[IA Series 11/n] The Grammar of Logic — A Term Sheet

Thu, 13 Aug 2026 12:59:30 +0200

This is a term sheet for the grammar of logic, in the same spirit as the original Intelligent Agents Term Sheet. Most terms come from Russell and Norvig’s ‘Artificial Intelligence: A Modern Approach’ (Chapters 7-8) and the standard literature on formal logic.

Introduction

The ontology work has kept using a vocabulary I never formally defined — predicates, derived rules, well-formedness — all of it written in the language of logic. Before going further I wanted a term sheet for the grammar of that language: what the symbols are, what strings count as well-formed, and what a well-formed formula means. This post is that term sheet. It is deliberately lean; the aim is a shared vocabulary, not a textbook.

The Grammar — Core Terminology

The symbols

Formal language: A set of strings built from an alphabet according to formation rules.
Alphabet: The symbols a formal language is built from — logical constants, variables, predicates, functions, and punctuation.
Connectives: The logical constants that combine sentences.
- Negation (¬): not — flips a sentence’s truth value.
- Conjunction (∧): and — true only when both sides are true.
- Disjunction (∨): or — true when at least one side is true.
- Implication (⇒): if … then … — false only when the antecedent is true and the consequent is false.
- Biconditional (⇔): if and only if — true when both sides agree in truth value.
Quantifiers: The logical constants that range over objects.
- Universal (∀): for all.
- Existential (∃): there exists.
Equality (=): a predicate asserting two terms denote the same object.
Parentheses: grouping, and the grammar’s only punctuation.

Terms

Term: An expression that refers to an object — a constant, a variable, or a function applied to terms.
- Constant symbol: Names a specific object (e.g., pr-42).
- Variable symbol: Ranges over objects (e.g., x).
- Function symbol: Maps objects to objects (e.g., parent(x)).

Predicates and sentences

Predicate symbol: A relation over objects. Applied to terms it yields an atomic sentence.
Atomic sentence (atom): A predicate applied to terms — the smallest well-formed sentence (e.g., merged(pr-42)).
Compound sentence: Atoms combined with connectives and quantifiers.
Well-formed formula (wff): A string that obeys the formation rules — the grammar’s definition of a legal sentence.
Sentence: A well-formed formula with no free variables (a closed formula) — a claim that has a definite truth value in a model.

Scope and binding

Scope: The part of a formula governed by a quantifier.
Bound variable: A variable within the scope of its own quantifier.
Free variable: A variable not bound by any quantifier. A formula with free variables is open; it has no definite truth value until the variables are assigned.

The Grammar, stated

The formation rules, in BNF — this is the grammar of logic proper:

Sentence         → AtomicSentence | ComplexSentence
AtomicSentence   → Predicate(Term, ...) | Term = Term
ComplexSentence  → ( Sentence )
                 | ¬ Sentence
                 | Sentence ∧ Sentence
                 | Sentence ∨ Sentence
                 | Sentence ⇒ Sentence
                 | Sentence ⇔ Sentence
                 | Quantifier Variable, ... Sentence
Term             → Function(Term, ...) | Constant | Variable

Everything else is not a sentence of the language — no matter how plausible it reads.

This grammar declares facts — predicates and the sentences built from them. Actions, operations with preconditions and effects, are the planning half, written over this grammar’s sentences rather than inside it.

Semantics — Meaning

Interpretation: An assignment of meaning to a language — a domain plus a mapping that names the constants, predicates, and functions in it.
Model: An interpretation that makes a given sentence (or set of sentences) true.
Determination: A sentence has no intrinsic truth. The grammar determines what is well-formed; truth is determined relative to an interpretation, and which interpretation is intended is attributed — chosen by a human modeling the world, not discovered. No truth without a model; no model without an attributed interpretation.
Domain (universe): The set of objects an interpretation ranges over.
Truth value: True or false — what a sentence receives relative to an interpretation.
Truth table: A complete enumeration of a connective’s truth behaviour for all combinations of its arguments.
Satisfaction: A formula is satisfied by an interpretation when it is true in it.
Entailment (⊨): KB ⊨ α — every model of the knowledge base is also a model of α. The semantic notion of logical consequence.
Validity: A sentence true in every interpretation (a tautology).
Satisfiability: A sentence true in some interpretation.
Contingency: A sentence true in some interpretations and false in others — neither valid nor contradictory.
Contradiction (unsatisfiability): A sentence true in no interpretation.
Logical equivalence (≡): Two sentences true in exactly the same interpretations.

Reasoning — Using the Grammar

Knowledge base (KB): The set of sentences an agent holds to be true.
Inference: Deriving new sentences from existing ones.
Derivation (⊢): KB ⊢ α — α is reachable from the KB by applying inference rules. The syntactic notion of proof.
Sound inference: Never derives a false conclusion — every derived sentence is entailed.
Complete inference: Can derive every entailed sentence.
Inference rules: The grammar’s rewrite moves — e.g., modus ponens (if α ⇒ β and α then β), conjunction introduction (α, β ⊢ α ∧ β), resolution (a complete rule for propositional and first-order clauses).
Entailment vs derivation: Entailment (⊨) says what is true given the KB; derivation (⊢) says what is provable from it.
Soundness theorem: If ⊢ then ⊨ — the proof system never lies.
Completeness theorem: If ⊨ then ⊢ — everything true is provable.
Decidability: Whether a proof procedure is guaranteed to terminate with an answer. Propositional entailment is decidable; first-order entailment is not in general.

Closing

The grammar is the substrate the world ontology is written in: predicates are atomic sentences, Kinds classify how their truth is determined, and a derived predicate is an entailment stated by a rule. This term sheet names the pieces so that work can be described precisely.

logic on Context may be all you need