[2.11.] “Material Conditionals.”

 

The fourth connective we’ll learn symbolizes the English language phrase “If ... then .....”.

 

E.g., “If Murphy has a bone, then Murphy is happy.”

 

A sentence of this form is called a conditional (it is also sometimes called a hypothetical): “A compound sentence that expresses an ‘If ... then’ relationship between its component sentences.” (p.47)

 

In a conditional, the component sentence between the “if” and the “then” is the antecedent; the component sentence after the “then” is called the consequent. In the example, “Murphy has a bone” is the antecedent and “Murphy is happy” is the consequent.

 

Important: a conditional can be expressed in English without using the word “then”

§         “If Murphy has a bone, Murphy is happy.”

and without using the words “if” or “then”:

§         “As long as Murphy has a bone, he is happy.”

§         “Murphy is happy, assuming that he has a bone.”

 

The symbol for the conditional is the horseshoe:   É

 

So to represent the sentence about Murphy in our system, we write:         B É H

 

(where “B” is a sentence constant representing “Murphy has a bone” and “H” is a sentence constant representing “Murphy is happy.”)

 

In our system of sentential logic, the conditional is truth-functional: the truth-value of a given material conditional is always completely determined by the truth-values of the atomic sentences of which it is composed (the antecedent and the consequent):

 

 

In our system, a conditional is false exactly when its antecedent is true and its consequent is false; in all other cases, the conditional is true.

 

While the connectives we have looked at so far (the dot, the tilde and the wedge) pretty closely capture the meanings of their English-language counterparts (“and,” “not” and “or”), the horseshoe captures only some of the meaning of the words “if...then...”

 

This is because the conditional in sentential logic is truth-functional, but most conditionals in ordinary language are not truth-functional. In other words, for most ordinary-language conditionals, you can’t always tell the truth-value of the entire conditional just from knowing the truth value of its antecedent and consequent.

 

For instance, consider “If it rains tonight, then I will stay home.” (Compare the Notre Dame vs. Miami example on p.38):

 

It rains tonight.	I will stay home.	If it rains tonight, then I will stay home.
T	T	T -- if it actually does rain, and I actually do stay home, then the conditional sentence I uttered is true.
T	F	F -- if it actually does rain, but I go out, then the conditional sentence I uttered is false.
F	T	??? -- if it doesn’t rain tonight and I stay home, it’s not clear whether the conditional is true or false (since it’s unknown what would have happened had it rained)
F	F	??? -- if it doesn’t rain tonight and I go out, it’s not clear whether the conditional is true or false (since it’s unknown what would have happened had it rained)

 

So we can’t always tell whether a conditional in ordinary English is true or false, even if we know the truth values of its antecedent and consequent. This is what is meant by saying that conditional sentences in ordinary English are not always truth-functional.

 

But in sentential logic, all connectives are truth-functional. We must be able to assign a “T” or an “F” to every conditional, based on its component sentences. So in the two problem cases above, where the antecedent is false, we have to assign either a “T” or an “F”.

 

Although it is not obviously correct to say that the conditional in this example is true in the case that it doesn’t rain, it seems less accurate to say that it is false than to say that it is true. It is this sort of thinking that grounds the assignment of “T” to conditionals with false antecedents:

 

 

Because there is this difference in meaning between (mostly non truth-functional) conditionals in ordinary language and (always truth-functional) conditionals in sentential logic, conditionals in sentential logic are called material conditionals. The truth-function expressed by a material conditional is called material implication.

 

truth-function	symbol	name of symbol	English equivalent(s)
conjunction	·	dot	And
negation	~	tilde	not ; it is not the case that
disjunction	Ú	wedge (a.k.a vel)	or ; either...or...
material implication	É	horseshoe	if …, then …

 

EXAMPLES p.40 -- We won’t go through these in class. Look at them closely to be sure you understand that all of them express the same conditional and can all be symbolized the same way.

 

 

[2.11.1.] Oddities of Material Implication.

 

Because material implication is truth-functional, it results in some oddities:

 

Any material conditional with a false antecedent or a true consequent is true, even if the antecedent and the consequent have nothing to do with one another: e.g.,

§         “If Dick Cheney is Vice President [T], then DNA takes the form of a double helix [T].”

§          “If George W. Bush is a robot [F], then Atlanta is the capitol of Georgia [T].”

§         “If the Nazis won WWII [F], then Cuba is a representative democracy [F].”

All three of these conditionals are true, if they are interpreted as material conditionals.

 

However, this rarely has an effect on our translations of actual, ordinary language arguments into symbolization. This is because people rarely assert conditionals the antecedent and consequent of which have nothing to do with each other, and they rarely assert conditionals with antecedents they know to be false.

 

 

[2.12.] “Material Biconditionals.”

 

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

 

 

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

 

material equivalence: 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 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”[1]   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

 

 

[2.13.] “‘Only if’ and ‘Unless’”.

 

[2.13.1.] “Only if.”

 

p only if q         =          if p, then q                    =          p É q

 

Example:

 

“McCain wins only if he takes Florida.” This means--

 

“If McCain wins, then he takes Florida.” In other words, there is no way that McCain will win without taking Florida. McCain taking Florida is a necessary condition of McCain winning. So if McCain does in fact win, that is a guarantee that he has actually taken Florida.

 

So “McCain wins only if he takes Florida” should be symbolized “W É F” (where “W” symbolizes “McCain wins” and “F” symbolizes “McCain takes Florida.”

 

 

[2.13.2] ‘Unless.’

 

p unless q         =          if not-q, then p              =          ~q É p    =   q v p    =   p v q

 

Example:

 

“McCain wins unless Clinton runs.” This means--

 

“If Clinton does not run, then McCain wins.” In other words, the only condition in which McCain does not win is if Clinton runs. Clinton not running is a sufficient condition of McCain winning. So if Clinton does not run, that is a guarantee that McCain will win.

 

So “McCain wins unless Clinton runs” should be symbolized  “~C É M” (where “C” symbolizes “Clinton runs” and “M” symbolizes “McCain wins”).

 

“Unless” can also be directly translated with the wedge (Ú), although this might seem less obviously correct than translating it with the horseshoe. “McCain wins unless Clinton runs” is equivalent to “Either McCain wins or Clinton runs” and so can also be symbolized as “M Ú C”.

 

Note that sometimes “unless” and “only if,” when used in everyday English, are intended by their speakers to express a biconditional, rather than a conditional. As your textbook notes, you should translate these phrases as expressing conditionals according to the above guidelines, unless there is some clear indication in what you are translating that the speaker or writer intends to express a biconditional.

 

EXAMPLES (p.43)

 

EXERCISE 2-5 (p.43)

·         Complete all of these and check your answers to the even-numbered problems to the answers in the back of the book. We will review the odd-numbered problems at the beginning of the next class.

 

 

[2.14.] “Symbolizing Complex Sentences.”

 

So far we have symbolized only relatively simple sentences. In this section we began to symbolize more complex sentences. Let’s start with this example:

 

“Either Art and Bob are bachelors, or Carl and Dennis are widowers.”

 

Your textbook recommends the following strategy:

 

Step 1: Identify the main connective of the sentence to be symbolized.

 

tips:

1.       Sentences that begin with “If” are usually conditionals, so the main connective will be the horseshoe.

2.       Sentences that begin with “Either” are usually disjunctions, so the main connective will be the wedge.

3.       Pay attention to punctuation. A single comma, or a single semi-colon, frequently marks the division between the component sentences that will go on either side of the main connective.

 

The example sentence illustrates the second and third tips: it should be symbolized as a disjunction with the wedge as its main connective, and the disjuncts are separated by the comma:

 

(____) Ú (____)

 

Step 2: Translate each of the individual component sentences.

 

In our example, the two component sentences are:

 

 “Art and Bob are bachelors”     =                      A · B

 

“Carl and Dennis are bachelors “ =                     C · D

 

Plugging these into the form identified in the previous step gives us:

 

(A · B) Ú (C · D)

 

 

Step 3: To double-check your symbolization, translate it back into natural language.

 

This will help you tell whether your symbolization is saying the same thing (expressing the same meaning) as the sentence of which it is supposed to be a translation.

 

 

EXAMPLES, pp.46-47

 

Exercises 2-6, 2-7, 2-8 (pp.47-50).

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

 

 

Stopping point for Wednesday January 23. For next time, complete exercises 2-5, 2-6, 2-7, and 2-8, then read chapter 3 secs. 1-2 (pp.55-64).