English forms of the material conditional

We can come fairly close to an adequate English rendering of the material

conditional P ’ Q with the sentence If P then Q. At any rate, it is clear that

if . . . then

Chapter 7

Material conditional symbol: ’ / 179

this English conditional, like the material conditional, is false if P is true and

Q is false. Thus, we will translate, for example, If Max is home then Claire is

at the library as:

Home(max) ’ Library(claire)

In this course we will always translate if. . . then. . . using ’, but there

are in fact many uses of the English expression that cannot be adequately

expressed with the material conditional. Consider, for example, the sentence,

If Max had been at home, then Carl would have been there too.

This sentence can be false even if Max is not in fact at home. (Suppose the

speaker mistakenly thought Carl was with Max, when in fact Claire had taken

him to the vet.) But the ¬rst-order sentence,

Home(max) ’ Home(carl)

is automatically true if Max is not at home. A material conditional with a

false antecedent is always true.

We have already seen that the connective because is not truth functional

since it expresses a causal connection between its antecedent and consequent.

The English construction if. . . then. . . can also be used to express a sort of

causal connection between antecedent and consequent. That™s what seems to

be going on in the above example. As a result, many uses of if. . . then. . .

in English just aren™t truth functional. The truth of the whole depends on

something more than the truth values of the parts; it depends on there being

some genuine connection between the subject matter of the antecedent and

the consequent.

Notice that we started with the truth table for ’ and decided to read

it as if. . . then. . . . What if we had started the other way around, looking for

a truth-functional approximation of the English conditional? Could we have

found a better truth table to go with if. . . then. . . ? The answer is clearly no.

While the material conditional is sometimes inadequate for capturing sub-

tleties of English conditionals, it is the best we can do with a truth-functional

connective. But these are controversial matters. We will take them up further

in Section 7.3.

Necessary and su¬cient conditions

Other English expressions that we will translate using the material conditional

P ’ Q include: P only if Q, Q provided P, and Q if P. Notice in particular only if, provided

that P only if Q is translated P ’ Q, while P if Q is translated Q ’ P. To

Section 7.1

180 / Conditionals

understand why, we need to think carefully about the di¬erence between only

if and if.

In English, the expression only if introduces what is called a necessary

necessary condition

condition, a condition that must hold in order for something else to obtain.

For example, suppose your instructor announces at the beginning of the course

that you will pass the course only if you turn in all the homework assignments.

Your instructor is telling you that turning in the homework is a necessary

condition for passing: if you don™t do it, you won™t pass. But the instructor is

not guaranteeing that you will pass if you do turn in the homework: clearly,

there are other ways to fail, such as skipping the tests and getting all the

homework problems wrong.

The assertion that you will pass only if you turn in all the homework

really excludes just one possibility: that you pass but did not turn in all the

homework. In other words, P only if Q is false only when P is true and Q is

false, and this is just the case in which P ’ Q is false.

Contrast this with the assertion that you will pass the course if you turn

in all the homework. Now this is a very di¬erent kettle of ¬sh. An instructor

who makes this promise is establishing a very lax grading policy: just turn in

the homework and you™ll get a passing grade, regardless of how well you do

on the homework or whether you even bother to take the tests!

In English, the expression if introduces what is called a su¬cient condition,

su¬cient condition

one that guarantees that something else (in this case, passing the course) will

obtain. Because of this an English sentence P if Q must be translated as

Q ’ P. The sentence rules out Q being true (turning in the homework) and

P being false (failing the course).

Other uses of ’

In fol we also use ’ in combination with ¬ to translate sentences of the form

Unless P, Q or Q unless P. Consider, for example, the sentence Claire is at the

unless

library unless Max is home. Compare this with the sentence Claire is at the

library if Max is not home. While the focus of these two sentences is slightly

di¬erent, a moment™s thought shows that they are false in exactly the same

circumstances, namely, if Claire is not at the library, yet Max is not home

(say they are both at the movies). More generally, Unless P, Q or Q unless

P are true in the same circumstances as Q if not P, and so are translated as

¬P ’ Q. A good way to remember this is to whisper if not whenever you see

unless. If you ¬nd this translation of unless counterintuitive, be patient. We™ll

say more about it in Section 7.3.

It turns out that the most important use of ’ in ¬rst-order logic is not

in connection with the above expressions at all, but rather with universally

Chapter 7

Biconditional symbol: ” / 181

quanti¬ed sentences, sentences of the form All A™s are B™s and Every A is a

B. The analogous ¬rst-order sentences have the form:

For every object x (A(x) ’ B(x))

This says that any object you pick will either fail to be an A or else be a B.

We™ll learn about such sentences in Part II of this book.

There is one other thing we should say about the material conditional,

which helps explain its importance in logic. The conditional allows us to reduce

the notion of logical consequence to that of logical truth, at least in cases reducing logical

consequence to

where we have only ¬nitely many premises. We said that a sentence Q is a

logical truth

consequence of premises P1 , . . . , Pn if and only if it is impossible for all the

premises to be true while the conclusion is false. Another way of saying this

is that it is impossible for the single sentence (P1 § . . . § Pn ) to be true while

Q is false.

Given the meaning of ’, we see that Q is a consequence of P1 , . . . , Pn if

and only if it is impossible for the single sentence

(P1 § . . . § Pn ) ’ Q

to be false, that is, just in case this conditional sentence is a logical truth. Thus,

one way to verify the tautological validity of an argument in propositional

logic, at least in theory, is to construct a truth table for this sentence and see

whether the ¬nal column contains only true. In practice, this method is not

very practical, since the truth tables quickly get too large to be manageable.

Remember

1. The following English constructions are all translated P ’ Q: If P

then Q; Q if P; P only if Q; and Provided P, Q.

2. Unless P, Q and Q unless P are translated ¬P ’ Q.

3. Q is a logical consequence of P1 , . . . , Pn if and only if the sentence

(P1 § . . . § Pn ) ’ Q is a logical truth.

Section 7.2

Biconditional symbol: ”

Our ¬nal connective is called the material biconditional symbol. Given any

sentences P and Q there is another sentence formed by connecting these by

Section 7.2

182 / Conditionals

means of the biconditional: P ” Q. A sentence of the form P ” Q is true if

and only if P and Q have the same truth value, that is, either they are both

true or both false. In English this is commonly expressed using the expression

if and only if. So, for example, the sentence Max is home if and only if Claire

if and only if

is at the library would be translated as:

Home(max) ” Library(claire)

Mathematicians and logicians often write “i¬” as an abbreviation for “if

i¬

and only if.” Upon encountering this, students and typesetters generally con-

clude it™s a spelling mistake, to the consternation of the authors. But in fact it

is shorthand for the biconditional. Mathematicians also use “just in case” as a

just in case

way of expressing the biconditional. Thus the mathematical claims n is even

i¬ n2 is even, and n is even just in case n2 is even, would both be translated

as:

Even(n) ” Even(n2 )

This use of “just in case” is, we admit, one of the more bizarre quirks of

mathematicians, having nothing much to do with the ordinary meaning of

this phrase. In this book, we use the phrase in the mathematician™s sense,

just in case you were wondering.

An important fact about the biconditional symbol is that two sentences

P and Q are logically equivalent if and only if the biconditional formed from

them, P ” Q, is a logical truth. Another way of putting this is to say that

P ” Q is true if and only if the fol sentence P ” Q is logically necessary.

So, for example, we can express one of the DeMorgan laws by saying that the

following sentence is a logical truth:

¬(P ∨ Q) ” (¬P § ¬Q)

This observation makes it tempting to confuse the symbols ” and ”. This

” vs. ”

temptation must be resisted. The former is a truth-functional connective of

fol, while the latter is an abbreviation of “is logically equivalent to.” It is

not a truth-functional connective and is not an expression of fol.

Semantics and the game rule for ”

The semantics for the biconditional is given by the following truth table.

P Q P”Q

T

t t

truth table for ” F

t f

F

f t

T

f f

Chapter 7

Biconditional symbol: ” / 183

Notice that the ¬nal column of this truth table is the same as that for

(P ’ Q) § (Q ’ P). (See Exercise 7.3 below.) For this reason, logicians often

treat a sentence of the form P ” Q as an abbreviation of (P ’ Q) § (Q ’ P).

Tarski™s World also uses this abbreviation in the game. Thus, the game rule game rule for ”

for P ” Q is simple. Whenever a sentence of this form is encountered, it is

replaced by (P ’ Q) § (Q ’ P).

Remember

1. If P and Q are sentences of fol, then so is P ” Q.

2. The sentence P ” Q is true if and only if P and Q have the same truth

value.

Exercises

For the following exercises, use Boole to determine whether the indicated pairs of sentences are tauto-

logically equivalent. Feel free to have Boole build your reference columns and ¬ll them out for you. Don™t

forget to indicate your assessment.

7.1 7.2

A ’ B and ¬A ∨ B. ¬(A ’ B) and A § ¬B.

‚ ‚

7.3 7.4

A ” B and (A ’ B) § (B ’ A). A ” B and (A § B) ∨ (¬A § ¬B).

‚ ‚

7.5 7.6

(A § B) ’ C and A ’ (B ∨ C). (A § B) ’ C and A ’ (B ’ C).

‚ ‚

7.7 7.8

A ’ (B ’ (C ’ D)) and A ” (B ” (C ” D)) and

‚ ‚

((A ’ B) ’ C) ’ D. ((A ” B) ” C) ” D.

7.9 (Just in case) Prove that the ordinary (nonmathematical) use of just in case does not express

a truth-functional connective. Use as your example the sentence Max went home just in case

Carl was hungry.

7.10 (Evaluating sentences in a world) Using Tarski™s World, run through Abelard™s Sentences, eval-

‚ uating them in Wittgenstein™s World. If you make a mistake, play the game to see where you

have gone wrong. Once you have gone through all the sentences, go back and make all the false

ones true by changing one or more names used in the sentence. Submit your edited sentences

as Sentences 7.10.

Section 7.2