Symbolizing English Sentences

When symbolizing English sentences in propositional logic, it is often helpful to work backwards, reversing the order in which well-formed formulas (WFFs) are constructed. Rather than trying to identify the simplest components of the sentence and build up from them, first try to determine whether the sentence as a whole is a negation, a conjunction, a disjunction, a conditional, or a biconditional. In other words, try to identify the main connective. Symbolize the main connective first, then check each of the components and try to identify their main connectives, and so on. Eventually, you’ll find some components that contain no truth-functional connectives. Symbolize those simple propositions last, using sentence letters.

Consider the following sentence:

If you like logic and you want to master it, then you’re neither foolish nor stupid.

The sentence as a whole is a conditional (an “if-then” statement), which is represented by the horseshoe symbol. So, we begin by making a “⊃” with parentheses around it, leaving some space between the parentheses on each side of the horseshoe to save room for the component propositions, like this: (             ⊃             )

Next, let’s symbolize the antecedent—the component on the left side of the horseshoe: You like logic and you want to master it. That statement is a conjunction, which is represented by a dot. The “•” gets its own set of parentheses, which will be inside the parentheses that belong to the “⊃,” like this: ((    •    ) ⊃             ) The two conjuncts are You like logic and You want to master logic. These are both simple propositions (they contain no truth-functional connectives), so we represent them with sentence letters. Let’s use “L” to represent You like logic and “W” to represent You want to master logic: ((L • W) ⊃             ) We’re done with the antecedent of the conditional. Now, let’s symbolize the consequent—the proposition on the right side of the horseshoe: You’re neither foolish nor stupid. The word “neither” means not either. In other words, this proposition says it’s not true that you are either foolish or stupid. This is a negation, symbolized by a “~” (tilde). We attach the “~” to the left side of the proposition being negated, namely, the false claim that you are either foolish or stupid. That “either-or” claim is a disjunction, symbolized with a “∨” (wedge) enclosed in parentheses. So, using sentence letters “F” and “S” to stand for the two disjuncts, the statement You’re neither foolish nor stupid can be symbolized like this: ~(F ∨ S). The proposition “~(F ∨ S)” is the consequent of the conditional, so it belongs on the right side of the horseshoe: ((L • W) ⊃ ~(F ∨ S)) That’s it! We’ve symbolized the logical structure of the whole sentence using a WFF of propositional logic.

Below are some further points of advice. Please pay careful attention to these tips, as they will help you avoid common mistakes when symbolizing sentences in propositional logic.