Logical operation

A logical operation transforms individual propositions into compound propositions. Most logical operations are analogues to natural English expressions such as "not," "and," "or," and "if."

Here's the order in which logical operations should be evaluated:

1. Parentheses
2. Negation (\(\lnot\))
3. Conjunction (\(\land\))
4. Disjunction (\(\lor\))
5. Any other logical operation (Parentheses should indicate order)

Logical operations can be unary or binary, meaning they operate on one or two propositions, respectively.

Negation

Negation is a unary logical operation that reverses the truth value of a proposition. The negation of a proposition \(p\) is denoted \(\lnot p\) and is read as "not \(p.\)"

Conjunction

Conjunction is a binary logical operation that results in a compound proposition which is true if and only if the starting propositions are both true. The conjunction of propositions \(p\) and \(q\) is the proposition \(p \land q\) and is read "\(p\) and \(q\)". It is true only if both \(p\) and \(q\) are true, and false if \(p\) is false, \(q\) is false, or both \(p\) and \(q\) are false.

"\(p\) and \(q\)," "\(p\) but \(q\)," "although \(p\), \(q\)," and "despite \(p\), \(q\)" are all conjunctions of \(p\) and \(q.\)

Disjunction

Disjunction is a binary logical operation that results in a compound proposition which is false if and only if the starting propositions are both false. The disjunction of propositions \(p\) and \(q\) is the proposition \(p \lor q\) and is read "\(p\) or \(q\)". It is false only if both \(p\) and \(q\) are false, and true if \(p\) is true, \(q\) is true, or both \(p\) and \(q\) are true.

The disjunction operation corresponds with the notion of an "inclusive or."

Exclusive disjunction

In conversation, when we say "or," we sometimes mean that only one of two options is true. But in a disjunction, both options are allowed to be true. To get rid of this ambiguity, the exclusive disjunction was created. Its truth table resembles disjunction with just one notable exception.

Exclusive disjunction is a binary logical operation that results in a compound proposition which is true if and only if the starting propositions have opposite truth values. The exclusive disjunction of propositions \(p\) and \(q\) is the proposition \(p \oplus q\) and is read "either \(p\) or \(q\)". It is true only if just \(p\) is true or just \(q\) is true, and false if both are true, or both are false.

The exclusive disjunction operation corresponds with the notion of an "exclusive or."

The conditional operation

The conditional operation, a.k.a. implication, is a directed binary logical operation that results in a compound proposition (a conditional proposition) which is false if and only if the first proposition, the hypothesis, is true, while the second proposition, the conclusion, is false. The conditional operation applied to propositions \(p\) and \(q\), in that order, results in the proposition \(p \rightarrow q\) and is read "if \(p\), then \(q\)". The proposition \(p \rightarrow q\) is false if and only if \(p\) is true and \(q\) is false, otherwise, it's true.

The final two rows of the truth table are known as vacuous truths where the conclusion is shown to be true or false even though the hypothesis never took effect.

Notice that the choice of which proposition is first and which is second can affect the truth value of a conditional proposition. In other words, order matters here. \(p \rightarrow q\) and \(q \rightarrow p\) are not the same proposition.

For a conditional proposition \(p \rightarrow q\), its converse is \(q \rightarrow p\), its (logically equivalent) contrapositive is \(\lnot q \rightarrow \lnot p\), and its inverse is \(\lnot p \rightarrow \lnot q.\)

"If \(p\), then \(q\)," "\(q\) if \(p\)," "\(p\) implies \(q\)," "\(p\) only if \(q\)," "\(p\) is sufficient for \(q\)," and "\(q\) is necessary for \(p\)" are all implications where \(p\) is the hypothesis and \(q\) is the conclusion.

Example: "If it's a fish, then it can swim."

The biconditional operation

The biconditional operation is a binary logical operation that results in a compound proposition which is true if and only if the starting propositions have the same truth values. The biconditional operation applied to propositions \(p\) and \(q\) results in the proposition \(p \leftrightarrow q\) which is read "\(p\) if and only if \(q\)". The proposition \(p \leftrightarrow q\) is true if and only if \(p \equiv q.\)

"\(p\) if and only if \(q\)," "\(p\) if \(q\) and \(q\) if \(p\)," "\(p\) is necessary and sufficient for \(q\)," "if \(p\) then \(q\), and conversely," and "\(p\) iff \(q\)" are all correct ways to write \(p \leftrightarrow q.\) The least ambiguous of these options is "\(p\) if \(q\) and \(q\) if \(p\)," which has the added benefit of showing you that if \(p \leftrightarrow q\) is true, \(p \rightarrow q\) and its converse \(q \rightarrow p\) are true. The biconditional operation is sort of like the conditional operation applied in both directions.

If \(p \leftrightarrow q\) is a tautology, then \(p\) causes \(q\), and \(q\) causes \(p.\) So then, one cannot be true without the other also being true. This notion of "if and only if" is used a lot in precise mathematical definitions, as you may have noticed!

Example: "I am breathing if and only if I am alive." Note that this is the same as "I'm alive if I'm breathing and I'm breathing if I'm alive."

Alternative denial

Alternative denial, a.k.a. non-conjunction, is a binary logical operation that results in a compound proposition which is false if and only if the starting propositions are both true. The alternative denial of propositions \(p\) and \(q\) is the proposition \(p \uparrow q\) and is read "not both \(p\) and \(q\)". It is false only if both \(p\) and \(q\) are true, and true if \(p\) is false, \(q\) is false, or both \(p\) and \(q\) are false.

This operation is logically equivalent to the negation of conjunction. In particular, \(p \uparrow q \equiv \lnot (p \land q).\) Its symbol, \(\uparrow\) or \(|\), may be referred to as the Sheffer stroke.