Logic and Symbolic Reasoning
Propositional through first-order logic, deontic and defeasible logic, Prolog, argumentation frameworks and neurosymbolic methods. Your most differentiated asset, and the only approach that answers the duty to give reasons by construction.
Listen
0. Why this chapter is your strongest card
Almost nobody working in computational antitrust has formal logic. The field is dominated by people who came from machine learning or from doctrine, and the symbolic tradition in AI and Law is a largely separate literature that neither group reads.
1. Propositional logic
The simplest system. Whole statements are atoms, each true or false, combined with connectives. You cannot look inside a statement — *Acme fixed prices* is just a letter.
| Connective | Read as | True when |
|---|---|---|
| Negation, ¬ | not P | P is false |
| Conjunction, ∧ | P and Q | Both are true |
| Disjunction, ∨ | P or Q | At least one is true. Inclusive, unlike much ordinary speech |
| Implication, → | if P then Q | Whenever P is false, or Q is true. This is the one that trips everyone |
| Biconditional, ↔ | P if and only if Q | Both have the same truth value |
| Term | Definition |
|---|---|
| Valuation | An assignment of true or false to every atom |
| Satisfiable | There is at least one valuation making the formula true |
| Valid, tautology | True under every valuation |
| Entailment | Every valuation making the premises true also makes the conclusion true. Written with ⊨ |
| Conjunctive normal form | A conjunction of disjunctions. The standard input shape for automated solvers |
| SAT solver | A program that decides satisfiability. NP-complete in theory, and startlingly effective in practice on millions of variables |
2. First-order logic
Propositional logic cannot say *all undertakings*. First-order logic adds the internal structure of statements: objects, properties of objects, relations between them, and quantifiers.
| Element | Meaning | Example |
|---|---|---|
| Constant | A specific thing | acme, art101 |
| Variable | Stands for some thing | X, Y |
| Predicate | A property or relation, true or false of its arguments | undertaking(acme), agreedWith(acme, beta) |
| Function | Maps things to things, not true or false | parentOf(acme) |
| Universal quantifier, ∀ | For all | ∀X (cartel(X) → prohibited(X)) |
| Existential quantifier, ∃ | There exists at least one | ∃X (bidder(X) ∧ withdrew(X)) |
| Property | Definition |
|---|---|
| Model | An interpretation under which a set of formulas is true. Confusingly, nothing to do with machine learning models |
| Soundness | Anything the proof system derives is genuinely entailed. It cannot prove falsehoods |
| Completeness | Anything genuinely entailed can be derived. Gödel proved first-order logic is complete in this sense |
| Decidability | First-order logic is semi-decidable: if something follows you can eventually prove it, but if it does not you may search forever. This is why practical systems use restricted fragments |
3. How proofs are actually found
| Method | What it does |
|---|---|
| Natural deduction | Introduction and elimination rules per connective. Closest to how humans argue on paper |
| Resolution | Convert everything to clauses, then repeatedly combine pairs that contain a literal and its negation. The engine behind most automated provers |
| Unification | Finding the substitution of variables that makes two expressions identical. Matching agreement(X, beta) against agreement(acme, Y) yields X equals acme and Y equals beta |
| Tableaux | Try to build a counterexample systematically. If every branch closes, the formula is valid |
| Forward chaining | Start from known facts and derive everything that follows. Good when facts are few and you want all consequences |
| Backward chaining | Start from the goal and work out what would have to be true. Good when you have one specific question. This is what Prolog does |
4. The logics built for law
Standard logic describes what is. Law is mostly about what ought to be, what is permitted, and what holds at what time. Those need extensions.
| Logic | Adds | Operators |
|---|---|---|
| Modal | Necessity and possibility | □ for necessarily, ◇ for possibly |
| Deontic | Obligation, permission, prohibition. The logic of law | O for obliged, P for permitted, F for forbidden |
| Temporal | Time: always, eventually, until, next | Needed because provisions are amended and conduct has duration |
| Epistemic | Knowledge and belief. Who knew what | K for knows. Directly relevant to intent |
5. Defeasible reasoning, which is what law actually is
Classical logic is monotonic: adding premises never destroys a conclusion. Law is emphatically not like that. Every rule has exceptions, and new facts routinely overturn what followed before.
| Mechanism | What it provides |
|---|---|
| Defeasible rules | Rules that normally hold but can be defeated by a more specific one |
| Priority between rules | Which rule wins when two conflict. Law has its own principles for this: *lex specialis* favours the specific, *lex posterior* the later, *lex superior* the higher |
| Rebutting defeat | A counter-rule reaching the opposite conclusion |
| Undercutting defeat | Attacking the connection rather than the conclusion — not *that is false* but *that does not follow here* |
| Burden of proof | Who must establish what. First-class in legal reasoning and absent from classical logic entirely |
6. Description logic, ontologies and knowledge graphs
A deliberately restricted fragment of first-order logic, chosen so that reasoning stays decidable and fast. The trade is expressive power for guaranteed termination.
| Term | Definition |
|---|---|
| Concept, class | A set of things. Undertaking, Decision, InformationRequest |
| Role, property | A relation. parentOf, cites, annuls |
| Individual | A specific thing. heidelbergCement |
| Subsumption | One concept being a subtype of another. Every Cartel is an Agreement |
| TBox and ABox | The TBox holds the general vocabulary and rules; the ABox holds the facts about individuals |
| OWL, RDF, SPARQL | The web standards for writing ontologies, stating facts as triples, and querying them |
| Reasoner | A program computing subsumption and consistency. Asks: does this ontology contradict itself, and what else follows? |
7. Logic programming and Prolog
Prolog is logic as a programming language. You state facts and rules; the engine finds derivations by backward chaining. On your skills list, and worth being able to discuss concretely.
| Concept | Meaning |
|---|---|
| Fact | An unconditional truth. undertaking(acme). |
| Rule | A conditional. prohibited(X) :- cartel(X). — read the :- as *if* |
| Horn clause | At most one positive conclusion per rule. The restriction that makes Prolog efficient |
| SLD resolution | The specific proof search Prolog performs |
| Query | A goal to prove. ?- prohibited(acme). |
| Backtracking | When a branch fails, undo and try the next alternative |
| Unification | The pattern-matching that binds variables during the search |
| Closed-world assumption | Anything not derivable is treated as false |
| Negation as failure | \+ P succeeds when P cannot be proved — which is not the same as P being false |
| Relative | What it adds |
|---|---|
| Datalog | A restricted Prolog with guaranteed termination. The basis of several deductive database systems |
| Answer Set Programming | Handles multiple alternative solutions and non-monotonic reasoning more cleanly. Good for defeasible rules |
| Constraint logic programming | Adds arithmetic and constraint solving, so you can reason about thresholds and quantities as well as predicates |
| ProbLog | Prolog with probabilities on facts and rules. This is the one that matters most for you — see the research section |
| LegalRuleML, Akoma Ntoso | XML standards for marking up legal rules and legal documents so they can be processed |
8. The AI and Law tradition, which is a separate literature
Decades of work on formalising legal reasoning that the current machine learning wave largely does not read. Knowing it exists is itself a signal.
| Idea | What it says |
|---|---|
| Rule-based versus case-based reasoning | Statutes are rules you apply. Precedent is cases you compare. Different machinery, and common law systems need both |
| Factors and dimensions | Cases compared on named factors that favour one side or the other, rather than on a single similarity score. Systems like HYPO and CATO built on this |
| The isomorphism principle | The formal representation should mirror the structure of the legal source — one rule per provision, in the same order. So that when the law is amended, you know which line to change |
| Open texture | Hart's point: legal concepts have deliberately indeterminate boundaries, settled by adjudication rather than definition. *Reasonable*, *necessary*, *appreciable* |
| Hohfeld's relations | A careful taxonomy of right, duty, privilege, power, liability — more precise than ordinary legal speech |
9. Neurosymbolic methods
Combining learned components with symbolic ones, so that each does what it is good at. Five recognised patterns.
| Pattern | Division of labour |
|---|---|
| Neural perception, symbolic reasoning | The model extracts facts from messy input; the logic layer derives conclusions from those facts. The pattern that fits legal work best |
| Symbolic constraints on neural output | Logic restricts what the model may emit — constrained decoding, or a penalty for violating known rules |
| Neural guidance of symbolic search | A learned heuristic chooses which branch a prover explores, making search tractable |
| Differentiable or probabilistic logic | Attach probabilities or gradients to logical formulas, so you can learn weights and still get a derivation. ProbLog, DeepProbLog, Markov logic networks |
| Knowledge graph embeddings | Represent symbolic structure as vectors to allow approximate matching, at the cost of exactness |
10. How this connects to ATLANTIS
| Strand | What logic contributes |
|---|---|
| Fairness | The direct answer to the reason-giving problem. Article 296 wants reasons a court can retrace; a derivation is retraceable by construction, where a score and a post-hoc explanation are not |
| Accuracy, the data problem | Consistency checking. Given extracted facts and formalised provisions, you can ask mechanically whether a decision contradicts itself or whether a necessity claim is supported |
| Institutional arrangements | Formalising a provision forces the ambiguity into the open. Writing Article 18 as a rule makes you decide what *necessary* ranges over, which is the question the project exists to answer |
11. What the panel brings to this chapter
| Panel member | Where logic meets their work |
|---|---|
| VU Amsterdam itself | This is the strongest institutional point. The summer school you took, Logic as a Tool for Modelling, is their course on their campus, and you scored 9.0 in it. VU has a serious formal methods and knowledge representation tradition. You are not proposing an exotic method to a sceptical audience — you are proposing one you learned from them |
| Thibault Schrepel | Complexity science is his lens, which is dynamical rather than logical, so do not oversell. But his knowledge graph of Commission decisions is one TBox away from an ontology, and his insistence that oversight *clings to prose and PDFs, not computable data* is an argument for formal representation whether or not he frames it that way |
| Catalina Goanta | Her legal compliance API proposal is the same instinct in a different vocabulary: make legal requirements machine-checkable rather than narratively assessed. That is a formalisation argument |
| Tijmen Wisman | SyRI failed partly because its risk indicators and operation were not knowable from outside. A rule base is inspectable in precisely the way a weight matrix is not — so this chapter is a direct answer to the defect that case identified |
12. What is unexplored, and five projects
13. Your CV, mapped onto this chapter
| What you have | Where it lands | Why it fits |
|---|---|---|
| VU summer school, Logic as a Tool for Modelling, grade 9.0 | Everything here | Their course, their campus, first and second-order logic applied to legal reasoning. It is the single most on-point credential you hold for this chapter and it came from them |
| First-order modelling of actus reus and mens rea | Projects 1, 2 and 4 | You have already separated the elements of an offence into predicates and made classification depend on derivation. That is this chapter applied to criminal law |
| SWI-Prolog and logic programming | Projects 1, 2 and 3 | ProbLog is Prolog with probabilities, so Project 1 is a short step rather than a new field |
| Knowledge representation | Project 5 | Ontologies, TBox and ABox, and reasoners are this skill by name |
| Probability and statistical inference | Project 1 | Weighted logic needs both halves, and the weights have to be calibrated to mean anything |
| NLP and entity extraction — regex, POS tagging, embeddings, the hybrid paper | Projects 3 and 4 | The neural half of the neurosymbolic pipeline. Your hybrid instinct, rules where surface form is reliable and learning where meaning is open, is already the right architecture |
| Legal training in statutory interpretation and evidence | Projects 2 and 3 | Open texture, burden of proof and the isomorphism principle are legal problems first. Most people who can write the logic cannot see them |
14. If you remember ten things
- Propositional logic treats statements as atoms; first-order logic adds objects, predicates and quantifiers, so it can express *all undertakings*.
- Material implication is true whenever the condition fails — which is exactly what a conditional statutory provision means.
- First-order logic is sound and complete but only semi-decidable, which is why practical systems use restricted fragments.
- Soundness is the property no statistical model has: the conclusion follows necessarily, and every step can be printed.
- Deontic logic is the logic of obligation, permission and prohibition — the logic of law — and it has known paradoxes.
- Law is defeasible and non-monotonic. Conclusions flipping as facts arrive is legal reasoning, not a bug.
- Abstract argumentation models parties attacking each other's reasoning, which fits adversarial procedure far better than deriving truth from axioms.
- Prolog's closed-world assumption conflates *not proven* with *false*, and the presumption of innocence is the law's more careful version of the same distinction.
- The isomorphism principle: encode rules so they mirror the provisions, which is what makes a derivation a statement of reasons rather than a trace.
- The defensible claim is narrow: keep extraction learned, make the inference step reviewable. A derivation cannot omit a load-bearing premise, because the conclusion would not follow without it.