Conjunction

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Section 1.3 Conjunction

We will spend a lot of our time combining statements together to get new statements. What allows us to do that are things called logical connectives. The first logical connective we have to contemplate is “and.”

Definition 1.3.1.

If \(\phi\) and \(\psi\) are statements, their conjunction is the statement \(\quine{\phi \text{ and } \psi}\text{.}\) We will use the notation \(\quine{\phi \wedge \psi}\) for the conjunction of \(\phi\) and \(\psi\text{.}\) For example, if \(\phi\) is the statement “\(2\) is even” and \(\psi\) is the statement “\(57\) is prime,” the statement \(\quine{\phi \wedge \psi}\) asserts that “\(2\) is even and \(57\) is prime.” The truth-value of the conjunction operates like this:

\(\quine{\phi \wedge \psi}\) is true exactly when both \(\phi\) is true and \(\psi\) is true

To unpack this, we use a truth table. When we combine statements \(\phi\text{,}\) \(\psi\text{,}\) \(\chi\text{,}\) etc. using logical connectives to get a statement \(\alpha\text{,}\) a truth table is then a list of the truth-values of \(\alpha\) depending on the truth-values of \(\phi\text{,}\) \(\psi\text{,}\) \(\chi\text{,}\) etc.. Here is the truth table for the conjunction:

\(\phi\)	\(\psi\)	\(\phi \wedge \psi\)

T	T	T
T	F	F
F	T	F
F	F	F

Question 1.3.2. A quick check.

What is the truth-value of “\(2\) is even and \(57\) is prime”?

It is false, because \(57\) is not prime.