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⭐ Core Definition: First-order theory

In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, giving rise to a formal system that combines the language with deduction rules. An element $\phi \in T$ of a deductively closed theory $T$ is then called a theorem of the theory. In many deductive systems there is usually a subset $\Sigma \subseteq T$ that is called "the set of axioms" of the theory $T$ , in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.

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⭐ Core Definition: First-order theory
First-order theory in the context of Sequent calculus
First-order theory in the context of Mereotopology

First-order theory in the context of Sequent calculus

In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than David Hilbert's earlier style of formal logic, in which every line was an unconditional tautology. More subtle distinctions may exist; for example, propositions may implicitly depend upon non-logical axioms. In that case, sequents signify conditional theorems of a first-order theory rather than conditional tautologies.

Sequent calculus is one of several extant styles of proof calculus for expressing line-by-line logical arguments.

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First-order theory in the context of Mereotopology

In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts.

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First-order theory in the context of "Deductively closed"

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⭐ Core Definition: First-order theory

First-order theory in the context of Sequent calculus

First-order theory in the context of Mereotopology