunction	disjunction	material implication	material equivalence
operator	~	·	Ú	Ì	º
English term	not	and	or	if...then	if and only if; just in case
number of inputs	1	2	2	2	2

 

 

Exercises 2-7 (pp.46-47) and 2-8 (pp.47-48)

·         Do all of these exercises. Check your even-numbered answers against the book. We will go through the odd-numbered problems next time.

 

 

 

[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:

§  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 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

 

~[(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    FT     FT   F

 

~[(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

 

 

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 the odd-numbered problems in class next time.

 

 

 

Stopping point for Tuesday January 24. For next time:

·         complete exercises 2-7, 2-8, and 3-1

·         read secs. 3-3 through  3-4 (pp.63-76)

·         Heads up: your first test is Thursday February 2.