[2.9.] “Disjunctions.”

 

Disjunction is the third truth-function we’ll examine:

 

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...

 

Examples:

 

            A Ú B               =                      Amy loves logic or Bert loves logic.

 

C Ú D               =                      You’re either crazy or demented.

 

The two sentences connected by a wedge are called disjuncts.

 

 

[2.9.1.] Exclusive Disjunction vs. Inclusive Disjunction.

 

There are two different senses of the English word “or”:

 

exclusive	inclusive (a.k.a. nonexclusive)
“p or q” means   Either p or q, but not both.       E.g., “I will get an ‘A’ or a ‘B’ on the test.”   Since this is an exclusive “or,” the speaker is implying that she will not get both an A and a B on her test. She means that she will get one or the other, not both.	“p or q” means   Either p, or q, or both p and q   (This is sometimes expressed by “and/or”)   E.g., “Tonight I’ll drink beer or wine.”   Since this is the inclusive “or”, the speaker may have both wine and beer. The sentence is not made false by her having both wine and beer.

 

In sentential logic, the wedge symbolizes “or” in the INCLUSIVE sense, not in the exclusive sense. So “p Ú q” is true when “p” and “q” are both true. “p Ú q” is false only when both “p” and “q” are false.

 

This is demonstrated by the truth-table definition for disjunction:

 

 

Notice that the only case in which a disjunction is false is when both of its disjuncts are false; in all other cases, the disjunction is true.

 

But there is still a way to symbolize an English-language sentence containing the exclusive “or”:

 

I will get an “A” or a “B” on the test.

 

A               =          I will get an ‘A’.

B                =          I will get a ‘B’.

A Ú B        =          I will get an ‘A’ or I will get a ‘B’ (inclusive).

 

To express the exclusive sense, we need to say:

 

(A Ú B) · ~(A · B)    which means:

 

“I will get an ‘A’ or a ‘B’, and I will not get both an ‘A’ and a ‘B’.”

 

In other words, we need to translate from our initial English-language statement:

 

I will get an ‘A’ or a ‘B’ (exclusive).

 

into a new English sentence that makes it explicit that we mean “or” in the exclusive sense:

 

I will get an ‘A’ or a ‘B’, but I won’t get both an ‘A’ and a ‘B’.

 

and then translate this into symbolization:

 

(A Ú B) · ~(A · B)

 

[You should be able to answer these questions:

§  what is the main connective in this sentence?

§  how many component sentences does that connective have in its scope?

§  which component sentence(s) is it / are they?]

 

When you are translating English disjunctions into symbolization, you should translate them as inclusive disjunctions unless it is obvious that the exclusive “or” is intended.

 

 

Exercise 2-3 (pp.32-33)

·         Complete all problems. Check your answers to even-numbered questions to those in the back of the book. We will go over the odd-numbered problems at the beginning of the next class.

·         Make sure to think about whether each disjunction is supposed to be an inclusive or an exclusive disjunction.

 

 

[2.10.] “‘Not Both’ and ‘Neither ... Nor’”.

 

[2.10.1.] “Not Both.”

 

As we saw, it is possible to deny one conjunct in a conjunction...

 

“Amy loves logic but Bert does not.”  =          A · ~B

 

and it is possible to deny both conjuncts...

 

“Amy does not love logic, and neither does Bert.”      =          ~A · ~B

 

But sometimes what we mean to deny is the entire conjunction:

 

“It is not the case that both Amy and Bert love logic.”

 

The point of these statements is that one of them might love logic, but it is false that both of them do: either Amy does not, or Bert does not. So this sentence is false only if BOTH Amy and Bert love logic; otherwise it is true.

 

We can symbolize this in two different ways:

 

~(A · B)                       or                     ~A Ú ~B

 

These are equivalent symbolizations; each of them is an accurate translation.

 

But notice that

 

~A · ~B

 

is not an accurate symbolization of “It is not the case that both Amy and Bert love logic.” [See the next section…]

 

 

[2.10.2.] “Neither Nor.”

 

A different example: “Neither Amy nor Bert loves logic.”

 

You may be tempted to symbolize this using the wedge, perhaps because “nor” sounds like “or.” But this English sentence is not a disjunction, and should not be symbolized as such.

 

Rather it is a conjunction of two negations and can be symbolized as:

 

~A · ~B

 

I.e., “It is not the case that Amy loves logic, and it is not the case that Bert loves logic.” For this sentence to be true, it must be the case that neither of them loves logic.

 

We can also accurately symbolize this sentence as:

 

~(A Ú B)

 

This reads: “It is not the case that either Amy or Bert loves logic.” Again, for this to be true, it must be the case that neither of them loves logic. These two symbolizations

 

~A · ~B           and      ~(A Ú B)

 

are both accurate and equally good.

 

 

Notice that

 

            ~A Ú ~B

 

is not an accurate symbolization of “It is not the case that either Amy or Bert loves logic.” it means that either Amy doesn’t love logic or Bert doesn’t love logic, which would be true if Amy but not Bert loved logic. It would also be true if Bert but not Amy loved logic.

 

Examples (p.34)

 

 

 

Exercise 2-4 (pp.34-35)

·         Complete all problems. Check your answers to even-numbered questions to those in the back of the book. We will go over the odd-numbered problems at the beginning of the next class.

 

 

[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.51)

 

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: conditionals are frequently 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 many conditionals in ordinary language are not truth-functional. In other words, for many ordinary-language “if-then” sentences, you can’t always tell the truth-value of the entire sentence 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.36):

 

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 must be 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 (frequently non truth-functional) conditionals in ordinary language and (always truth-functional) conditionals in sentential logic, conditionals in sentential logic are not just called conditionals; they are given a more specific name: “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.38 -- 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 Joe Biden is Vice President [T], then DNA takes the form of a double helix [T].”

·         “If Lady Gaga 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.] “‘Only if’ and ‘Unless’”.

 

[2.12.1.] “Only if.”

 

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

 

Example:

 

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

 

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

 

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

 

 

[2.12.2] ‘Unless.’

 

The most intuitive translation of “unless” into an English form that can be easily translated into our system is: if not.

 

p unless q   =   p if not-q

 

Once you replace “unless” with “if not”, you can then shift the “if” portion to the beginning of the sentence…

 

p if not-q  =   if not-q, then p

 

From here, the symbolization is easy:

 

if not-q, then p   =   ~q É p

 

Now, notice that the symbolization (which basically means “if q is not true, then p is true”) is equivalent to “either q is true or p is true”: q Ú p … which is equivalent to p Ú q. So

 

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

 

 

Example:

 

“Romney wins unless Palin speaks.” This means--

 

“If Palin does not speak, then Romney wins.” In other words, the only condition in which Romney does not win is if Palin speaks. Palin not speaking is a sufficient condition of Romney winning. So if Palin does not speak, that is a guarantee that Romney will win.

 

So “Romney wins unless Palin speaks” should be symbolized  “~P É M” (where “P” symbolizes “Palin speaks” and “M” symbolizes “Romney wins”).

 

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

 

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 (i.e., if there is not) some clear indication in what you are translating that the speaker or writer intends to express a biconditional.

 

EXAMPLES (p.41)

 

EXERCISE 2-5 (p.41)

·         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 begin 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.44-45

 

Exercise 2-6 (pp.45-46)

·         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 Thursday January 19. For next time:

·         complete exercises 2-3, 2-4, 2-5, 2-6

·         read ch.3:1-2 (pp.53-62)