Syntax vs Semantics

Definition

Syntax is the formal structure of an expression.

Semantics is what that expression means under an interpretation.

Example:

x + 1

Syntax asks:

Is this expression formed correctly?

Semantics asks:

What does it mean when x has a value?

Syntax = form. Semantics = meaning.

---

Syntax

Syntax is about whether symbols are arranged correctly.

Valid form:

x + 1

Invalid form:

+ x 1 )

The invalid form may contain recognizable symbols.

But the arrangement does not follow the rules of the language.

Syntax controls which expressions are well-formed.

---

Semantics

Semantics gives meaning to a well-formed expression.

Example:

x + 1

If:

x = 5

then:

x + 1 = 6

The syntax did not change.

The meaning became concrete once an interpretation was supplied.

Semantics depends on interpretation.

---

Same Syntax, Different Semantics

The same expression can mean different things in different contexts.

Example:

x + 1

Context 1:

x = 2
result = 3

Context 2:

x = 10
result = 11

The syntax is the same.

The semantic value changes.

Same form can produce different meaning under different assignments.

---

Syntax Without Semantics

An expression can have correct syntax without a known meaning.

Example:

P AND Q

This is syntactically valid.

But until P and Q are interpreted, we do not know whether the expression is true or false.

A well-formed expression may still need interpretation.

---

Why This Matters For Reasoning

Reasoning has two layers.

First, expressions must be formed correctly.

Second, the expressions must be interpreted correctly.

Example:

If P, then Q
P
Therefore Q

The structure is syntactic.

Whether the statements are true depends on semantics.

Logic needs both valid form and meaningful interpretation.

---

Final Summary

• Syntax is symbolic form.
• Semantics is meaning under interpretation.
• Syntax determines whether an expression is well-formed.
• Semantics determines what the expression refers to or whether it is true.
• The same syntax can have different semantics.
• Reasoning requires keeping form and meaning separate.