We have five logical connectors:
- and: $\wedge$
- or: $\vee$
- not: $\neg$ (or $\sim$)
- if, then: $\rightarrow$
- if and only if: $\leftrightarrow$
Truth values of the compound statements constructed by these using the truth tables of these connections. (Wittgenstein)
Given two statements $p$ and $q$:
1. Conjunction
The conjunction of $p$ and $q$ is the statement $p \wedge q$ and is read as “$p$ and $q$”.
$\rightarrow$ The truth table of $p \wedge q$ is given by:
$$ \begin{array}{|c|c|c|} \hline p & q & p \wedge q \\ \hline \text{T} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{F} \\ \text{F} & \text{T} & \text{F} \\ \text{F} & \text{F} & \text{F} \\ \hline \end{array} $$
2. Disjunction
The disjunction of $p$ and $q$ is the statement $p \vee q$ and is read as “$p$ or $q$”.
$\rightarrow$ The truth table of $p \vee q$ is given by:
$$ \begin{array}{|c|c|c|} \hline p & q & p \vee q \\ \hline \text{T} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{T} \\ \text{F} & \text{T} & \text{T} \\ \text{F} & \text{F} & \text{F} \\ \hline \end{array} $$
3. Conditional
The conditional from $p$ to $q$ is the statement $p \rightarrow q$ and is read as:
- “if $p$, then $q$”
- “$q$ whenever $p$”
- “$q$ if $p$”
- “$p$ only if $q$”
- “$q$ provided that $p$”
- “$q$ given that $p$”
- “$p$ is sufficient for $q$”
- “$q$ is necessary for $p$”
$\rightarrow$ The truth table is:
$$ \begin{array}{|c|c|c|} \hline p & q & p \rightarrow q \\ \hline \text{T} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{F} \\ \text{F} & \text{T} & \text{T} \\ \text{F} & \text{F} & \text{T} \\ \hline \end{array} $$
4. Biconditional
The biconditional from $p$ and $q$ is the statement of $p \leftrightarrow q$ and read as:
- “$p$ if and only if $q$”
- “$p$ is necessary and sufficient for $q$”
$\rightarrow$ The truth table is given by:
$$ \begin{array}{|c|c|c|} \hline p & q & p \leftrightarrow q \\ \hline \text{T} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{F} \\ \text{F} & \text{T} & \text{F} \\ \text{F} & \text{F} & \text{T} \\ \hline \end{array} $$
5. Negation
The negation of $p$ is given by $\neg p$ and is read as “not $p$”.
- “it is not the case that $p$ holds”
$$ \begin{array}{|c|c|} \hline p & \neg p \\ \hline \text{T} & \text{F} \\ \text{F} & \text{T} \\ \hline \end{array} $$
Examples
Write the truth tables of the statements:
- $p \rightarrow (q \rightarrow p)$
- $(p \wedge R) \rightarrow (\neg(R \rightarrow S))$
- $((p \rightarrow q) \wedge p) \wedge \neg q$
1. $p \rightarrow (q \rightarrow p)$
$$ \begin{array}{|c|c|c|} \hline p & q & p \rightarrow (q \rightarrow p) \\ \hline \text{T} & \text{T} & \text{T} \\ \text{F} & \text{F} & \text{T} \\ \text{T} & \text{F} & \text{T} \\ \text{F} & \text{T} & \text{T} \\ \hline \end{array} $$
2. $(p \wedge R) \rightarrow (\neg(R \rightarrow S))$
$$ \begin{array}{|c|c|c|c|} \hline p & R & S & (p \wedge R) \rightarrow (\neg(R \rightarrow S)) \\ \hline \text{T} & \text{T} & \text{T} & \text{T} \\ \text{T} & \text{T} & \text{F} & \text{T} \\ \text{T} & \text{F} & \text{T} & \text{F} \\ \text{T} & \text{F} & \text{F} & \text{F} \\ \text{F} & \text{T} & \text{T} & \text{F} \\ \text{F} & \text{T} & \text{F} & \text{F} \\ \text{F} & \text{F} & \text{T} & \text{F} \\ \text{F} & \text{F} & \text{F} & \text{F} \\ \hline \end{array} $$
3. $((p \rightarrow q) \wedge p) \wedge \neg q$
$$ \begin{array}{|c|c|c|} \hline p & q & ((p \rightarrow q) \wedge p) \wedge \neg q \\ \hline \text{T} & \text{T} & \text{F} \\ \text{T} & \text{F} & \text{F} \\ \text{F} & \text{T} & \text{F} \\ \text{F} & \text{F} & \text{F} \\ \hline \end{array} $$
This is an example of contradiction.
Definitions
A statement that is always true is called a tautology.
A statement that is always false is called a contradiction.