[2.] Symbolizing in Sentential Logic

 

[2.1.] “Atomic and Compound Sentences.”

 

Examples, p.20:

·         All three of these sentences are compound sentences: “[s]entences built from shorter sentences by means of sentence connectives such as ‘and’ or ‘or’” (p.20)

·         If a sentence is not compound, it is an atomic sentence (a.k.a. simple sentence).

·         Each of these three examples is a compound sentence composed of two atomic sentences, plus a sentence connective: “A term or phrase used to make a larger sentence from [one or] two smaller ones.” (p.51)

·         In example #1, the sentence connective is “and.”

·         In example #2, the sentence connective is “or.”

·         In example #3, the sentence connective is “if … then …”

 

The first part of the system of symbolic logic we will cover (in chapters two through five) is sentential logic (a.k.a. sentence calculus, or propositional logic): “The logic that deals with relationships holding between sentences, atomic or compound, without dealing with the interior structure of atomic sentences.” (p.52)

 

 

[2.2.] “Truth Functions.”

 

Some (but not all) sentence connectives express truth functions.

 

Your textbook introduces the concept of a truth function by making an analogy between truth functions and arithmetical functions, like addition, subtraction, multiplication and division. These mathematical functions are “like number-transforming machines” (21) that work in exactly the same way every time they are used. When you “put numbers into” these functions, you “get numbers out” of them. For example, when you put the numbers 2 and 5 into the addition function, you get 7; when you put them into the multiplication function, you get 10; and so on.

 

Notice these two traits of mathematical functions:

·         the output of the function is completely determined by the input. Given the same input into the function addition, you will always get the same output (for example, 2 added to 5 always equals 7). For this reason…

·         you get the same output every time you use the same input.

 

A truth-function works the same way, except that the input and the output are not numbers; rather, they are truth values.

 

In the system of symbolic logic we will be learning in this class, there are only two truth values: true and false. In other words, every sentence in this system will either take the value “true” or take the value “false.” There is no third truth value in this system.

 

So in our system of logic, each truth function can take only true and false as inputs and can yield only true or false as outputs.

 

 

[2.3.] “Conjunctions.”

 

The first truth-function we will examine is conjunction.

 

This truth function known as conjunction is typically indicated in English by the word “and.”

 

Consider these two atomic sentences:

 

“Obama is President.”

“Biden is Vice President.”

 

Each of these sentences has the truth-value true. If you connect these two atomic sentences with the sentence connective “and,” you get the following conjunction:

 

“Obama is President and Biden is Vice President.”

 

This new, compound sentence is true; in other words, it has the truth value true.

 

Whenever you use conjunction to create a compound sentence out of two true sentences, the compound sentence you create will be true. In other words, when you put two trues into the truth-function of conjunction, the output will be true.

 

But consider this compound sentence:

 

“Romney is President and Biden is Vice President.”

 

This compound sentence has the truth value false. This is because one of the atomic sentences it contains is false. Whenever you use conjunction to create a compound sentence out of two sentences, one of which is true and the other of which is false, the compound sentence you create will be false.

 

And finally, consider:

 

“Romney is President and Tom Cruise is Vice President.”

 

This conjunction has the truth value false -- because both of the atomic sentences it contains are false. Whenever you use conjunction to create a compound sentence out of two false sentences, the compound sentence you create will be false.

 

--

 

Note that the word “conjunction” is used to refer to both (1) a truth-function and (2) the type of compound sentence formed by that truth function. So the three compound sentences displayed above are all called “conjunctions.”

 

 

[2.4.] Symbolizing Conjunctions.

 

Consider once again the following conjunction:

 

“Romney is President and Tom Cruise is Vice President.”

 

To symbolize this compound sentence, first choose a different upper-case letter to represent each atomic sentence:

 

Symbolize “Romney is President” with the letter “R”

 

Symbolize “Tom Cruise is Vice President.” with the letter “C”

 

Finally, symbolize the truth-function called conjunction with “·“

 

So the entire compound sentence is symbolized:

 

R · C

 

The dot symbol is a truth-functional operator (a symbol that represents a truth-function and that “operates” on one or more sentences to form a new, compound sentence). Each of the five truth functions in our system will be represented by its own symbol. We will learn each of them as we go along.

 

Conjunction is a truth function. Because of this, if we are given the truth values of R and C (i.e., if we are told whether it is true that Romney is President and whether it is true that Cruise is Vice President), we can figure out the truth value of the conjunction R · C.

 

The following table demonstrates all possible combinations of truth values for the sentences R and C (see p.24):

 

 

The only case in which the conjunction R · C is true is when both of its conjuncts are true (the sentences connected by the dot to form a conjunction are called conjuncts); in all other cases, the conjunction is false. As we’ll see soon, the same is true for all conjunctions.

 

 

 

 

 

[2.5.] Difficulties with “And.”

 

Consider this conjunction: “Amy and Bert are cheaters.” This can be “translated” into a sentence that is more obviously a compound sentence: “Amy is a cheater and Bert is a cheater.” If we represent “Amy is a cheater” with “A” and “Bert is a cheater” with “B”, then we can symbolize this sentence in a way similar to the Romney/Cruise example: “A · B”.

 

But the following sentence is not a conjunction: “Amy and Bert are siblings.” (Compare this to the example on p.23: “Art and Betsy are lovers.”) What this sentence is saying is that Amy and Bert are each other’s siblings, i.e., that Amy is Bert’s sister and that Bert is Amy’s brother. This is different than the combined meaning of “Amy is a sibling” and “Bert is a sibling.” Even though it contains the word “and,” this sentence is not a conjunction. It is an atomic sentence that asserts that a particular relationship holds between Amy and Bert.

 

At this point, the best that we can do to represent this sentence is to represent it with a single letter, like “S.” (Later in the semester, we will learn how better to represent this sort of atomic sentence.)

 

EXAMPLES p.24: All of the following words/phrases are used to join two atomic sentences to form a conjunction:

·         and

·         both...and

·         but [as your book indicates, some meaning is lost when “but” is symbolized with the dot; but that meaning is not relevant to the arguments in which the word might occur, so it is fine to symbolize it that way]

·         in spite of the fact that

·         although

·         while

·         also

 

 

EXERCISE 2-1 (p.24)

§  Complete ALL; check your answers to the even-numbered problems against the back of the book; we will go through the odd numbered problems at the beginning of the next class.

 

 

 

[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 so that the truth-table says something, not just about that specific sentence, but about any conjunction whatsoever:

 

 

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)

 

 

 

 

[2.8.] “Parentheses and Brackets.”

 

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

A · B               =          Amy and Bert love logic.

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

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

~A · ~B           =          Amy doesn’t love logic, and neither does Bert (i.e., neither Amy 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 Amy and Bert love logic.

 

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

 

“~(A · B)” means that not both Amy 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 Amy 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 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 Amy, Bert and Cole all love logic.”

 

~{A · ~[B · (C · ~D)]}            =          “It is not the case both that Amy 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 that are connected by truth-functional operators 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.51)

 

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.30 [these should have come at the end of section 2.7]

 

Exercise 2-2 (p.30)

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

 

 

 

Stopping point for Tuesday January 17. For next time, complete exercises 2-1 and 2-2, then read ch.2 secs. 10-14 [this is the remainder of ch.15 minus sec.15, which we won’t be covering].