The Lexicon · concepts in two tongues
The vocabulary of logic
The core concepts, each defined precisely, in English and in Arabic. Rigorous technical Arabic for logic is scarce; this is a small attempt to set it down.
- Proposition القضية p, q
- A declarative statement that is either true or false, the basic bearer of truth in logic. in Basic Concepts
- Validity الصلاحية
- A property of arguments: the conclusion is true in every interpretation that makes all the premises true. in Basic Concepts
- Soundness الإحكام
- A calculus is sound when everything it proves is valid: it never derives a falsehood from truths. in Mathematical Logic
- Completeness الاكتمال
- A calculus is complete when every valid statement can be proved within it (Gödel’s completeness theorem, for first-order logic). in Mathematical Logic
- Consistency الاتساق
- A theory is consistent if it proves no contradiction, never both a statement and its negation. in Mathematical Logic
- Quantifier المُسوِّر ∀ ∃
- A symbol binding a variable over a domain: the universal ∀ (“for all”) and the existential ∃ (“there exists”). in Predicate Logic
- Syllogism القياس
- Aristotle’s form of deductive inference: two premises sharing a middle term yield a conclusion. in Ancient Logic
- Modus ponens قياسُ الوضع
- The fundamental rule of detachment: from A and “A implies B”, infer B. in Propositional Logic
- Truth table جدولُ الصدق
- A table giving a compound proposition’s truth value for every assignment of truth to its atoms. in Propositional Logic
- Tautology قضيةٌ واجبةُ الصدق
- A proposition true under every interpretation: a logical truth. in Propositional Logic
- Excluded middle الثالثُ المرفوع
- The classical law that every proposition is either true or false, rejected by intuitionistic logic. in Non-Classical Logics
- Natural deduction الاستنباطُ الطبيعي
- Gentzen’s proof system built from introduction and elimination rules that mirror ordinary reasoning. in Proof Theory
- Sequent calculus حسابُ المتواليات
- Gentzen’s calculus operating on sequents Γ ⊢ Δ, the spine of structural proof theory. in Proof Theory
- Cut elimination حذفُ القطع
- Gentzen’s Hauptsatz: every sequent proof can be rewritten without the cut rule, yielding the subformula property. in Proof Theory
- Model النموذج · التأويل
- A structure assigning meaning to a language’s symbols, under which its sentences come out true or false. in Mathematical Logic
- Compactness التراصّ
- A set of first-order sentences has a model iff every finite subset of it has a model. in Mathematical Logic
- Löwenheim–Skolem لوفنهايم–سكولم
- If a countable first-order theory has an infinite model, it has models of every infinite cardinality. in Mathematical Logic
- Diagonal (fixed-point) lemma قضيةُ النقطة الثابتة
- In a strong enough theory, for any predicate ψ there is a sentence asserting its own ψ-ness: the engine of Gödel and Tarski. in Mathematical Logic
- Incompleteness عدمُ الاكتمال
- Gödel: any consistent formal system strong enough for arithmetic contains true statements it cannot prove. in Mathematical Logic
- Gödel numbering ترقيمُ غودل
- A coding of a formal language’s symbols and proofs as natural numbers, letting arithmetic talk about itself. in Mathematical Logic
- Possible worlds العوالمُ الممكنة □ ◇
- Kripke’s semantics for modal logic: necessity is truth in all accessible worlds, possibility in some. in Modal Logic
- λ-abstraction تجريدُ لامدا
- Church’s notation for a function defined by an expression: the atom of functional computation. in Computational Logic
- Curry–Howard تناظرُ كَري–هاوَرد
- The correspondence under which propositions are types and proofs are programs: logic and computation, identified. in Category Theory
- Decidability القابليةُ للبَتّ
- A problem is decidable if an algorithm settles every instance in finite time; logic abounds in undecidable ones. in Computational Logic
- Cardinality العددُ الأصلي
- The size of a set, generalized to the infinite by Cantor: ℵ₀, the continuum, and beyond. in Set Theory & Foundations
- The ZFC axioms بديهيّاتُ ZFC
- Zermelo–Fraenkel set theory with Choice: the standard axiomatic foundation of modern mathematics. in Set Theory & Foundations
- Boolean algebra الجبرُ البولياني
- The algebraic structure of classical propositions: meet, join, and complement obeying the distributive laws. in Algebraic Logic
- Univalence الأحادية
- Voevodsky’s axiom: equivalent types are equal, the heart of homotopy type theory. in Cutting-Edge Research