Symbolizing English Sentences

Symbolizing the logical structure of English sentences using truth-functional connectives can be challenging. With practice, you can master this foundational skill. Please study the following examples and points of advice before practicing exercises 2, 3, and 4. Pay careful attention to these tips, and review them frequently as you work through the exercises, because they will help you avoid common mistakes.

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

  The letter “E” can be used to represent a proposition with no truth-functional connectives, such as 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.

• 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.

  For example, the sentence Logic is fun, but it’s not easy can be symbolized (F • ~E). In this symbolization, F stands for the simple proposition Logic is fun. E stands for the simple proposition Logic is easy, so we use ~E to represent the negation Logic is not easy.

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

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

  For instance, the sentence Unless you practice logic, you won’t master it can be symbolized (P ∨ ~M), using P to represent You practice logic and M to represent You will master logic.

  Similarly, the sentence You won’t master logic unless you practice it can be symbolized (~M ∨ P).

• The word “neither” means “not either.” So, a sentence of the form “neither A nor B” should be understood as the negation of a disjunction: ~(A ∨ B).

  The sentence Neither Socrates nor Plato studied logic can be symbolized ~(S ∨ P), using S for Socrates studied logic and P for Plato studied logic.

• A disjunction, symbolized by the wedge, represents the inclusive sense of “or.” In other words, (A ∨ B) means either A or B or both. To symbolize the exclusive sense of “or”—either A or B but not both—we need to recognize that this expression contains several more truth-functional connectives: “but” is symbolized by the dot, “not” is symbolized by the tilde, and “both” means both A and B, which is symbolized with another dot. Putting this all together, the statement either A or B, but not both A and B is symbolized like this: ((A ∨ B) • ~(A • B))

  The sentence You may have either cake or pie for dessert, but not both can be symbolized ((C ∨ P) • ~(C • P)), using C for You may have cake and P for You may have pie.

• Conditionals are especially tricky to symbolize, because the antecedent (left side of the ‘⊃’) and consequent (right side of the ‘⊃’) sometimes appear in reverse order in an English sentence. Regardless of where it appears in a 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” by itself indicates the antecedent	If P then Q.
  	Q if P.
  “provided” indicates the antecedent	Provided P, Q.
  	Q, provided P.

  For example, in the sentence If academics always argue then bookworms become brilliant, the proposition academics always argue is the antecedent because it appears immediately after the word “if” (without the word “only”). So, it belongs on the left side of the horseshoe: (A ⊃ B).

  However, in the sentence Academics always argue if bookworms become brilliant, the antecedent is bookworms become brilliant. Since it appears immediately after the word “if,” this time B belongs on the left side of the horseshoe: (B ⊃ A).

  In the sentence Academics always argue, provided bookworms become brilliant, the proposition bookworms become brilliant is the antecedent because it appears immediately after the word “provided,” which is a synonym of “if.” So, this sentence is symbolized the same way as the previous one: (B ⊃ A).

  In the sentence Academics always argue only if bookworms become brilliant, the proposition bookworms become brilliant is the consequent because it appears immediately after the phrase “only if.” Thus, B belongs on the right side of the horseshoe: (A ⊃ B).

  In the sentence Only if academics always argue, bookworms become brilliant, the proposition academics always argue is the consequent, because it appears immediately after the phrase “only if.” In this case, A belongs on the right side of the horseshoe: (B ⊃ A).

  To recap: the word “if” by itself always indicates the antecedent. The phrase “only if” indicates the consequent. The word “provided” indicates the antecedent, just like “if” does.

• The phrase “if and only if” indicates the biconditional, so it is symbolized with the triple bar, not the horseshoe. The phrase “just in case” also indicates the biconditional.

  The sentence Academics always argue if and only if bookworms become brilliant is symbolized (A ≡ B).

  The sentence Academics always argue just in case bookworms become brilliant is symbolized the same way: (A ≡ B).

When symbolizing sentences that contain multiple truth-functional connectives, 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 treat each of its components as new sentences 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.