Predicate Logic · step 8 of 13

Interpretations, structures & satisfaction

A structure supplies a non-empty domain plus a meaning for every symbol in the signature: each constant names an element, each n-ary predicate is a set of n-tuples, each function symbol is an actual function on the domain. Satisfaction, written 𝔄 ⊨ φ, is then defined recursively — and this is where the semantics of first-order logic actually lives.

Practise interpretations, structures & satisfaction free → All Predicate Logic subtopics

✓ Unlimited questions · marked criterion by criterion · no card needed

Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

List the signatureNote every constant, function and predicate symbol appearing in the formula, with its arity.
Fix the domain and interpret each symbolGive the domain as a small explicit set, then a table for each predicate and function. Constants get a specific element.
Evaluate terms firstCompute the value of each closed term under the interpretation before looking at any predicate.
Work outward through the connectives and quantifiers∀x φ holds iff φ holds for every element assigned to x; ∃x φ iff for at least one. Show the case check.

Worked example

Let D = {1, 2, 3}, R = {(1,2), (2,3)}. Does 𝔄 ⊨ ∀x ∀y ∀z ((R(x,y) ∧ R(y,z)) → R(x,z))?

The formula asserts transitivity of R.
Search for a violating triple: we need R(x,y) and R(y,z) both in R.
Take x = 1, y = 2, z = 3: (1,2) ∈ R and (2,3) ∈ R.
For transitivity we would need (1,3) ∈ R, but it is not.

Answer. No — the structure does not satisfy the formula, witnessed by x = 1, y = 2, z = 3.

Where marks get dropped

These are the specific errors that cost credit on interpretations, structures & satisfaction questions — QED's rubric penalises each of them separately.

Allowing an empty domain. First-order structures require a non-empty domain, which is why ∀x P(x) → ∃x P(x) is valid.
Interpreting a function symbol partially. Every function symbol must be total on the domain, or the structure is not well-defined.
Evaluating quantifiers without checking every element. Over a three-element domain a ∀ claim needs three checks, not one.

Practise this until it is automatic

Unlimited fresh questions

QED generates new interpretations, structures & satisfaction problems on demand at warm-up, exam and challenge level, so you can drill this one skill until it stops costing you marks.

Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.

Saved to your library

Every question you generate is kept and re-takeable as a timed exam, and your Predicate Logic mastery is tracked so you know when this is exam-ready.

Interpretations, structures & satisfaction — frequently asked questions

What is the difference between a model and a structure?

A structure is any interpretation of the signature. It is a model of a formula (or theory) when it satisfies it. Every model is a structure; most structures are not models of a given sentence.

Why must the domain be non-empty?

The standard semantics assumes it, which keeps ∀x P(x) → ∃x P(x) valid and simplifies the proof rules. Free logics drop this assumption at the cost of extra machinery.

Does an assignment matter for sentences?

No. A sentence has no free variables, so its truth value depends only on the structure. Assignments matter only for formulas with free variables.

The rest of Predicate Logic

Quantifiers, predicates, binding, and validity. Each subtopic below has its own method, worked example and mark-losing traps.

Ready to make interpretations, structures & satisfaction exam-proof?

Generate your first questions free — no card, no setup, no personal data stored. Practise until the method is second nature.

Start practising free →