1.6: Rules of Formation and Rules of Valuation

We can summarize many important points discussed so far by giving explicit rules which tell us what counts as a sentence of sentence logic and how to determine the truth values of compound sentences if we are given the truth values of the components.

Formation Rules

1. Every capital letter 'A', 'B', 'C' ... is a sentence of sentence logic. Such a sentence is called an Atomic Sentence or a Sentence Letter.
2. If X is a sentence of sentence logic, so is (~X), that is, the sentence formed by taking X, writing a '~' in front of it, and surrounding the whole by parentheses. Such a sentence is called a Negated Sentence.
3. If X and Y are sentences of sentence logic, so is (X&Y), that is, the sentence formed by writing X, followed by '&', followed by Y, and surrounding the whole with parentheses. Such a sentence is called a Conjuncton, and X and Y are called its Conjuncts.
4. If X and Y are sentences of sentence logic, so is (XvY), that is, the sentence formed by writing X, followed by 'v', followed by Y, and surrounding the whole with parentheses. Such a sentence is called a Disjunction, and X and Y are called its Disjuncts.
5. Until further notice, only expressions formed by using rules i) through iv) are sentences of sentence logic.

If you wonder why I say "until further notice," I want you to digest the present and some new background material before I introduce two new connectives, corresponding to the expressions "If . . . then" and "if and only if." When I introduce these new connectives, the formation rules will need to be extended accordingly.

As I explained earlier, we agree to cheat on these strict rules in two ways (and in these two ways only!). We omit the outermost parentheses, and we omit parentheses around a negated sentence even when it is not the outermost sentence, because we agree to understand '~' always to apply to the shortest full sentence which follows it.

I should also clarify something about formation rule i). In principle, sentence logic can use as many atomic sentences as you like. It is not limited to the 26 letters of the alphabet. If we run out of letters, we can always invent new ones, for example, by using subscripts, as in '$$A_1$$' and '$$C_{37}$$'. In practice, of course, we will never need to do this.

Rules of Valuation

1. The truth value of a negated sentence is t if the component (the sentence which has been negated) is f. The truth value of a negated sentence is f if the truth value of the component is t.
2. The truth value of a conjunction is t if both conjuncts have truth value t. Otherwise, the truth value of the conjunction is f.
3. The truth value of a disjunction is t if either or both of the disjuncts have truth value t. Otherwise, the truth value of the disjunction is f.

Note that these rules apply to any compound sentence. However, they only apply if somehow we have been given a truth value assignment to the atomic sentence letters. That is, if we have been given truth values for the ultimate constituent atomic sentence letters, then, using the rules of valuation, we can always calculate the truth value of a compound sentence, no matter how complex. Once again, this is what we mean when we say that the connectives are truth functional.

How does one determine the truth value of atomic sentences? That's not a job for logicians. If we really want to know, we will have to find out the truth value of atomic sentences from someone else. For example, we'll have to consult the physicists to find out the truth value of "light always travels at the same speed." As logicians, we only say what to do with truth values of atomic constituents once they are given to us. And when we do truth tables, we do not have to worry about the actual truth values of the atomic sentence letters. In truth tables, like those in the following exercises, we consider all possible combinations of truth values which the sentence letters could have.

The truth table definitions of the connectives give a graphic summary of these rules of valuation. I'm going to restate those truth table definitions here because, if truth be told, I didn't state them quite right. I gave them only for sentence letters, 'A' and 'B'. I did this because, at that point in the exposition, you had not yet heard about long compound sentences, and I didn't want to muddy the waters by introducing too many new things at once. But now that you are used to the idea of compound sentences, I can state the truth table definitions of the connectives with complete generality.

Suppose that X and Y are any two sentences. They might be atomic sentence letters, or they might themselves be very complex compound sentences. Then we specify that:

Truth table definition of '~'

 Case X ~X 1 t f 2 f t

Truth table definition of '&'

 Case X Y X&Y 1 t t t 2 t f f 3 f t f 4 f f f

Truth table definition of 'v'

 Case X Y XvY 1 t t t 2 t f t 3 f t t 4 f f f

The difference between my earlier, restricted truth table definitions and these new general definitions might seem a bit nitpicky. But the difference is important. You probably understood the intended generality of my first statement of the truth table definitions. However, a computer, for example, would have been totally confused. Logicians strive, among other things, to give very exact statements of everything. They enjoy exactness for its own sake. But exactness has practical value too, for example, when one needs to write a program that a computer can understand.

Exercise $$\PageIndex{1}$$

1-5 Which of the following expressions are sentences of sentence logic and which are not?

a) A&~B

b) A~&B

c) Gv(~B&~H))

d) A&(C&~(DvH))

e) (A&B)v(C&D)

f) (AvB)&CvD

1~6. Construct a complete truth table for each of the following sentences. The first one is done for you: 1~7. Philosopher's problem: Why do I use quotation marks around sentences, writing things like and '~(CV~ A)' but no quotation marks about boldface capital letters, writing X, Y, XVY, etc. when I want to talk about sentences generally?

 A B ~B ~BvA t t f t t f t t f t f f f f t t

a) ~BvA

b) ~(BvA)

c) (QvT)&(~Qv~T)

d) (D&~G)v(G&D)

e) Av(~BvC)

f) Kv[~P&(~PvM)]

g) [(Dv~~B)&(Dv~B)]&(DvB)

h) L&[Mv[~N&(~Mv~L)]}

1-7. Philosopher's problem: Why do I use quotation marks around sentences, writing things like

'B'

and

'~(Cv~A)'

but no quotation marks about boldface capital letters, writing

X,Y,XvY,etc.

when I want to talk about sentences generally?

This section has also illustrated another thing worth pointing out. When I talked about sentences generally, that is, when I wanted to say something about any sentences, X and Y, I used boldface capital letters from the end of the alphabet. I'm going to be doing this throughout the text. But rather than dwell on the point now, you will probably best learn how this usage works by reading on and seeing it illustrated in practice.

Contributors

• Paul Teller (UC Davis). The Primer was published in 1989 by Prentice Hall, since acquired by Pearson Education. Pearson Education has allowed the Primer to go out of print and returned the copyright to Professor Teller who is happy to make it available without charge for instructional and educational use.