Introduction to Mathematical Logic - Mathematical Logic

An overview of the fundamental concepts of mathematical logic, including propositional logic, predicate logic, and their applications in mathematics and computer science.

Introduction to Mathematical Logic
Key concepts, definitions, and examples for a first‑time learner

1. What Is Mathematical Logic?

Purpose – Provides a formal language to express mathematical statements and a systematic way to reason about them.
Two main branches
1. Proof Theory – Studies the structure of mathematical proofs.
2. Model Theory – Studies the interpretation of logical languages in mathematical structures.

2. Formal Languages

Component	Description	Example
Alphabet	Finite set of symbols (variables, logical connectives, quantifiers, parentheses, etc.)	`p, q, r, ∧, ∨, ¬, →, ∀, ∃, ( , )`
Terms	Built from variables and function symbols; denote objects of the domain.	`x`, `f(x, y)`
Atomic formulas	Predicates applied to terms; the simplest statements.	`P(x)`, `R(f(a), b)`
Formulas	Constructed from atomic formulas using logical connectives and quantifiers.	`¬P(x) ∨ ∀y R(x,y)`
Sentences	Formulas with no free variables (all variables bound).	`∀x ∃y R(x,y)`

3. Logical Connectives (Propositional Logic)

Symbol	Name	Truth‑functional definition
`¬φ`	Negation	True iff φ is false
`φ ∧ ψ`	Conjunction	True iff both φ and ψ are true
`φ ∨ ψ`	Disjunction	True iff at least one of φ, ψ is true
`φ → ψ`	Implication	False only when φ is true and ψ is false
`φ ↔ ψ`	Biconditional	True iff φ and ψ have the same truth value

Note: Truth tables give the precise semantics for each connective.

4. Quantifiers (Predicate Logic)

Symbol	Name	Informal reading
`∀x φ(x)`	Universal quantifier	“For every x, φ(x) holds.”
`∃x φ(x)`	Existential quantifier	“There exists an x such that φ(x) holds.”

Scope – The part of a formula to which a quantifier applies.
Bound vs. free variables – A variable is bound if it lies inside the scope of a quantifier; otherwise it is free.

5. Semantics: Structures & Interpretations

A structure (or model) 𝔐 for a language L consists of:

A non‑empty domain ( D ) (the set of objects we talk about).
An interpretation of each constant symbol as an element of ( D ).
An interpretation of each n‑ary function symbol as a function ( D^n → D ).
An interpretation of each n‑ary predicate symbol as a subset of ( D^n ) (i.e., a relation).

Evaluation – Given a structure 𝔐 and a variable assignment ( s ), we define the truth value of any formula recursively:

Atomic: (𝔐 \models P(t_1,\dots,t_n)[s]) iff ((\llbracket t_1 \rrbracket_s,\dots,\llbracket t_n \rr

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Introduction to Mathematical Logic

Propositional Logic

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Predicate Logic

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Inference Rules and Proofs

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Logical Equivalences

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Formal Proofs

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Set Theory and Logic

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Applications of Mathematical Logic

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Gödel's Incompleteness Theorems

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