Given the truth-table definitions of the logical connectives we can now go on to calculate quite mechanically the truth-value of any complex formula of PL for any assignment of truth-values to its constituent atomic formulas.
This kind of semantic investigation reveals some interesting properties of certain formulas of PL. But, before we go on to look at some actual cases, let’s first consider how truth-tables are constructed via three simple examples.
First, consider the simplest possible kind of case, a negated sentence-letter. Suppose that we are interested in the overall truth-values of ~P, for example. How should we proceed to construct the truth-table? Well, the very first thing to do is to draw a horizontal line at least one and a half times as long as the formula involved and write out the formula to the right of that line. So, for example, in the case of ~P we write:
~ P
Next, we identify each and every sentence-letter involved in the formula, list these in alphabetical order to the left of the formula and separate the two with a vertical line. In the present case we only have P to worry about so:
We must now assign truth-values to the sentence-letter listed on the left. The number of truth-value assignments required, i.e. the number of rows of
P ~P
assignments to the sentence-letter(s) on the left, is just 2n where ‘n’ is the number of sentence-letters involved. So, if only one sentence-letter is involved, as in the present case, the number of rows of assignments is 21, i.e. 2.
Having done so, we construct the required number of truth-value assignments as follows: under the sentence-letter nearest the formula itself list the required number of truth-value assignments (in this case 2) beginning with T and alternating with F to the relevant number, e.g.:
To complete the truth-table (i.e. to complete the blank box under the formula itself) consider each complete truth-value assignment, i.e. each row of the table (in this case rows numbered 1–2) very carefully, then, in each case, begin by simply writing the truth-value assigned to the sentence-letter by that assignment under each and every occurrence of the relevant sentence-letter in the formula itself. So, for example, because row 1 assigns T to P, we enter T under the occurrence of P on that line in the formula, and because row 2 assigns F to P we enter F under P on that assignment, like so:
Next, we identify the main connective and highlight the column below it by enclosing it in a box with ‘m.c.’ written underneath. Thus, we emphasise that this column is the main column, i.e. the column in which the overall truth-value of the whole formula will be recorded under each assignment.
Finally, for each assignment, consider the truth-value assigned to P itself and then simply exploit the truth-table for the relevant connective (in this case ‘~’) to compute the overall values of the formula. According to the table for ‘~’, of course, when a formula is true its negation is false and when a formula is false its negation is true. Hence, we complete the
table by entering just those overall values under the main connective as follows:
We have considered the simplest possible case here and you should not be surprised to learn that your logic exam will contain slightly more complex cases. None the less, the procedure is very similar, if a little more articulated. For example, consider the formula (P & Q) → (Q v P). Again, we write the formula on a line, identify every sentence-letter involved in the formula and list these in alphabetical order on the left, separating the two with a vertical line:
Next, we assign truth-values to the sentence-letters listed on the left.
The number of truth-value assignments required is 2n where ‘n’ is the number of sentence-letters involved. So, when two sentence-letters are involved, as in the present case, the number of rows of assignments is 22, i.e. 4. Again, we begin with the sentence-letter nearest the formula itself (in this case Q) and assign truth-values beginning with T and alternating with F to the requisite number (in this case 4).
Under the remaining sentence-letter (in this case P) we again list the required number of truth-value assignments but this time beginning with two Ts and alternating with two Fs to the required number:
As before, we begin to complete the table simply by writing the truth-value assigned to the sentence-letter by that assignment under each and every occurrence of the relevant sentence-letter in the formula itself. So, for example, because row 1 assigns T to P and to Q, we enter T under each and every occurrence of P and Q on that line like so:
Next, we must again identify the main connective and highlight the column below it by enclosing it in a box marked ‘m.c.’ to emphasise that this column is the main column, i.e. the column in which the overall truth-value of the whole formula will be recorded under each assignment:
The main connective in this formula is the ar row. So, the whole formula is a conditional. However, its antecedent is a conjunction and its consequent is a disjunction. In order to calculate the overall truth-value then we must first calculate the value of the antecedent using the table for ‘&’, then calculate the value of the consequent using the table for ‘v’, and write these values under the relevant connectives. To that end, consider the first assignment carefully. Because both P and Q are true the conjunction P & Q is also true, and we record that fact by entering T under the connective ‘&’. Further, because both P and Q are true, the disjunction P v Q is also true, and we record that fact by writing T under the disjunction symbol. So far, then, line 1 looks like this:
In turn, these two results now provide our input truth-values for the final calculation, which we make simply by using the truth-table for ‘→’. Consider line 1, for example; the antecedent (P & Q) is true and so is the consequent (P v Q). When that is so the truth-table for the conditional assures us that the whole conditional is true. Therefore, the overall value of the whole formula (P & Q) → (P v Q) under the first assignment of truth-values is T.
So, we may complete line 1 as follows:
To complete the truth-table in its entirety, we simply follow the same procedure for the remaining assignments 2–4. The completed truth-table looks like this:
By now, you will have a fair idea of how to go about constructing truth-tables but, before we go on to tackle an exercise, I will sum up the procedure and give one last illustration, namely, the table for the formula P & (Q v R). So, to construct a truth-table observe the following procedure carefully:
1 Draw a horizontal line at least one and a half times as long as the formula(s) involved and enter the formula(s) to the right of that line, e.g.
2 Identify all the sentence-letters involved in the formula(s), list these in alphabetical order to the left of the formula and separate the two with a vertical line, e.g.
3 Assign truth-values to the sentence-letters listed on the left. The number of truth-value assignments required, i.e. the number of rows of assignments to the sentence-letters on the left, is just 2n where ‘n’
is the number of letters involved. So, if only one sentence-letter is involved, the number of rows of assignments is 21, i.e. 2. If two sentence-letters are involved, the number of rows of assignments is 22, i.e. 4. If, as in the present case, three sentence-letters are involved, the number of rows of assignments is 23, i.e.
(2×2)×2, so 8. Construct the required number of truth-value assignments as follows:
(i) Under the sentence-letter nearest the formula itself (in this caseR) list the required number of truth-value assignments (in this case 8) beginning with T and alternating with F to the relevant number, e.g.:
(ii) Under the next sentence-letter to the left (in this case Q) again list the required number of truth-value assignments but this time beginning with two Ts and alternating with two Fs to the required number, thus:
(iii) Under the next sentence-letter to the left (in this case P) again list the required number of truth-value assignments but this time beginning with four Ts and alternating with four Fs to the required number, thus:
4 To complete the truth-table consider each truth-value assignment, i.e.
each row of the table (in this case rows numbered 1–8), then:
(i) In each case, begin by entering the truth-value assigned to each sentence-letter by that assignment under each and every occurrence of the relevant sentence-letter in the formula itself.
In this case, because row 1 assigns T to P, Q and R, we can enter T under each and every occurrence of P, Q and R in the formula like so:
(ii) Identify the main connective and highlight the column below it by enclosing it in a box with ‘m.c.’ written underneath it thus:
(iii) Identify the scope of every other connective involved and, for each assignment, using the truth-values already assigned to the sentence-letters involved and the truth-tables for the relevant connectives, establish the overall truth-value of each sub-formula.
In the present case the formula is a conjunction whose first conjunct is P and whose second conjunct is (Q v R). On assignment 1 every sentence-letter is assigned T and so we simply appeal to the truth-table for disjunction to establish that when both disjuncts are true the disjunction is true.
Hence, the overall value of the formula (Q v R) is also T:
(iv) Next, using the truth-values of the sub-formulas we have just worked out as inputs, we simply use the appropriate truth-table for the main connective to compute the overall value of the whole formula under each assignment of truth-values. In the case above, the conjunction has two true conjuncts, namely, P and (Q v R). A conjunction with two true conjuncts is itself true, so we complete this line by entering T in the column belonging to the main connective like so:
(v) Finally, repeat steps (i)–(iv) for every other assignment in-volved.
Before attempting Exercise 4.1 note carefully that while the main connective in the formula ~P & (Q v R) is ‘&’ the main connective in the formula ~(P & (Q v R)) is ‘~’. Therefore, in that latter case, and in the case of every other negated formula, the overall truth-value of the formula must be stated under that occurrence of the connective. (Anyone who wants a refresher course on the notion of the scope of a connective might like to reread Section II of Chapter 2 before attempting the following exercise.) EXERCISE 4.1
1 Construct and complete truth-tables for the following formulas of PL: