# What is the difference between a model and an interpretation in logic?

On page 319 of Irving M. Copi's 'Symbolic Logic', he states, "if we want our logical system to be applicable to any possible universe, regardless of the exact number of individuals it contains there is no N_j such that we want N_j or ~N_j provable as theorems, where N_j is the statement that there are at most j individuals.

Then on page 320 he states "instead of continuing to speak of 'possible universes', we shall talk about models, where a model is any non-empty collection of elements each of which is thought of as an individual. And instead of speaking of our system of logic as being applied to a possible universe, we shall speak of a model as constituting an interpretation of our formal system."

Then he gives a rather complicated definition of an interpretation.

This is all done to prove the completeness of the first order function calculus RS_1. What I'm wondering, is whether or not there is a simpler definition of interpretation than the one offered by Copi, and whether his definition of the term 'model' is the same one that appears in the statement, "A logical constant is a symbol of symbolic logic that has the same meaning in all models"?

• Model is relative to a set T of formulas: it is an interpretation that satisfy all formulas of T. Commented May 11 at 5:56
• The terms are often confused. A model/interpretation for a Language of Predicate Calculus is an Ordered Pair (X,I) where X is a set and I is an interpretation function. A model of a Theory (Set of Sentences) is an interpretation of a Language, which satisfies every sentence in the Theory. Commented May 12 at 3:02
• If the interpretation function isn't mentioned explicitly, most likely they mean the standard interpretation. I.E. if it's in the context of Set Theory, we interpret ∈ in the usual way. ( such models are called ∈-models) If it's Arithmetic, we interpret 1,0, <,+ in the usual way etc... Commented May 12 at 3:23
• Yes, it is the same notion of model as used in "A logical constant is a symbol of..." Commented May 12 at 3:31

Unfortunately, those terms are used in different ways by different logicians. It is fairly standard to say that an interpretation of a sentence in a formal language is an assignment of meanings to its terms, i.e. an interpretation assigns a referent to names, classes to predicates and functions, truth values to propositional variables. A model of a sentence is an interpretation under which the sentence is true.

By extension we can also speak of interpretations and models of entire theories, i.e. of classes of sentences. So an interpretation of a theory is one that provides an interpretation for all its sentences, and a model of a theory is an interpretation under which all of its sentences are true.

So for example, a sentence Fa may have many interpretations. If we take our domain to be the real world, and interpret the predicate F to mean "is a professional football player", and the name a to mean Lionel Messi, then that interpretation is a model of that sentence, since the sentence is true under that interpretation. If instead we interpret a to mean Gordon Ramsey, the sentence would be false under that interpretation and so the interpretation would not be a model of it.

However, some writers use model and interpretation interchangeably, which is rather confusing. Also, interpretation is sometimes used in a broader sense to refer to the relation under which one structure can be mapped into another.

• In your rather learned opinion, is interpretation nothing more than an attempt at rational (as in consistent) variable binding of tokens to meanings, which themselves construed as linearized collections of tokens masquerading as definienda and definientia?
– J D
Commented May 10 at 22:55
• In the extended sense, an interpretation, or sometimes a relative interpretation, refers to the ability of one structure to interpret another, or one theory to interpret another. This amounts to saying that there exists a syntactic relation that provides a mapping in such a way that definitions and theorems are preserved. A simple example is that a complex number can be interpreted as an ordered pair of real numbers. Commented May 11 at 14:50
• Graham Priest says in p. 121 that model, interpretation, structure and situation are the same notion. Commented Jun 5 at 8:57

In Ebbinghaus 2021, the four important terms regarding the semantics of a language are a structure, an assignment, an interpretation, and a model.

An alphabet consists of eight distinct groups of symbols: quantifiers, variables, constants, functions, relations, punctuation, logical symbols, and equality.

When we supply rules for constructing terms and formula from those symbols, we have a language. This is basically like a grammar. In model theory, variables, constants, and functions are terms; and relations are formula.

A structure is a map from the function symbols, relation symbols, and constant symbols of the alphabet to some actual functions, relations and values, in some semantic domain.

When we have a structure, we are now closer to knowing what referents the terms of the language refer to.

An assignment is a mapping of the variables to specific values. (I am now realizing I do not know how this interacts with quantifiers).

An interpretation is a structure with an assignment. When we have an interpretation, we now know everything the language is saying, so we can determine if a given formula written in the language says something true or not.

A model of a formula just means a structure which makes the formula true (for all possible variable assignments). You can think of it intuitively as a structure for which the formula is universally true!

We recycle the same term ‘model’ for sets of formulae: if we find a structure (by interpreting components of a language as referring to particular things); and if all the formulae in a set of formulae are true (for all possible values of the variables); we say that that structure is a model of that set of formula. The model is just an example of something which the formula describe meaningfully.

Copi calls a “model” what Ebbinghaus calls a “structure”. Ebbinghaus considers a model a structure which actually interprets a set of formula well. But Copi considers a model “any semantic assignment for the language, whether it is a good one or not.”

• An interpretation would be valid with respect to a WFF, not a language, wouldn't it? Commented May 10 at 20:28
• @david gudeman, on page 320 Copi states Now we define interpretation of a WFF S with respect to a given model as an assignment of meanings such that, etc. Then further down he says, "An interpretation of a system with respect to a given model is an assignment of meanings which provides normal interpretations for all WFFS of the system. Commented May 10 at 20:48
• @julius, if a structure maps constant symbols of the language to actual values, would that apply to the universal and existential quantifiers as well? They are constants but they don't map to values. Does he mean," map individual constants to values"? Commented May 10 at 20:59
• @DavidGudeman I am definitely missing something; still learning. I realized I don’t know where the axioms of a theory enter the picture; and a model is something that satisfies a theory, not a language (I think). I’ll keep reading Ebbinghaus. Commented May 10 at 21:08
• @julius, I understood everything you said, except one thing. You said, when you choose a structure you are going to have to decide what 'Socrates'' refers to - and at that point the meaning of 'Socrates'' will not change. Don't you mean, "at that point the referent will not change. The referrer 'Socrates'' is meaningless Commented May 10 at 22:36