Term, Variable, and Constant

Context: FIT1058_MOC · what goes into a predicate’s argument slots · free vs bound variables · a predicate becomes a proposition once all free variables are assigned

Quick Revision

• 🎯 Objective: the inputs a predicate consumes ➔ constant / variable / function-applied-to-terms.
• 📦 Core Components: variable (domain-ranging) ➔ constant (fixed) ➔ term (recursive).
• ⚡ Key Constraint: a predicate becomes a proposition only when every free variable is assigned or bound.

📝 Core

1. The Three Objects

• Variable ➔ a name for an unspecified object from a domain.
• Constant ➔ a specific value ($3$, $\pi$, Alan Turing).
• Term ➔ a constant, a variable, or a function applied to (valid) terms (recursive).

2. Free vs Bound

• Free ➔ no value yet, open to assignment ➔ a predicate with a free variable is not a proposition.
• Bound ➔ captured by a quantifier ➔ no longer assignable.
• Becomes a proposition ➔ when every free variable is assigned from its domain (or bound).

3. Type-Checking Terms

• Domain fit ➔ a term must make sense in its argument’s domain.
• Available functions only ➔ codomains must match the next argument’s domain (Function Composition of types).

Key identities:

$\text{terms}:-5,22/7,y,3\times x+1,2\times \pi \times x$ $\text{not terms}:2+3+,\log x(\text{unavailable}),x\wedge y(\text{connective, not a function})$ $\text{Parent}(\text{Turing},X)(\text{free }X)\stackrel{X:=\text{Sara Turing}}{\to }\text{Parent}(\text{Turing},\text{Sara Turing})(\text{proposition})$

⚖️ Core Decision Matrix

Object	Ranges / fixed	Quantifiable?	Role
variable	ranges over a domain	yes	assigned or bound
constant	fixed object	no	a specific value
term	names an object	—	predicate input
predicate application	a statement	—	has a truth value

When It Flips: math favours single letters ($x,\theta$) because juxtaposition means multiplication (myVar reads as a product). Term ≠ formula: a term names an object; a predicate application is a statement.

📊 Exam Execution Trace

Manual Execution Trace

Classify expressions and variable occurrences:

Step / State	Expression	Term?	Variable status
0 (Init)	—	—	—
1	$3\times x+1$	term	$x$ free
2	$\log x$	not (unavailable fn)	—
3	$x\wedge y$	not (connective)	—
4	$\forall x(x<y)$	—	$x$ bound, $y$ free

⚠️ Common Mistakes

• 💡 $x\wedge y$ is not a term ➔ $\wedge$ combines statements, not objects; and constants can’t be quantified ($\exists 5$, $\forall \text{Annie}$ are errors).

🧠 Active Recall

What is a term, and which of $3\times x+1$, $\log x$, $x\wedge y$ qualify?

• Hint: Constant/variable/function-applied.

Answer

• Short answer: A term is a constant, variable, or function applied to terms; only $3\times x+1$ qualifies.
• Why: Object vs statement ➔ $\log$ is unavailable; $\wedge$ combines statements, not objects.

Distinguish free vs bound variables, and when does a predicate become a proposition?

• Hint: Assignment / binding.

Answer

• Short answer: Free = open to assignment (no truth value); bound = captured by a quantifier; a predicate becomes a proposition when every free variable is assigned or bound.
• Why: No truth value while free ➔ $\text{Parent}(\text{Turing},X)$ is not a proposition until $X$ is given a person.