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Maths › 03. Logic and Foundations ›

4. Structures and Models

Model theory studies mathematical structures through the formal languages used to describe them, and it asks how much information about a structure can be expressed by first order formulas. A structure consists of a domain together with interpretations of the symbols in a language, and this framework allows groups, rings, fields, ordered sets, graphs, and many other mathematical objects to be treated within one common logical setting. The chapter begins...
logic model theory structures models May 2, 2026
Maths › 03. Logic and Foundations ›

2.4 Satisfaction

Satisfaction is the formal relation that connects syntax with semantics in first order logic, and it tells us when a formula is true in a structure under a given assignment of objects to its free variables. Satisfaction Let $\mathcal{M}$ be a structure with domain $M$, and let $s$ be an assignment: $$ s : \mathrm{Var} \to M $$ The notation: $$ \mathcal{M}, s \models A $$ means that the formula...
logic first order logic satisfaction models theories May 2, 2026
Maths › 03. Logic and Foundations ›

2.5 Validity and Entailment

Validity and entailment are semantic notions that compare formulas across all structures, rather than inside one fixed structure. Satisfaction tells us whether a formula is true in a particular structure under a particular assignment, while validity and entailment ask whether truth is forced in every possible interpretation of the language. Validity A sentence is valid if it is true in every structure for its language. Definition 2.41 (Validity) Let $A$...
logic first order logic validity entailment models May 2, 2026
Maths › 03. Logic and Foundations ›

2.3 Structures

In first order logic, formulas do not have meaning by themselves, and meaning is provided by specifying a structure in which the symbols of the language are interpreted as actual objects, functions, and relations. Structures A structure consists of a domain together with interpretations of all symbols in the signature, and it serves as the semantic environment in which formulas are evaluated. Definition 2.19 (Structure) A structure $\mathcal{M}$ for a...
logic first order logic structures interpretations models May 2, 2026
Maths › 00. General Mathematics ›

04.1 Structure vs Instance

4.1 Structure vs Instance A structure is a pattern. An instance is a concrete realization of that pattern. This distinction is central to mathematical reasoning. It lets us prove results once at the structural level and then apply them to many examples. A group is a structure. The integers under addition are an instance of that structure. The nonzero rational numbers under multiplication are another instance. The symmetries of a...
mathematics structure abstraction instances models May 2, 2026
Maths › 00. General Mathematics ›

02.2 Formal Systems and Semantics

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. Component Role Alphabet Symbols used to write expressions Syntax Rules for forming valid expressions Axioms Initial statements assumed true Inference rules Methods to derive new statements These components define the syntactic layer....
mathematics logic formal-systems semantics models May 2, 2026