PHIL 4160: Symbolic Logic
Dr. Robert Lane
Lecture Notes: Thursday January 16, 2014

 

[2.6] “Variables and Constants.

 

The letters we’ve used in previous sections have all been abbreviations for specific sentences:

 

R = “Romney is President.”

C = “Tom Cruise is Vice President”

etc.

 

These upper-case letters are sentence constants: “A capital letter abbreviating an English sentence, atomic or compound.” (51) They are called “constants” because, once it has been established, in a given context, that a capital letter stands for a specific sentence, then the meaning of that letter is fixed (constant, unchanging) within that context.

 

But frequently, we want to be able to talk about, not some specific sentence or sentences, but sentences in general. To represent sentences in general, we will use lower-case letters, beginning with “p” and continuing through “z”: p, q, r ... z. These are sentence variables: placeholders for abbreviations of individual sentences.[1]

 

Using such placeholders, we can recreate the truth-table we made for the sentence R · C (“Romney is President and Cruise is Vice President”) so that the truth-table says something, not just about that specific sentence, but about any conjunction whatsoever:

 

p

q

p · q

T

T

T

T

F

F

F

T

F

F

F

F

 

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

 

This sort of table, the top row of which contains sentence variables rather than sentence constants, is called a truth-table definition: it defines conjunction by showing all of the possible combinations of truth values for the sentences joined by ·, along with the corresponding truth value of the entire conjunction.

 

 

[2.7.] “Negations.”

 

The second truth-functional operator of our system:               ~

 

·         is called the tilde;

·         symbolizes the function called negation;

·         and is read: “It is not the case that.”

 

Let’s continue to let “C” be a sentence constant for “Tom Cruise is Vice President.”

 

In that case, “~C” represents “It is not the case that Tom Cruise is Vice President” (which means the same thing as “Tom Cruise is not Vice President”).[2]

 

Examples: p.28

 

Unlike conjunction, which requires two sentences to create a compound sentence, negation requires only one sentence. (The tilde is still called a “connective,” even though it does not literally connect two sentences.)

 

In other words, the truth-function of negation takes only one truth value as input. Sometimes this is put as follows: negation operates on only individual sentences, unlike conjunction, which operates on two sentences at a time.

 

“~” is the only operator in our system of sentential logic that forms a new sentence by combining with only one sentence. All the other operators combine with two sentences to form a new sentence.

 

The truth-table definition of negation (p.27)

 

p

~p

T

F

F

T

 

 

[2.8.] “Parentheses and Brackets.”

 

Let “A” represent “Anne loves logic” and “B” represent “Bert loves logic.” We can combine negation and conjunction to form longer sentences:

 

A · B               =          Anne and Bert love logic.

~A · B             =          Anne does not love logic, but Bert does.

A · ~B             =          Anne loves logic, and Bert doesn’t.

~A · ~B           =          Anne doesn’t love logic, and neither does Bert (i.e., neither Anne nor Bert loves logic.)

 

Sometimes you will want to apply negation, not just to the individual conjuncts in a conjunction, but to the entire conjunction itself. For example, you might want to apply negation to “A · B”. But simply adding a tilde to the front of this sentence...

 

~A · B

 

won’t do, since that only applies negation to “A”.

 

To indicate that the tilde applies to the entire conjunction, you must use parentheses:

 

~(A · B)           =          It is not the case that both Anne and Bert love logic.

 

Notice that “~(A · B)” has a different meaning than “~A · ~B”:

 

“~(A · B)” means that not both Anne and Bert love logic (this leaves open the possibility that one of them does love logic) [we will come back to this example soon];

 

“~A · ~B” means that neither Anne nor Bert loves logic (this does not leave open the possibility that one of them loves logic).

 

As sentences become more complicated, you may need to add more grouping devices: first brackets, and then braces. Let “C” stand for “Cole loves logic” and “D” stand for “Drew loves logic”:

 

~[(A · B) · C]              =          “It is not the case that Anne, Bert and Cole all love logic.”

 

~{A · ~[B · (C · ~D)]}            =          “It is not the case both that Anne loves logic and that it is not the case that Bert loves logic and that Cole does but Drew does not.”

 

 

Notice these two points:

1.      You should never add parentheses, brackets or braces around an entire sentence: in “(A · B)”, the parentheses are unnecessary and should be left off.

2.      A tilde operates on only the shortest complete sentence that it precedes; for this reason, you should not add parentheses, brackets or braces to demarcate the scope of a tilde when that tilde is operating on a single sentence constant. In “~(A) · B”, the parentheses are not necessary and should be left off.

 

A sentence that violates either of these rules of grammar is said not to be well-formed. A sentence that does not violate any of the grammatical rules of our system is a well-formed formula (wff).

 

Some new terms to help you think about all this:

 

component sentences: within a compound sentence, the shorter sentences upon which truth-functional operators operate in order to make longer sentences; a component sentence can be either atomic or compound.

 

In “A · B” both component sentences are atomic.

 

In “(A · B) · C” the component sentence “A · B” is compound and the component sentences “A”, “B” and “C” are atomic.

 

scope: “The scope of an operator is the component sentence or sentences that the operator operates on. The negation operator, ‘~’, operates on a single component sentence. All of the other operators operate on two component sentences.” (p.53)

 

in the symbolization...

 

~A

§  the scope of the tilde is “A”

A · B

§  the scope of the dot is “A” and “B”

~A · ~B

§  the scope of the first tilde is “A”

§  the scope of the second tilde is “B”

§  the scope of the dot is “~A” and “~B”

~(A · B)

§  the scope of the dot is “A” and “B”

§  the scope of the tilde is “(A · B)”

 

main connective: “the main connective of a sentence is the truth-functional connective whose scope encompasses the entire remainder of the sentence.” (p.29) It is the connective that has the widest scope in the sentence. IT IS A MUST THAT YOU LEARN TO PICK OUT THE MAIN CONNECTIVE OF A GIVEN SENTENCE.

 

Examples: p.29

 

Exercise 2-2 (p.30)

·         Complete all of these problems; we’ll go through them at the beginning of the next class.

 

 

[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               =                      Anne 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., “Today I’ll drink Coke or Pepsi.”

 

Since this is the inclusive “or”, the speaker may have both Coke and Pepsi. The sentence is not made false by her having both drinks.

 

 

 

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:

 

p

q

p Ú q

T

T

T

T

F

T

F

T

T

F

F

F

 

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 against 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...

 

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

 

and it is possible to deny both conjuncts...

 

“Anne 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 Anne 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 Anne does not, or Bert does not. So this sentence is false only if BOTH Anne and Bert love logic; otherwise it is true.

 

We can symbolize this in two different ways, each of which is equally good:

 

~(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 Anne and Bert love logic.” [See the next section…]

 

 

[2.10.2.] “Neither Nor.”

 

A different example: “Neither Anne 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 Anne 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 Anne 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 Anne or Bert loves logic.” it means that either Anne doesn’t love logic or Bert doesn’t love logic, which would be true if Anne but not Bert loved logic. It would also be true if Bert but not Anne 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.

 

 

--

Stopping point for Thursday January 16. For next time:

·         complete exercises 2-2, 2-3, and 2-4;

·         no new reading; read Ch.2:10-14 (originally assigned for today) if you haven’t already done so.

 

HEADS UP: Your first exam is scheduled for two weeks from today. The study guide is now on the class website. Even though we have fallen behind the original schedule, I don’t yet want to change the date for the first exam. I will decide by next Thursday (1/23) whether the date of the first test needs to be moved back.

 

Today or tomorrow I will update the online syllabus to reflect where we now stand with regard to assigned readings and exercises.

 

 



[1] The textbook gives the following, more technical definition: sentence variable: “A lowercase letter p through z used as a placeholder in a sentence form (or other linguistic form) such that if all the sentence variables in a sentence form are replaced by capital letters (sentence constants), then the resulting expression is a sentence.” (52)

 

[2] The difference between internal negation (e.g, “Cruise is not vice president”) and external negation (e.g., “It is not the case that Cruise if vice president”), although not important in this example, is sometimes very important. In the case of allegedly meaningless sentences, it can make a big difference. Haack, Philosophy of Logics, 1978 (p.35) gives the following example. “Virtue is not triangular” has been taken by some to be meaningless (just like “Virtue is triangular”), whereas “It is not the case that virtue is triangular” has been taken to be true (and therefore meaningful). It also makes a big difference with sentences having subject terms containing “some” or “all,” e.g., “Some of the horses are not gray” means something different than “It is not the case that some of the horses are gray.” The latter sentence implies that no horse is gray, while the former sentence does not imply that.

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