Definition1.1.1Proposition
A proposition is a sentence that is either true or false.
Section1.1Propositions and Connectives
¶In this chapter we introduce classical logic which has two truth values, True and False. Every proposition takes on a single truth value.
Definition1.1.2Conjunction, Disjunction, Negation
Given propositions \(P\) and \(Q\text{,}\) the
- conjunction
- of \(P\) and \(Q\text{,}\) denoted \(P \wedge Q\text{,}\) is the proposition “\(P\) and \(Q\text{.}\)” \(P \wedge Q\) is true exactly when both \(P\) and \(Q\) are true.
- disjunction
- of \(P\) and \(Q\text{,}\) denoted \(P \vee Q\text{,}\) is the proposition “\(P\) or \(Q\text{.}\)” \(P \vee Q\) is true exactly when at least one \(P\) or \(Q\) is true.
- negation
- of \(P\text{,}\) denoted \(\neg P\text{,}\) is the proposition “not \(P\text{.}\)” \(\neg P\) is true exactly when \(P\) is false.
Note that although the disjunction \(P \vee Q\) is translated into English as “\(P\) or \(Q\)”, it means something slightly different than the way we use “or” in everyday speech. In particular, disjunction is inclusive, which means that it is true whenever at least one of \(P\) or \(Q\) is true. On the other hand, in English, “or” is often exclusive, which means that it is true whenever exactly one of the alternatives is true. For example, if I said “today, I will either go to the park or to the pool,” it would generally be understood that I will do one or the other, but not both. However, if \(P\) is the proposition “I will go to the park” and \(Q\) is the proposition, “I will go to the pool”, then \(P \vee Q\) means “I will go to the park or to the pool or both.” Throughout the remainder of this course, whenever we say “or”, we mean the inclusive version corresponding to disjunction.
Definition1.1.3Well-formed formula
A well-formed formula is an expression involving finitely many logical connective symbols and letters representing propositions which is syntactically (i.e., grammatically) correct.
For example, \(\neg (P \vee Q)\) is a well-formed formula, but \(\vee PQ\) is not.
Definition1.1.4Equivalent forms
Two well-formed formulas are equivalent if and only if they have the same truth tables.
From the defintion, we can see that \(P \vee Q\) and \(Q \vee P\) are equivalent forms. Similarly, \(P \wedge Q\) and \(Q \wedge P\) are equivalent forms. However, there are less obvious examples such as \(\neg((P \vee Q) \wedge R)\) and \(((\neg P) \vee (\neg R)) \wedge ((\neg Q) \vee (\neg R))\text{.}\)
Definition1.1.5Tautology
A tautology is a well-formed formula that is true for every assignment of truth values to its components.
For example, the Law of Excluded Middle which states that, for any proposition \(P\text{,}\) the disjunction \(P \vee (\neg P)\) is a tautology. The name refers to the fact that every proposition is either true or false, there are no possibilities in-between, i.e., the middle is excluded.
| \(P\) | \(\neg P\) | \(P \vee (\neg P)\) |
| True | False | True |
| False | True | True |
Definition1.1.7Contradiction
A contradiction is a well-formed formula that is false for every assignment of truth values to its components.
For example, for any proposition \(P\text{,}\) the conjunction \(P \wedge (\neg P)\) is a contradiction.
| \(P\) | \(\neg P\) | \(P \wedge (\neg P)\) |
| True | False | False |
| False | True | False |
Definition1.1.9Denial
A denial of a proposition \(P\) is any proposition equivalent to \(\neg P\text{.}\)
For example, for any propositions \(P\) and \(Q\text{,}\) the statement
\begin{gather*} ((\neg P) \wedge Q) \vee ((\neg P) \wedge (\neg Q)) \end{gather*}is a denial of \(P\text{.}\)
Make a truth table for each of the following propositions, and determine whether any of them are contradictions or tautologies.
\begin{gather*} (P \vee (\neg Q)) \wedge (\neg R),\\ ((\neg P) \vee (\neg Q)) \wedge ((\neg P) \vee Q),\\ (P \wedge Q) \vee (P \wedge (\neg Q)) \vee ((\neg P) \wedge Q) \vee ((\neg P) \wedge (\neg Q)),\\ (P \vee Q) \wedge (P \vee (\neg Q)) \wedge ((\neg P) \vee Q) \wedge ((\neg P) \vee (\neg Q)). \end{gather*}