2.2 Formal Systems and Semantics

Mathematics operates through formal systems. A formal system specifies how statements are written and how proofs are constructed. Semantics explains what those statements mean inside a structure.

A formal system has four components.

These components define the syntactic layer. They determine which sequences of symbols count as valid formulas and which derivations count as proofs.

A simple example appears in propositional logic. The alphabet includes symbols such as P, Q, ¬, ∧, ∨, →. Syntax specifies how to build formulas:

If P and Q are formulas, then (P ∧ Q), (P ∨ Q), (P → Q) are formulas.
If P is a formula, then ¬P is a formula.

Axioms provide base truths. Inference rules, such as modus ponens, allow derivation:

From P and (P → Q), infer Q

A proof is a finite sequence of formulas where each step is either an axiom or follows from previous steps by inference rules.

Semantics assigns meaning to these symbols. A structure interprets the language by mapping symbols to objects and operations.

For propositional logic, a structure assigns truth values to variables:

P = true
Q = false

Then compound formulas are evaluated according to truth tables.

For first-order logic, semantics is richer. A structure includes:

A statement such as:

∀x (P(x) → Q(x))

is evaluated by checking all elements in the domain. It is true if every element that satisfies P also satisfies Q.

The satisfaction relation connects syntax and semantics:

M ⊨ φ

This reads as “structure M satisfies statement φ.” It defines truth inside a model.

Formal systems manipulate symbols without reference to meaning. Semantics interprets those symbols inside a model. The interaction between the two layers is central.

A system is designed so that syntactic derivations correspond to semantic truth. This leads to soundness and completeness, discussed earlier.

Different systems use different languages. Arithmetic uses symbols for numbers, addition, multiplication, and order. Group theory uses a binary operation and identity. Set theory uses membership ∈.

Each language restricts what can be expressed. First-order logic cannot directly quantify over subsets or functions without encoding them. Higher-order systems allow richer expressions but are more complex.

Models provide flexibility. A single formal language can have many models.

A statement may be true in some models and false in others. This leads to model-dependent truth.

A theory is a set of axioms in a given language. Its models are structures where all axioms hold.

M ⊨ T   means   M satisfies all axioms in T

A theory describes a class of structures. For example, the axioms of group theory describe all groups. The axioms of real closed fields describe structures similar to real numbers.

Semantics allows comparison between theories and structures. It answers questions such as:

Formal systems also support interpretation between theories. One theory can be translated into another by mapping symbols and preserving structure. This is common in algebra and logic.

There is a practical distinction between formalization and informal reasoning. Mathematicians rarely write fully formal proofs. They rely on shared understanding of semantics. Formal systems provide a foundation that ensures rigor when needed.

Proof assistants make this explicit. Every step must follow syntactic rules. Every object must have a type. Semantics is encoded through definitions and structures.

A working pattern:

This pattern appears in all areas of mathematics. Algebra defines operations and axioms, then studies their consequences. Analysis defines limits and continuity, then studies functions. Geometry defines spaces and relations, then studies their properties.

Formal systems provide precision. Semantics provides meaning. Mathematics requires both.