Truth-functional connectives

Logical connectives (also called logical operators) are words or phrases that join propositions together to form longer propositions. Connectives can be thought of as sentences with “blanks” that can be filled with complete sentences.

“It is not true that______________.”
“Either ____________ or ____________.”
“If _______________, then ______________.”

If you put complete sentences into these blanks, the result will likewise be a complete sentence.

Some connectives are truth-functional, which means that the truth or falsity of any proposition built from them depends only on the truth or falsity of the propositions that are inserted into the blanks.

“It is false that _________________.” Whether this proposition is true depends only on whether the proposition inserted into the blank is true. (The proposition built from this connective will be true when the inserted proposition is false; it will be false when the inserted proposition is true.)

In contrast, consider: “Aristotle says that ________________.” Whether this proposition is true depends on what Aristotle says. Its truth or falsity doesn’t depend on whether the proposition in the blank is true.

There are 5 commonly used types of truth-functional connectives:

A negation is a proposition asserting that another proposition is false.
A conjunction is a proposition asserting that two other propositions are both true.
A disjunction is a proposition asserting that at least one of two propositions is true.
A conditional is a proposition asserting that if one proposition is true, then so is another.
A biconditional is a proposition asserting that two propositions are either both true or both false. (In other words, one is true if and only if the other is true.)

A simple proposition is one that contains no truth-functional connectives. A proposition is compound if it contains one or more truth-functional connectives. Negations, conjunctions, disjunctions, conditionals, and biconditionals are compound propositions.

Here are a few examples of simple propositions:

Logic is important.
Studying logic can help you learn to reason well.
Philosophers should aspire to master the tools of propositional logic.

None of those sentences contains any truth-functional connectives, so they are all regarded as simple propositions. In contrast, the following are examples of compound propositions:

Logic isn’t easy. (This means it is false that logic is easy, so the sentence is a negation.)
Logic and philosophy are important. (This sentence asserts that logic is important and that philosophy is important, so it is a conjunction.)
You should study either logic or philosophy. (This sentence is a disjunction, since it asserts that at least one of the following two propositions is true: you should study logic or you should study philosophy.)
If you want to reason well, you should study logic. (This sentence is a conditional: it says that if it’s true that you want to reason well, then the proposition you should study logic is also true.)

In propositional logic, we use capital letters (A, B, C, etc.) called sentence letters to stand for simple propositions. To symbolize compound propositions, we put sentence letters together with various symbols.

Negations are symbolized with a “~” (tilde), conjunctions are symbolized with a “•” (dot), disjunctions are symbolized with a “∨” (wedge), conditionals are symbolized with a “⊃” (horseshoe), and biconditionals are symbolized with a “≡” (triple bar). In the following table, those symbols are used with sentence letters P and Q (representing simple propositions) to show how compound propositions are symbolized.

types of connectives	examples in English	symbolization
negations	It is not true that P. It is false that P. It is not the case that P.	~P
conjunctions	P and Q. P but Q. P, although Q.	(P • Q)
disjunctions	Either P or Q (or both). P unless Q.	(P ∨ Q)
conditionals	If P then Q. P only if Q. Q if P.	(P ⊃ Q)
biconditionals	P if and only if Q. P just in case Q.	(P ≡ Q)

The sentence If philosophers ponder profound problems then quandaries quicken quotidian questions can be symbolized: (P ⊃ Q) The sentence Either academics always argue or bookworms become brilliant can be symbolized: (A ∨ B)

Truth-functional connectives can also be used to join compound propositions, to symbolize sentences that contain more than one truth-functional connective.

The sentence If philosophers ponder profound problems, then either academics always argue or bookworms become brilliant can be symbolized: (P ⊃ (A ∨ B)) The sentence It is false that if philosophers ponder profound problems, then either academics always argue or bookworms become brilliant can be symbolized: ~(P ⊃ (A ∨ B))

The next two sections will explain some rules and strategies for symbolizing English sentences in the notation of propositional logic.