An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.
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. In these contexts an interpretation is a function that provides the extension of symbols and strings of an object language. For example, an interpretation function could take the predicate symbol and assign it the extension . All our interpretation does is assign the extension to the non-logical symbol , and does not make a claim about whether is to stand for tall and for Abraham Lincoln. On the other hand, an interpretation does not have anything to say about logical symbols, e.g. logical connectives "", "" and "". Though we may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.
An interpretation often (but not always) provides a way to determine the truth values of sentences in a language. If a given interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentence or theory.
Formal languages
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. The inventory from which these letters are taken is called the alphabet over which the language is defined. To distinguish the strings of symbols that are in a formal language from arbitrary strings of symbols, the former are sometimes called well-formed formulæ (wff). The essential feature of a formal language is that its syntax can be defined without reference to interpretation. For example, we can determine that (P or Q) is a well-formed formula even without knowing whether it is true or false.
Example
A formal language can be defined with the alphabet , and with a word being in if it begins with and is composed solely of the symbols and .
A possible interpretation of could assign the decimal digit '1' to and '0' to . Then would denote 101 under this interpretation of .
Logical 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. The idea behind this terminology is that logical symbols have the same meaning regardless of the subject matter being studied, while non-logical symbols change in meaning depending on the area of investigation.
Logical constants are always given the same meaning by every interpretation of the standard kind, so that only the meanings of the non-logical symbols are changed. Logical constants include quantifier symbols ∀ ("all") and ∃ ("some"), symbols for logical connectives ∧ ("and"), ∨ ("or"), ¬ ("not"), parentheses and other grouping symbols, and (in many treatments) the equality symbol =.
General properties of truth-functional interpretations
Many of the commonly studied interpretations associate each sentence in a formal language with a single truth value, either True or False. These interpretations are called truth functional; they include the usual interpretations of propositional and first-order logic. The sentences that are made true by a particular assignment are said to be satisfied by that assignment.
In classical logic, no sentence can be made both true and false by the same interpretation, although this is not true of glut logics such as LP.Priest, Graham, 2008. An Introduction to Non-Classical Logic: from If to Is, 2nd ed. Cambridge University Press. Even in classical logic, however, it is possible that the truth value of the same sentence can be different under different interpretations. A sentence is consistent if it is true under at least one interpretation; otherwise it is inconsistent. 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 connectives
Some of the logical symbols of a language (other than quantifiers) are truth-functional connectives that represent truth functions — functions that take truth values as arguments and return truth values as outputs (in other words, these are operations on truth values of sentences).
The truth-functional connectives enable compound sentences to be built up from simpler sentences. In this way, the truth value of the compound sentence is defined as a certain truth function of the truth values of the simpler sentences. The connectives are usually taken to be logical constants, meaning that the meaning of the connectives is always the same, independent of what interpretations are given to the other symbols in a formula.
This is how we define logical connectives in propositional logic: ¬Φ is True iff Φ is False. (Φ ∧ Ψ) is True iff Φ is True and Ψ is True. (Φ ∨ Ψ) is True iff Φ is True or Ψ is True (or both are True). (Φ → Ψ) is True iff ¬Φ is True or Ψ is True (or both are True). (Φ ↔ Ψ) is True iff (Φ → Ψ) is True and (Ψ → Φ) is True.
So under a given interpretation of all the sentence letters Φ and Ψ (i.e., after assigning a truth-value to each sentence letter), we can determine the truth-values of all formulas that have them as constituents, as a function of the logical connectives. The following table shows how this kind of thing looks. The first two columns show the truth-values of the sentence letters as determined by the four possible interpretations. The other columns show the truth-values of formulas built from these sentence letters, with truth-values determined recursively.
+ Logical connectives Interpretation Φ Ψ ¬Φ (Φ ∧ Ψ) (Φ ∨ Ψ) (Φ → Ψ) (Φ ↔ Ψ)#1 T T F T T T T#2 T F F F T F F#3 F T T F T T F#4 F F T F F T T
Now it is easier to see what makes a formula logically valid. Take the formula F: (Φ ∨ ¬Φ). If our interpretation function makes Φ True, then ¬Φ is made False by the negation connective. Since the disjunct Φ of F is True under that interpretation, F is True. Now the only other possible interpretation of Φ makes it False, and if so, ¬Φ is made True by the negation function. That would make F True again, since one of Fs disjuncts, ¬Φ, would be true under this interpretation. Since these two interpretations for F are the only possible logical interpretations, and since F comes out True for both, we say that it is logically valid or tautologous.
Interpretation of a theory
An interpretation of a theory is the relationship between a theory and some subject matter when there is a many-to-one correspondence between certain elementary statements of the theory, and certain statements related to the subject matter. If every elementary statement in the theory has a correspondent it is called a full interpretation, otherwise it is called a partial interpretation.
Interpretations for propositional logic
The formal language for propositional logic consists of formulas built up from propositional symbols (also called sentential symbols, sentential variables, propositional variables) and logical connectives. The only non-logical symbols in a formal language for propositional logic are the propositional symbols, which are often denoted by capital letters. To make the formal language precise, a specific set of propositional symbols must be fixed.
The standard kind of interpretation in this setting is a function that maps each propositional symbol to one of the truth values true and false. This function is known as a truth assignment or valuation function. In many presentations, it is literally a truth value that is assigned, but some presentations assign truthbearers instead.
For a language with n distinct propositional variables there are 2n distinct possible interpretations. For any particular variable a, for example, there are 21=2 possible interpretations: 1) a is assigned T, or 2) a is assigned F. For the pair a, b there are 22=4 possible interpretations: 1) both are assigned T, 2) both are assigned F, 3) a is assigned T and b is assigned F, or 4) a is assigned F and b is assigned T.
Given any truth assignment for a set of propositional symbols, there is a unique extension to an interpretation for all the propositional formulas built up from those variables. This extended interpretation is defined inductively, using the truth-table definitions of the logical connectives discussed above.
First-order logic
Unlike propositional logic, where every language is the same apart from a choice of a different set of propositional variables, there are many different first-order languages. Each first-order language is defined by a signature. The signature consists of a set of non-logical symbols and an identification of each of these symbols as either a constant symbol, a function symbol, or a predicate symbol. In the case of function and predicate symbols, a natural number arity is also assigned. The alphabet for the formal language consists of logical constants, the equality relation symbol =, all the symbols from the signature, and an additional infinite set of symbols known as variables.
For example, in the language of rings, there are constant symbols 0 and 1, two binary function symbols + and ·, and no binary relation symbols. (Here the equality relation is taken as a logical constant.)
Again, we might define a first-order language L, as consisting of individual symbols a, b, and c; predicate symbols F, G, H, I and J; variables x, y, z; no function letters; no sentential symbols.
Formal languages for first-order logic
Given a signature σ, the corresponding formal language is known as the set of σ-formulas. Each σ-formula is built up out of atomic formulas by means of logical connectives; atomic formulas are built from terms using predicate symbols. The formal definition of the set of σ-formulas proceeds in the other direction: first, terms are assembled from the constant and function symbols together with the variables. Then, terms can be combined into an atomic formula using a predicate symbol (relation symbol) from the signature or the special predicate symbol "=" for equality (see the section "Interpreting equality" below). Finally, the formulas of the language are assembled from atomic formulas using the logical connectives and quantifiers.
Interpretations of a first-order language
To ascribe meaning to all sentences of a first-order language, the following information is needed. A domain of discourse D, usually required to be non-empty (see below). For every constant symbol, an element of D as its interpretation. For every n-ary function symbol, an n-ary function from D to D as its interpretation (that is, a function Dn → D). For every n-ary predicate symbol, an n-ary relation on D as its interpretation (that is, a subset of Dn). An object carrying this information is known as a structure ( signature σ), or σ-structure, or L-structure (of language L), or as a "model".
The information specified in the interpretation provides enough information to give a truth value to any atomic formula, after each of its free variables, if any, has been replaced by an element of the domain. The truth value of an arbitrary sentence is then defined inductively using the T-schema, which is a definition of first-order semantics developed by Alfred Tarski. The T-schema interprets the logical connectives using truth tables, as discussed above. Thus, for example, is satisfied if and only if both φ and ψ are satisfied.