Recipe

Compute a compound's value on one row

Use when: you're given a sentence and one valuation, and asked whether the sentence is true or false there

Goal: the sentence's truth value on that valuation, with every intermediate part's value written down

Taught in Computing the value of a compound · a later section extends this recipe with more cases

1. 1

   Copy the valuation onto the sentence: write T or F under every occurrence of every letter.

   Example

   For (P \wedge Q) \to \neg R on P = T, Q = F, R = T: write T under P, F under Q, T under R.

   Why: every later step uses these values, and a letter has the same value everywhere it appears.

2. 2

   Locate the connective outside all brackets, or the leading \neg if the whole sentence is negated.

   Example

   In (P \wedge Q) \to \neg R the arrow sits outside the brackets, so it is the main connective.

   Why: it is the last step, so knowing it tells you which two parts you are aiming for.

3. 3

   Take each smallest compound part (a \neg on a letter, or two letters joined by one connective) and look its value up in that connective's table.

   Example

   P \wedge Q is T \wedge F, which is F. \neg R is \negT, which is F.

   Why: a connective can only be applied once its parts have values.

4. 4

   Replace each part you've finished with its value and repeat on the next part out, until only the main connective is left.

   • a part is a single letter — its value is already written under it.
   • a part is itself compound — evaluate its own parts first, then it.

   Why: the brackets group the sentence into nested parts, and each part feeds the one around it.

5. 5

   Look up the main connective's table with the values of its two parts (or one part, for \neg). That entry is the answer.

   Example

   The arrow with antecedent F and consequent F gives T. The sentence is T on this row.

   ⚡ Shortcut

   Before grinding through every part, check whether one part already settles the main connective: a false conjunct, a true disjunct, a false antecedent, or a true consequent. Section 6 of this chapter covers these.

   Why: the main connective's scope is the whole sentence, so its value is the sentence's value.