Interpretation (logic)

An interpretation is a way of giving meaning to the symbols used in a formal language. Many languages used in mathematics, logic, and theoretical computer science start with only rules about how symbols can be put together. Interpretation gives these symbols their meaning.

The most common types of logic studied are propositional logic, predicate logic, and their modal versions. In these logics, an interpretation works like a function. It decides what certain symbols stand for. For example, it might decide that a symbol like T stands for a specific idea or object. But the interpretation does not tell us what logical words like "and," "or," or "not" mean—those are already set in the rules of logic.

An interpretation can also help us know whether a sentence in the language is true or false. When an interpretation makes a sentence true, we call that interpretation a model of the sentence or theory. This idea helps us understand what it means for something to be true in a formal system.

Formal languages

Main article: Formal language

A formal language is a set of sentences made from a fixed group of symbols, called an alphabet. These sentences follow specific rules, so we can tell which ones are correctly formed without needing to know what they mean. For example, we can see that "(P or Q)" is a properly formed sentence even if we don’t know if it’s true.

In some formal languages, like those used in logic, the symbols are split into two groups: logical symbols and non-logical symbols. Logical symbols, such as "all", "some", "and", "or", and "not", always mean the same thing. Non-logical symbols change meaning based on what we’re studying. When we give meaning to these symbols, we call it an interpretation. For example, we might decide that one symbol means the number 1 and another means the number 0, so a sequence like "△ ◻ △" could stand for the number 101.

General properties of truth-functional interpretations

Many studies look at how sentences in a special kind of language can mean true or false. These are called truth functional ways of understanding. In regular logic, a sentence can only be true or false, not both at the same time.

Some symbols in these languages are special and help connect ideas. These symbols let us build more complicated sentences from simpler ones. For example, we can say a sentence is true if another sentence is false, or if two sentences are both true together. These rules help us figure out if sentences are always true, no matter how we understand the basic ideas.

Interpretation of a theory

Main article: Theory (mathematical logic)

An interpretation of a theory links ideas from the theory to real-world topics. It shows how statements in the theory relate to specific things. When every idea in the theory matches up, it is called a full interpretation. If some ideas do not match up, it is called a partial interpretation.

Interpretations for propositional logic

The formal language for propositional logic uses special symbols and connectives to build statements. These symbols, often capital letters, need meaning to be understood.

An interpretation gives each symbol a truth value, either true or false. This is called a truth assignment. For example, if there is one symbol, there are two possible meanings: true or false. With two symbols, there are four possible combinations of true and false. This helps us see if every statement made from these symbols is true or false.

First-order logic

First-order logic is different from propositional logic because it uses many different languages. Each language is defined by a signature. This includes symbols that stand for constants, functions, or predicates. These symbols help us understand what the language can talk about.

For example, in the language of rings, we have symbols like 0 and 1 for constants, and + and · for functions.

To give meaning to sentences in first-order logic, we need an interpretation. This includes a domain (a set of objects we’re talking about), meanings for constant symbols, functions for function symbols, and relations for predicate symbols.

An example interpretation might use a chess set as its domain. Here, constants could represent specific pieces, and predicates could describe properties like “is a pawn” or “can capture another piece.” This helps us decide which sentences are true or false in that interpretation.

Higher-order predicate logics

A formal language for higher-order predicate logic is like basic logic, but it has many kinds of variables. Some variables stand for elements of a group, like in basic logic. Others stand for more complex things: groups of elements, rules that connect elements, or even rules that work with groups of elements.

There are two main ways to give meaning to these complex variables. One way includes every possible group or rule. The other way sets up separate groups for each kind of variable. Understanding how these two ways relate to each other is important in studying higher-order logic.

Non-classical interpretations

There are many ways to understand logic that are different from the usual methods. Special kinds of logic, like intuitionistic logic and modal logic, use different ways to give meaning to ideas.

These include things like topological models, Boolean-valued models, and Kripke models. Kripke models are useful for studying modal logic.

Intended interpretations

Many formal languages have a special meaning that helps explain them. For example, in a system about numbers, the basic idea is that the numbers work in the way we usually think they do.

This special meaning is called a standard model. It was named by Abraham Robinson in 1960. In a system about numbers, the standard model uses the numbers we know, like 1, 2, 3, and their usual ways of adding and multiplying. Other models that are very similar to this one are also called standard. But there can be other models that are different, with ideas that don’t match our usual numbers.

When we make these formal systems, the special meaning affects how we build them. For example, the basic symbols must let us express the ideas we want to model. The rules must make sure that what we prove is true in this special meaning. This helps keep everything consistent and useful.

Other concepts of interpretation

The word "interpretation" can mean different things. In an area called model theory, one group of ideas can "interpret" another if some parts match up in a special way.

A group of ideas called a theory can also "interpret" another theory if it can be expanded to include the other theory’s ideas.

Main article: Interpretation (model theory)