“$x > 3$”

Example · High school

Terms

Term	This example	Why
predicate	✓	“$x > 3$” is a predicate because its truth depends on $x$.
formula logic	✓	“$x > 3$” is a formula, because “$x > 3$” applies the relation symbol $>$ to the terms $x$ and $3$.
expression	✓	“$x > 3$” is an expression in the logic sense, because “$x > 3$” is a well-formed string of symbols; the expression is a formula, not a term.
proposition	✗	“$x > 3$” is not a proposition, because it is not true or false until $x$ is given.
relation	✗	“$x > 3$” is not a relation, which is a set of pairs.
sentence	✗	“$x > 3$” is not a sentence, because $x$ is a free variable.
propositional formula	✗	“$x > 3$” is not a propositional formula, because its variable $x$ stands for a number, not for a proposition.
term	✗	“$x > 3$” is not a term, because “$x > 3$” makes a claim about $x$ instead of naming an object.

Distinctions using this example

• Formula vs. sentence: “$x > 3$” is a formula but not a sentence, because its variable $x$ is free.
• Proposition vs. predicate: “$x > 3$” is a predicate, and giving $x$ the value $5$ turns the predicate into the proposition “$5 > 3$,” which is true.
• Sentence vs. proposition
• Sentence vs. propositional formula: “$x > 3$” is neither a sentence nor a propositional formula: “$x > 3$” is a formula of first-order logic, but its variable $x$ is free, and $x$ stands for a number, not a proposition.
• Term vs. formula

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