## 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.

• Be sure that each sentence letter stands for a whole proposition, not just a word or phrase.

The letter “E” could be used to represent a proposition like Everyone should study logic. It should not be used to represent the word “everyone.”
Correct Incorrect
E = Everyone should study logic. E ≠ everyone

• Watch out for hidden negations. Sometimes the word “not” is hidden in a contraction like isn’t, doesn’t, can’t, etc.

The sentence Illogicality isn’t ideal means it is false that illogicality is ideal.
Correct Incorrect
I = Illogicality is ideal. I ≠ Illogicality isn’t ideal.
~I = Illogicality isn’t ideal.

• Be careful with conjunctions. Make sure that both conjuncts are complete propositions, and that the resulting conjunction means the same thing as the original sentence.

The sentence Plato and Aristotle are philosophers means Plato is a philosopher and Aristotle is a philosopher.

However, the sentence Plato and Aristotle are friends does not mean simply that Plato is a friend and Aristotle is a friend. It means they are friends with each other. Moreover, the sentence Plato is a friend of Aristotle means the same thing (expresses the same proposition) as Aristotle is a friend of Plato. So, the sentence Plato and Aristotle are friends doesn’t really express a compound proposition at all: it is a simple proposition, which we can symbolize with a single sentence letter like “F.”
Correct Incorrect
P = Plato is a philosopher.
A = Aristotle is a philosopher.
(P • A) = Plato and Aristotle are philosophers.
P ≠ Plato and Aristotle are philosophers.
F = Plato and Aristotle are friends. P = Plato is a friend.
A = Aristotle is a friend.
(P • A) ≠ Plato and Aristotle are friends.

• Regardless of where it appears in an English sentence, the phrase “only if” always indicates the consequent of a conditional, which must be written on the right side of the “⊃” in the WFF. The word “if” by itself (without the word “only”) always indicates the antecedent, which must be written on the left side of the horseshoe. The word “provided” is a synonym of “if,” and it likewise indicates the antecedent.

English sentence forms Symbolization
“only if” indicates the consequent P only if Q. (P ⊃ Q)
Only if Q, P.
“if” indicates the antecedent If P then Q.
Q if P.
“provided” indicates the antecedent Provided P, Q.
Q, provided P.

• The word “unless” indicates a disjunction, just like “or.”

English sentence forms Symbolization
P or Q. (P ∨ Q)
P unless Q.
Unless P, Q.

• The words “but,” “however,” and “although” indicate conjunctions, just like “and.”

English sentence forms Symbolization
P and Q. (P • Q)
P but Q.
P, although Q.
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