Model theory is the part of formal logic that studies how a formal language is interpreted in a model. It asks which sentences are true in which structures, and how that connects to proof and completeness in Formal Logic I.
Model theory is the study of how a formal language gets interpreted in a structure, or model, inside Formal Logic I. A sentence is not just a string of symbols here, because its truth depends on what the variables, predicates, and functions are taken to mean in a particular domain.
That means model theory sits on the semantic side of logic. Syntax is the system of symbols and proof rules, while semantics is about meaning and truth in an interpretation. If you write a formula like P(a) or \u2200x(P(x) \u2192 Q(x)), model theory asks what has to be true in the structure for that formula to come out true.
A model usually includes a domain of objects plus an interpretation for the nonlogical symbols in the language. The domain could be numbers, people, sets, or any other collection of objects the language is talking about. Predicates pick out subsets of the domain, function symbols name operations, and constants name particular elements.
The central idea is satisfaction. A structure satisfies a sentence when that sentence comes out true under the interpretation. If every model of a theory makes a sentence true, then the sentence is a semantic consequence of the theory. That is the bridge model theory builds toward completeness: if something is true in every model, then in a complete deductive system it can be proved from the axioms.
A simple example makes the idea clearer. Suppose a theory has axioms saying that every object has exactly one successor and that one object is the successor of no one. In one model, the domain might be the natural numbers. In another, it might be an infinite chain of abstract elements. Model theory looks at which sentences both structures satisfy, which sentences separate them, and what that tells you about the theory itself.
This is why model theory is more than a vocabulary list. It lets you compare different interpretations of the same formal language, test whether axioms really pin down a structure, and see where proof and truth line up. In Formal Logic I, that makes model theory the place where abstract symbol manipulation turns into a claim about meaning in a structure.
Model theory is the tool that lets Formal Logic I move from "Can I prove this?" to "Is this true in every structure that fits the axioms?" That shift matters because logic is not only about valid derivations, it is also about what a theory actually says about its models.
It gives you a way to test whether a set of axioms is strong enough. If the same axioms have many different models, then the theory may describe a pattern without fully fixing one unique structure. That is a big deal in logic, because it shows why some formal systems characterize a class of structures instead of a single object.
Model theory also clarifies why completeness is such a central result. When you see a statement that is semantically true in every model, model theory explains what that truth means before any proof is written down. Then completeness tells you the proof system can capture that truth syntactically.
This concept also helps when you compare finite and infinite models. A sentence can hold in one kind of structure and fail in another, even when the structures look similar at first glance. That kind of comparison shows up in class discussions about why formal languages are powerful, but not unlimited, in what they can express.
If you can think in terms of models, you can read logic problems more precisely. Instead of treating formulas as abstract symbols only, you start asking what kind of structure makes them true, which axioms constrain that structure, and which sentences follow from those constraints.
Keep studying Formal Logic I Unit 13
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view galleryInterpretation
An interpretation gives meaning to the symbols in a formal language, so model theory depends on it from the start. The same formula can be true in one interpretation and false in another, which is exactly the sort of comparison model theory studies. When you assign a domain and meanings to predicates and functions, you are building the structure that a sentence will be evaluated in.
Satisfaction
Satisfaction is the relation between a model and a sentence that says the sentence is true in that model. Model theory uses satisfaction as its main test for semantic truth. When your course asks whether a structure satisfies a set of axioms, you are doing model-theoretic work, even if the problem does not use that label every time.
Completeness
Completeness connects model theory to proof theory by saying that semantic truth implies provability in a complete system. If every model of a theory makes a sentence true, completeness says the sentence should be derivable from the theory's axioms. That is the exact bridge between meaning in structures and formal proof.
logical consequence
Logical consequence is the idea that a conclusion follows from premises in every model where the premises are true. Model theory gives you the semantic side of that idea. Instead of checking only whether a proof exists, you ask whether all structures satisfying the premises also satisfy the conclusion.
A problem set question may give you a structure and ask whether a sentence is satisfied, or it may describe a theory and ask which statements are true in all of its models. You might also need to explain why two different structures are both models of the same axioms, or why one sentence separates them. In a proof-based question, model theory shows up when you justify semantic consequence, not just when you manipulate symbols.
On a quiz, the usual move is to identify the domain, interpret the nonlogical symbols, and then check truth sentence by sentence. If the course asks about completeness, you may need to say that every semantically valid sentence is provable from the axioms. That is the kind of answer that shows you can connect models, truth, and derivation instead of treating them as separate topics.
Interpretation is the assignment of meanings to the symbols of a language, while model theory is the broader study of how those interpretations work and what sentences they make true. Think of interpretation as one part of the setup and model theory as the field that studies the setup, the truth relation, and the structural consequences.
Model theory studies how formal sentences are true or false in a structure, not just how they are written as symbols.
A model includes a domain plus interpretations for the language's symbols, and that setup determines satisfaction.
The big bridge in Formal Logic I is between semantic truth and syntactic proof, especially through completeness.
Different models can satisfy the same axioms, which shows that a theory may describe a class of structures instead of one unique object.
When you work model-theoretically, you are checking what follows in every interpretation, not only what can be derived mechanically.
Model theory is the study of how formal languages are interpreted in structures and how sentences are evaluated as true or false there. In Formal Logic I, it is the semantic side of logic, so it connects your formulas to a model's domain, symbols, and truth conditions.
Interpretation is the act of assigning meanings to the symbols of a language. Model theory is the larger area that studies those interpretations, the structures they create, and which sentences are satisfied in them. So interpretation is part of the machinery, while model theory is the theory of that machinery.
Completeness says that if a sentence is true in every model of a theory, then it can be proved from the theory's axioms. Model theory sets up the semantic side of that claim by describing what it means for a sentence to hold in all models. The two ideas fit together directly in Formal Logic I.
You might be given a structure with a domain of integers or abstract objects and asked whether a formula like a universal statement is satisfied. Another common task is comparing two models that satisfy the same axioms but differ on a specific sentence. That kind of problem shows how the same formal language can describe different structures.