# Validity (logic) > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Validity_(logic) > Markdown URL: https://mediated.wiki/source/Validity_(logic).md > Source: https://en.wikipedia.org/wiki/Validity_(logic) > Source revision: 1335834261 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Argument whose conclusion must be true if its premises are In [logic](/source/Logic), specifically in [deductive reasoning](/source/Deductive_reasoning), an [argument](/source/Argument) is **valid** [if and only if](/source/If_and_only_if) it takes a form that makes it impossible for the [premises](/source/Premise) to be [true](/source/Truth) and the conclusion nevertheless to be [false](/source/False_(logic)).[1] It is not required for a valid argument to have premises that are actually true,[2] but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called [well-formed formulas](/source/Well-formed_formula) (also called *wffs* or simply *formulas*). The **validity** of an argument can be tested, proved or disproved, and depends on its [logical form](/source/Logical_form).[3] ## Arguments Argument terminology used in logic In logic, an [argument](/source/Argument) is a set of related statements expressing the *premises* (which may consists of non-empirical evidence, empirical evidence or may contain some axiomatic truths) and a *necessary conclusion based on the relationship of the premises.* An argument is *valid* if and only if it would be contradictory for the conclusion to be false if all of the premises are true.[3] Validity does not require the truth of the premises, instead it merely [necessitates](/source/Logical_truth) that conclusion follows from the premises without violating the correctness of the [logical form](/source/Logical_form). If also the premises of a valid argument are proven true, this is said to be [sound](/source/Soundness).[3] The [corresponding conditional](/source/Corresponding_conditional) of a valid argument is a [logical truth](/source/Logical_truth) and the negation of its corresponding conditional is a [contradiction](/source/Contradiction). The conclusion is a [necessary consequence](https://en.wikipedia.org/w/index.php?title=Necessary_consequence&action=edit&redlink=1) of its premises. An argument that is not valid is said to be "invalid". An example of a valid (and [sound](/source/Soundness)) argument is given by the following well-known [syllogism](/source/Syllogism): - All men are mortal. (**True**) - Socrates is a man. (**True**) - Therefore, Socrates is mortal. (**True**) What makes this a valid argument is not that it has true premises and a true conclusion. Validity is about the tie in relationship between the two premises the necessity of the conclusion. There needs to be a relationship established between the premises i.e., a middle term between the premises. If you just have two unrelated premises there is no argument. Notice some of the terms repeat: men is a variation man in premises one and two, Socrates and the term mortal repeats in the conclusion. The argument would be just as valid if both premises and conclusion were false. The following argument is of the same [logical form](/source/Logical_form) but with false premises and a false conclusion, and it is equally valid: - All cups are green. (**False**) - Socrates is a cup. (**False**) - Therefore, Socrates is green. (**False**) No matter how the universe might be constructed, it could never be the case that these arguments should turn out to have simultaneously true premises but a false conclusion. The above arguments may be contrasted with the following invalid one: - All men are immortal. (**False**) - Socrates is a man. (**True**) - Therefore, Socrates is mortal. (**True**) In this case, the conclusion contradicts the deductive logic of the preceding premises, rather than deriving from it. Therefore, the argument is logically 'invalid', even though the conclusion could be considered 'true' in general terms. The premise 'All men are immortal' would likewise be deemed false outside of the framework of classical logic. However, within that system 'true' and 'false' essentially function more like mathematical states such as binary 1s and 0s than the philosophical concepts normally associated with those terms. Formal arguments that are invalid are often associated with at least one fallacy which should be verifiable. A standard view is that whether an argument is valid is a matter of the argument's logical form. Many techniques are employed by logicians to represent an argument's logical form. A simple example, applied to two of the above illustrations, is the following: Let the letters 'P', 'Q', and 'S' stand, respectively, for the set of men, the set of mortals, and Socrates. Using these symbols, the first argument may be abbreviated as: - All P are Q. - S is a P. - Therefore, S is a Q. Similarly, the third argument becomes: - All P's are not Q. - S is a P. - Therefore, S is a Q. An argument is termed formally valid if it has structural self-consistency, i.e. if when the operands between premises are all true, the derived conclusion is always also true. In the third example, the initial premises cannot logically result in the conclusion and is therefore categorized as an invalid argument. ## Valid formula A formula of a [formal language](/source/Formal_language) is a valid formula if and only if it is true under every possible [interpretation](/source/Interpretation_(logic)) of the language. In propositional logic, they are [tautologies](/source/Tautology_(logic)). ## Statements A statement can be called valid, i.e. logical truth, in some systems of logic like in Modal logic if the statement is true in all interpretations. In Aristotelian logic statements are not valid per se. Validity refers to entire arguments. The same is true in propositional logic (statements can be true or false but not called valid or invalid). ## Soundness Main article: [Soundness](/source/Soundness) Validity of deduction is not affected by the truth of the premise or the truth of the conclusion. The following deduction is perfectly valid: - All animals live on Mars. (**False**) - All humans are animals. (**True**) - Therefore, all humans live on Mars. (**False**) The problem with the argument is that it is not *sound*. In order for a deductive argument to be sound, the argument must be valid **and** all the premises must be true.[3] ## Satisfiability Main article: [Satisfiability](/source/Satisfiability) [Model theory](/source/Model_theory) analyzes formulae with respect to particular classes of interpretation in suitable mathematical structures. On this reading, a formula is valid if all such interpretations make it true. An inference is valid if all interpretations that validate the premises validate the conclusion. This is known as *semantic validity*.[4] ## Preservation In *truth-preserving* validity, the interpretation under which all variables are assigned a [truth value](/source/Truth_value) of 'true' produces a truth value of 'true'. In a *false-preserving* validity, the interpretation under which all variables are assigned a truth value of 'false' produces a truth value of 'false'.[5] - Preservation properties Logical connective sentences True and false preserving: Proposition • Logical conjunction (AND, ∧ {\displaystyle \land } ) • Logical disjunction (OR, ∨ {\displaystyle \lor } ) True preserving only: Tautology ( ⊤ {\displaystyle \top } ) • Biconditional (XNOR, ↔ {\displaystyle \leftrightarrow } ) • Implication ( → {\displaystyle \rightarrow } ) • Converse implication ( ← {\displaystyle \leftarrow } ) False preserving only: Contradiction ( ⊥ {\displaystyle \bot } ) • Exclusive disjunction (XOR, ⊕ {\displaystyle \oplus } ) • Nonimplication ( ↛ {\displaystyle \nrightarrow } ) • Converse nonimplication ( ↚ {\displaystyle \nleftarrow } ) Non-preserving: Negation ( ¬ {\displaystyle \neg } ) • Alternative denial (NAND, ↑ {\displaystyle \uparrow } ) • Joint denial (NOR, ↓ {\displaystyle \downarrow } ) ## See also - [Philosophy portal](https://en.wikipedia.org/wiki/Portal:Philosophy) - [Logical consequence](/source/Logical_consequence) - [Reductio ad absurdum](/source/Reductio_ad_absurdum) - [Mathematical fallacy](/source/Mathematical_fallacy) - [Soundness](/source/Soundness) - [Ω-validity](/source/%CE%A9-logic#Analysis) ## References 1. **[^](#cite_ref-1)** [Validity and Soundness – Internet Encyclopedia of Philosophy](http://www.iep.utm.edu/val-snd/) 1. **[^](#cite_ref-2)** Jc Beall and Greg Restall, ["Logical Consequence"](http://plato.stanford.edu/archives/fall2014/entries/logical-consequence/), The Stanford Encyclopedia of Philosophy (Fall 2014 Edition). 1. ^ [***a***](#cite_ref-:0_3-0) [***b***](#cite_ref-:0_3-1) [***c***](#cite_ref-:0_3-2) [***d***](#cite_ref-:0_3-3) Gensler, Harry J. (January 6, 2017). *Introduction to logic* (Third ed.). New York: Routledge. [ISBN](/source/ISBN_(identifier)) [978-1-138-91058-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-138-91058-4). [OCLC](/source/OCLC_(identifier)) [957680480](https://search.worldcat.org/oclc/957680480). 1. **[^](#cite_ref-4)** [L. T. F. Gamut](/source/L._T._F._Gamut), *Logic, Language, and Meaning: Introduction to Logic*, University of Chicago Press, 1991, p. 115. 1. **[^](#cite_ref-5)** Robert Cogan, *Critical Thinking: Step by Step*, University Press of America, 1998, [p. 48](https://archive.org/details/criticalthinking0000coga/page/48). ## Further reading - [Barwise, Jon](/source/Jon_Barwise); [Etchemendy, John](/source/John_Etchemendy). *Language, Proof and Logic* (1999): 42. - Beer, Francis A. "[Validities: A Political Science Perspective](https://www.tandfonline.com/doi/abs/10.1080/02691729308578683)", *Social Epistemology* 7, 1 (1993): 85–105. Wiktionary has definitions related to [***Validity***](https://en.wiktionary.org/wiki/validity). v t e Logic History Major fields Computer science Computational logic Formal semantics (natural language) Inference Philosophy of logic Proof Semantics of logic Syntax Logics Classical Informal Critical thinking Reason Mathematical Non-classical Philosophical Theories Argumentation Metalogic Metamathematics Set Foundations Abduction Analytic and synthetic propositions Antecedent Consequent Contradiction Paradox Antinomy Deduction Deductive closure Definition Description Dichotomy Entailment Linguistic Form Induction Logical truth Name Necessity and sufficiency Premise Probability Proposition Reference Statement Substitution Truth Validity Lists Topics Mathematical logic Boolean algebra Set theory Other Logicians Rules of inference Paradoxes Fallacies Logic symbols Category Outline Portal WikiProject changes v t e Mathematical logic General Axiom list Cardinality First-order logic Formal proof Formal semantics Foundations of mathematics Information theory Lemma Logical consequence Model Theorem Theory Type theory Theorems (list), paradoxes Gödel's completeness – incompleteness theorems Tarski's undefinability Banach–Tarski paradox Cantor's theorem – paradox – diagonal argument Compactness Halting problem Lindström's Löwenheim–Skolem Russell's paradox Logics Traditional Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞ Predicate First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus Set theory Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities Types of sets Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary Theories Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive Formal systems (list), language, syntax Alphabet Arity Automata Axiom schema Expression ground Extension by definition conservative Relation Formation rule Grammar Formula atomic closed ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃ ! ∀ rank Sentence atomic spectrum Signature String Substitution Symbol function logical/constant non-logical variable Term Theory list Example axiomatic systems (list) of true arithmetic Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of Boolean algebras canonical minimal axioms of geometry Euclidean Elements Hilbert's Tarski's non-Euclidean Principia Mathematica Proof theory Formal proof Natural deduction Logical consequence Rule of inference Sequent calculus Theorem Systems axiomatic deductive Hilbert list Complete theory Independence (from ZFC) Proof of impossibility Ordinal analysis Reverse mathematics Self-verifying theories Model theory Interpretation function of models Model atomic equivalence finite prime saturated spectrum submodel Non-standard model of non-standard arithmetic Diagram elementary Categorical theory Model complete theory Satisfiability Semantics of logic Strength Theories of truth semantic Tarski's Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably enumerable Computable function Computable set Decision problem decidable undecidable P NP P versus NP problem Kolmogorov complexity Lambda calculus Primitive recursive function Recursion Recursive set Turing machine Type theory Related Abstract logic Algebraic logic Automated theorem proving Category theory Concrete/Abstract category Category of sets History of logic History of mathematical logic timeline Logicism Mathematical object Philosophy of mathematics Supertask Mathematics portal --- Adapted from the Wikipedia article [Validity (logic)](https://en.wikipedia.org/wiki/Validity_(logic)) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Validity_(logic)?action=history)). 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