Bridge to Abstract Mathematics: Mathematical Proof and Structures

62 LOGIC, PART I: THE PROPOSITIONAL CALCULUS Chapter 2

Examples of conditionals include:

1. If I finish my work, I go out on the town.


2. If 2 + 2 = 5, then (0, 1) G (0, 1).


3. If 2 + 2 = 4, then 5 is not a prime number.

Figure 2.5 Truth tables for the conditional

and biconditional.

Examples of biconditionals include:

P f-' 4

T

F

F

T

-- P

T

T

F

F

4. 3 is odd if and only if 4 is even. (true)
   5.^7 +^6 =^14 if and only if^7 times^6 equals 41. (true)


5. A triangle has three sides if and only if a hexagon has (false)
   seven sides.

Students generally find the conditional to be the least intuitive of all the

connectives. One reason is that normal usage of "if... then" presupposes

a causal connection between the premise and the conclusion, as in (1). Sen-

tences such as (2) and (3) seem somehow unnatural, perhaps not even worthy

of being regarded as statements. But in applications of the propositional

calculus to mathematics, it is crucial that p -+ q be a statement whenever

p and q are (with no regard for any "cause and effect" relationship) with

its truth or falsehood totally a function of the (possibly independent) truth

values of p and q. Thus we consider (2) to be true (since its premise is

false, the truth value of the conclusion doesn't matter) and (3) to be false.

Another problem many students encounter with the truth table defining

the conditional lies in row 3. Why do we regard p + q as true when p is

false and q is true? Consider sentence (1). Most people would (justifiably)

regard this statement as true if, on a given evening, "I finished my work

and went out" (row 1) or if "I did not finish my work and didn't go out"

(row 4), and false (i.e., I lied) if "I finished my work and stayed home"

(row 2). But what if "I don't finish and still go out"? Given the impre-

cision with which language is used in everyday life, many would comment

that (1) is a false statement. Looking at what this statement actually says,

and not at any hidden meaning that is often read into such a statement,

I is this a fair comment? Not at all! The statement deals only with "what
I
I I

ir

.- -^4 -

T

F

T

F

P -* --- (^4) -.
T
F
T
T