Statements and Logical Connectives

Ordinary language is full of sentences whose truth cannot be settled — commands, questions, opinions and exclamations. Mathematical logic strips language down to just the sentences that can be judged true or false, and studies how such sentences combine. This is the foundation of every rigorous argument in mathematics, and — as the last section of this chapter shows — it also models the electrical switching circuits that run every calculator and computer.

Statement. A statement (or proposition) is a declarative sentence that is either true or false, but not both at the same time. Whether it is true or false is called its truth value, written $T$ (true) or $F$ (false). For example, "$3 + 5 = 8$" is a true statement, and "$7$ is an even number" is a false statement — both are statements because each has a definite truth value.

Sentences that are not statements include:

• Questions — "What is the time?" (cannot be true or false).
• Commands / requests — "Please close the door."
• Exclamations — "What a beautiful day!"
• Open sentences — "$x + 2 = 5$" is not a statement on its own, because its truth depends on the value of $x$; it becomes a statement only once $x$ is fixed.
• Opinions / paradoxes — "This sentence is false" is not a statement, since assuming it true forces it false and vice versa.

Simple statements are denoted by small letters $p, q, r, \ldots$ For instance we may write $p:$ "Mumbai is in Maharashtra" and note that its truth value is $T$.

Logical connectives. New statements are built from given ones using words called logical connectives. There are five, each with its own symbol.

A statement built using one or more connectives is a compound statement; a statement with no connective is a simple (or primitive) statement.

• Negation $\sim p$ reverses the truth value: if $p$ is true, $\sim p$ is false, and vice versa. If $p:$ "$5$ is prime", then $\sim p:$ "$5$ is not prime".
• Conjunction $p \wedge q$ ("$p$ and $q$") is true only when both parts are true.
• Disjunction $p \vee q$ ("$p$ or $q$") is true when at least one part is true. In logic "or" is always the inclusive or — it also allows both to be true.
• Conditional $p \rightarrow q$ ("if $p$ then $q$") — here $p$ is the antecedent (hypothesis) and $q$ the consequent (conclusion).
• Biconditional $p \leftrightarrow q$ ("$p$ if and only if $q$") asserts that $p$ and $q$ have the same truth value.

To write an ordinary compound sentence in symbols, first label each simple statement, then replace the connecting words by their symbols. For example, with $p:$ "It is raining" and $q:$ "The road is wet", the sentence "If it is raining then the road is wet" becomes $p \rightarrow q$.