6. Predicate logic basics

Predicate logic basics

Statements involving variables such as:

• $x>3$
• $x=y+4$
• Printer $x$ is networked to computer $y$.

Are quite often found in different places but these statements are neither true or false. As well as this, the variables $x$ and $y$ are not propositional variables but instead denote entities, which are things or objects.

We can formalise "$x>3$" as $\boxed{P(x)}$ where $x$ is a variable and $P$ is the predicate (property), which in this case is that it is “greater than 3”.

• By assigning a value (a concrete entity) to the variable, $P(x)$ becomes a proposition that has a truth value.
  • $P(5)$ means $5$ is greater than $3$ $⇝$ T
  • $P(1)$ means $1$ is greater than $3$ $⇝$ F
• We typically use upper case letters ($P,Q,A,B$) to denote predicates.

To formalise a statement such as "$x=y+4$", we use the predicate $Q(x,y)$, where:

• $x$ and $y$ are variables.
• $Q$ is the predicate connecting two things (otherwise called a binary predicate).
  • “the first number is the sum of the second number and $4$”
  • $Q(5,1):5=1+4⇝T$
  • $Q(6,6):6=6+4⇝F$