Interpretation (logic)

interpretationinterpretationsinterpretedtruth assignmentintended interpretationmodelintended modelintended semanticsEvaluationinterpret
An interpretation is an assignment of meaning to the symbols of a formal language.wikipedia
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Semantics of logic

formal semanticssemanticslogical semantics
The general study of interpretations of formal languages is called formal semantics.
In logic, the semantics of logic is the study of the semantics, or interpretations, of formal and (idealizations of) natural languages usually trying to capture the pre-theoretic notion of entailment.

Propositional calculus

propositional logicpropositionalsentential logic
The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation.
Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions.

Symbol (formal)

symbolsymbolsletters
An interpretation is an assignment of meaning to the symbols of a formal language.
Symbols of a formal language must be capable of being specified without any reference to any interpretation of them.

Satisfiability

satisfiablesatisfiability problemsatisfies
The sentences that are made true by a particular assignment are said to be satisfied by that assignment.
A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true.

First-order logic

predicate logicfirst-orderpredicate calculus
The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation.
An interpretation of a first-order language assigns a denotation to all non-logical symbols in that language.

Consistency

consistentinconsistentlogically consistent
A sentence is consistent if it is true under at least one interpretation; otherwise it is inconsistent.
The semantic definition states that a theory is consistent if and only if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true.

Logical constant

constantsconstantlogical constants
In the specific cases of propositional logic and predicate logic, the formal languages considered have alphabets that are divided into two sets: the logical symbols (logical constants) and the non-logical symbols.
In logic, a logical constant of a language \mathcal{L} is a symbol that has the same semantic value under every interpretation of \mathcal{L}.

Logical consequence

entailsentailmentfollows from
A sentence φ is said to be logically valid if it is satisfied by every interpretation (if φ is satisfied by every interpretation that satisfies ψ then φ is said to be a logical consequence of ψ).
Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation.

Sentence (mathematical logic)

sentencesentencesclosed formula
An interpretation often (but not always) provides a way to determine the truth values of sentences in a language. For example, primitive signs must permit expression of the concepts to be modeled; sentential formulas are chosen so that their counterparts in the intended interpretation are meaningful declarative sentences; primitive sentences need to come out as true sentences in the interpretation; rules of inference must be such that, if the sentence is directly derivable from a sentence, then turns out to be a true sentence, with meaning implication, as usual.
To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory.

Structure (mathematical logic)

structuremodelstructures
If a given interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentence or theory.
Logicians sometimes refer to structures as interpretations.

Formal language

formal language theoryformal languageslanguage
An interpretation is an assignment of meaning to the symbols of a formal language.
For instance, in mathematical logic, the set of possible formulas of a particular logic is a formal language, and an interpretation assigns a meaning to each of the formulas—usually, a truth value.

Well-formed formula

formulaformulaswell-formed
A formal language consists of a possibly infinite set of sentences (variously called words or formulas) built from a fixed set of letters or symbols.
A formula A in a language \mathcal{Q} is valid if it is true for every interpretation of \mathcal{Q}.

Löwenheim–Skolem theorem

(downward) Löwenheim–Skolem propertydownward Löwenheim–Skolem theoremLöwenheim-Skolem theorem
Second, if non-normal models are considered, then every consistent theory has an infinite model; this affects the statements of results such as the Löwenheim–Skolem theorem, which are usually stated under the assumption that only normal models are considered.
In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ.

Domain of discourse

universe of discoursedomainarea of interest
A domain of discourse D, usually required to be non-empty (see below). A model in the empirical sciences is an intended factually-true descriptive interpretation (or in other contexts: a non-intended arbitrary interpretation used to clarify such an intended factually-true descriptive interpretation.) All models are interpretations that have the same domain of discourse as the intended one, but other assignments for non-logical constants.
For example, in an interpretation of first-order logic, the domain of discourse is the set of individuals over which the quantifiers range.

Truth-bearer

truthbearerprimary bearerstruth bearer
In many presentations, it is literally a truth value that is assigned, but some presentations assign truthbearers instead.
In classical logic a sentence in a language is true or false under (and only under) an interpretation and is therefore a truth-bearer.

Truth

truetheory of truthtruth theory
For example, primitive signs must permit expression of the concepts to be modeled; sentential formulas are chosen so that their counterparts in the intended interpretation are meaningful declarative sentences; primitive sentences need to come out as true sentences in the interpretation; rules of inference must be such that, if the sentence is directly derivable from a sentence, then turns out to be a true sentence, with meaning implication, as usual.
Logicians use formal languages to express the truths which they are concerned with, and as such there is only truth under some interpretation or truth within some logical system.

Model theory

modelmodelsmodel-theoretic
Model theory
A set of sentences in a formal language is one of the components that form a theory. A model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory.

Non-standard model

nonstandard modelnon-standard modelsnonstandard logic
Most formal systems have many more models than they were intended to have (the existence of non-standard models is an example).
In model theory, a discipline within mathematical logic, a non-standard model is a model of a theory that is not isomorphic to the intended model (or standard model).

Intuitionistic logic

intuitionisticconstructive logicconstructive
In particular, there are other types of interpretations that are used in the study of non-classical logic (such as intuitionistic logic), and in the study of modal logic.
The interpretation of any intuitionistically valid formula in the infinite Heyting algebra described above results in the top element, representing true, as the valuation of the formula, regardless of what values from the algebra are assigned to the variables of the formula.

Valuation (logic)

valuationassignmentvaluations
A model in the empirical sciences is an intended factually-true descriptive interpretation (or in other contexts: a non-intended arbitrary interpretation used to clarify such an intended factually-true descriptive interpretation.) All models are interpretations that have the same domain of discourse as the intended one, but other assignments for non-logical constants.
In first-order logic and higher-order logics, a structure, (the interpretation) and the corresponding assignment of a truth value to each sentence in the language for that structure (the valuation proper). The interpretation must be a homomorphism, while valuation is simply a function.

Non-logical symbol

non-logical symbolsdescriptive signnon-logical
A model in the empirical sciences is an intended factually-true descriptive interpretation (or in other contexts: a non-intended arbitrary interpretation used to clarify such an intended factually-true descriptive interpretation.) All models are interpretations that have the same domain of discourse as the intended one, but other assignments for non-logical constants. The only non-logical symbols in a formal language for propositional logic are the propositional symbols, which are often denoted by capital letters.
A non-logical symbol only has meaning or semantic content when one is assigned to it by means of an interpretation.

Formal system

logical systemdeductive systemsystem of logic
While the intended interpretation can have no explicit indication in the strictly formal syntactical rules, it naturally affects the choice of the formation and transformation rules of the syntactical system.
According to model-theoretic interpretation, the semantics of a logical system describe whether a well-formed formula is satisfied by a given structure.

Formal proof

prooflogical prooflogical proofs
For example, primitive signs must permit expression of the concepts to be modeled; sentential formulas are chosen so that their counterparts in the intended interpretation are meaningful declarative sentences; primitive sentences need to come out as true sentences in the interpretation; rules of inference must be such that, if the sentence is directly derivable from a sentence, then turns out to be a true sentence, with meaning implication, as usual.
Such a language can be defined without reference to any meanings of any of its expressions; it can exist before any interpretation is assigned to it – that is, before it has any meaning.

Conceptual model

modelmodelsschema
Conceptual model
In logic, a model is a type of interpretation under which a particular statement is true.

Herbrand interpretation

interpretation
Herbrand interpretation
Interpretation (logic)