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Logical connective
The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol → {\displaystyle
Material_conditional
Family of logics for natural-language and counterfactual conditionals
conditionals more faithfully than the classical material conditional, which gives rise to well-known paradoxes. Conditional logics are used in philosophical logic
Conditional_logic
Conditionals that discuss what would have been if things were otherwise
as a challenge for the analysis of conditionals in terms of the material conditional, which would make all counterfactuals with false antecedents trivially
Counterfactual_conditional
Type of logical contradiction
The paradoxes of material implication are a group of classically true formulae involving material conditionals whose translations into natural language
Paradoxes of material implication
Paradoxes_of_material_implication
Topics referred to by the same term
consequent Material conditional, in propositional calculus, or logical calculus in mathematics Relevance conditional, in relevance logic Conditional logic
Conditional
Formal statement in logic
logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal
Strict_conditional
Topics referred to by the same term
A conditional statement may refer to: A conditional formula in logic and mathematics, which can be interpreted as: Material conditional Strict conditional
Conditional_statement
Natural-language "if" sentences about what may be the case
Linguistics portal Conditional logic Counterfactual conditional Logical consequence Material conditional Strict conditional Pragmatics Conditional probability
Indicative_conditional
Conditional statement which is true because the antecedent cannot be satisfied
vacuous truth is a conditional or universal statement (specifically a universal statement that can be converted to a conditional statement) that is true
Vacuous_truth
Logical connective
connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then")
If_and_only_if
Sentence expressing an 'if-then' relation
A conditional sentence is a sentence in a natural language that expresses that one thing is contingent on another, e.g., "If it rains, the picnic will
Conditional_sentence
Test in the study of deductive reasoning
the material conditional, since the natural language conditional is not the material conditional. (See also the paradoxes of the material conditional for
Wason_selection_task
Topics referred to by the same term
Material implication may refer to: Material conditional, a logical connective Material implication (rule of inference), a rule of replacement for some
Material_implication
In logic, the corresponding conditional of an argument (or derivation) is a material conditional whose antecedent is the conjunction of the argument's
Corresponding_conditional
If and only if relation
the only case where a logical biconditional is different from a material conditional is the case where the hypothesis (antecedent) is false but the conclusion
Logical_biconditional
Symbol connecting formulas in logic
expressions P, Q have side effects. Also, a conditional, which in some sense corresponds to the material conditional connective, is essentially non-Boolean
Logical_connective
Topics referred to by the same term
others Material conditional (also material implication), a logical connective and binary truth function typically interpreted as "If p, then q" Material implication
Implication
Rule of logical inference
classical two-valued logic, modus ponens can be interpreted from the material conditional (implication) operator. The notion of implication as a rigorous mathematical
Modus_ponens
Mathematical symbol
Horseshoe (⊃, \supset in TeX) is a symbol used to represent: Material conditional in propositional logic Superset in set theory It was used by Whitehead
Horseshoe_(symbol)
Theory of logic to account for observations from quantum theory
reject quantum logic as a basis for reasoning, because it lacks a material conditional; a common alternative is the system of linear logic, of which quantum
Quantum_logic
True when either but not both inputs are true
of material implication (a material conditional is equivalent to the disjunction of the negation of its antecedent and its consequence) and material equivalence
Exclusive_or
Topics referred to by the same term
connective introduced by Alonzo Church a rule in classical logic that the material conditional ¬p → q is equivalent to the disjunction p ∨ q, so that these two
Conditional_disjunction
Graphical symbol or pictogram used to point or indicate direction
decrease. In mathematical logic, a right-facing arrow indicates material conditional, and a left-right (bidirectional) arrow indicates if and only if
Arrow_(symbol)
Application of logical methods to philosophical problems
truth-functional interpretation of the material conditional by introducing the additional requirement of relevance: for the conditional to be true, its antecedent
Philosophical_logic
Topics referred to by the same term
Human condition, characteristics and key events of human existence Material conditional, a logical connective used to form "if...then..." statements Necessary
Condition
How one process influences another
antecedent is false. The ordinary indicative conditional has somewhat more structure than the material conditional. For instance, although the first is the
Causality
Mathematical symbol for "greater than"
operator Mathematical operators and symbols in Unicode Guillemet Material conditional Smith, Charles L. (1964). "On the origin of ">" and "<"". The Mathematics
Greater-than_sign
Proof strategy for showing a collection of statements are equivalent
equivalence of the statements then results from the transitivity of the material conditional. For n = 4 {\displaystyle n=4} the proofs are given for φ 1 ⇒ φ 2
Ringschluss
1879 book on logic by Gottlob Frege
in an idiosyncratic two-dimensional notation, based on negation, material conditional and universal quantification. Other connectives and existential quantification
Begriffsschrift
Grammatical mood
The conditional mood (abbreviated cond) is a grammatical mood used in conditional sentences to express a proposition whose validity is dependent on some
Conditional_mood
Materials with granular size 1 to 100 nm
as protective as a legally mandated OEL for the bulk form of the material (conditional recommendation, low-quality evidence). If specific OELs for MNMs
Nanomaterials
Object that exists in the imagination
are sometimes known as material conditionals. In contrast, indicative conditionals, sometimes known as non-material conditionals, attempt to describe if-then
Object_of_the_mind
Probability theory and statistics concept
In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome
Conditional probability distribution
Conditional_probability_distribution
argument is assumed in one of the premises. Philonian conditional Another term for the material conditional, emphasizing its use in propositional logic to represent
Glossary_of_logic
Mathematical table used in logic
then p ∨ q is p, otherwise p ∨ q is q. Logical implication and the material conditional are both associated with an operation on two logical values, typically
Truth_table
Form of reasoning
which happens when one of the premises has the form of a negative material conditional, as in "If the card does not have an A on the left, then it has a
Deductive_reasoning
Word used in English for conditional constructions
conjunction", a category covering a broad range of clause-connecting words. Material conditional §Discrepancies with natural language Huddleston, Rodney; Pullum,
If_(preposition)
Sentences of the form "if x, then y"
headings zero conditional, first conditional (or conditional I), second conditional (or conditional II), third conditional (or conditional III) and mixed
English_conditional_sentences
(P(x)\wedge \neg \exists y\,(P(y)\wedge y\neq x))} . ⇒ 1. Denotes material conditional, and is read as "implies". If P and Q are logical predicates, P ⇒
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Mathematical logic concept
contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated
Contraposition
Algebraic manipulation of "true" and "false"
operations for all four possible inputs. Material conditional The first operation, x → y, or Cxy, is called material implication. If x is true, then the result
Boolean_algebra
Test for the acceptability of conditionals via hypothetical belief revision
Ramsey test (also called the Ramsey test for conditionals) is a proposal for how to evaluate and accept conditional sentences of the form if p (then) q. Informally
Ramsey_test
Type of argument
true, and B is true, so A must be true"), but in this example the material conditional logical connective ("A implies B") in the formal fallacy does not
Argumentation_scheme
Process in logic
regression. Material inference should not be confused with the following concepts, which refer to formal, not material validity: Material conditional — the
Material_inference
Possessing negative truth value
implication vacuously true). In most logical systems, negation, material conditional and false are related as: ¬p ⇔ (p → ⊥) In fact, this is the definition
False_(logic)
Metatheorem
3, 5), each subformula is evaluated according to the rules for material conditional, the result being shown below its main operator. Column 6 shows that
Frege's_theorem
Principle of classical logic
thus also in classical logic, where the conditional operator " → {\displaystyle \rightarrow } " is taken as material implication. In the Curry-Howard correspondence
Import–export_(logic)
Refutation of a logical fallacy
not certain. Indeed, p implies q has the technical meaning of the material conditional: if p then q symbolized as p → q. That is, "if circumstance p is
Correlation does not imply causation
Correlation_does_not_imply_causation
Philosophical problem about what constitutes knowledge
counterfactual conditional), the justified true belief came about, if Smith's purported claims are disputable, as the result of entailment (but see also material conditional)
Gettier_problem
first, second or third conditional; there also exist "zero conditional" and mixed conditional sentences. A "first conditional" sentence expresses a future
Uses_of_English_verb_forms
Topics referred to by the same term
steamship route Horseshoe (symbol), "⊃", a logical connective meaning material conditional in propositional logic Horseshoe cloud, a meteorological phenomenon
Horseshoe_(disambiguation)
Logical biconditional; 10, q, Projection function; 11, if/then, Material conditional; 12, p, Projection function; 13, then/if, Converse implication; 14
List_of_rules_of_inference
Kind of quantifier in logic
denotes material implication in the metalanguage.) The minimal conditional logic M is characterized by the first six properties, and stronger conditional logics
Conditional_quantifier
Reasoning for mathematical statements
equivalence of the statements then results from the transitivity of the material conditional. A probabilistic proof is one in which an example is shown to exist
Mathematical_proof
Concept in first-order logic
such that Y" is interpreted as a universal generalization of the material conditional "if anything is such that X then it is such that Y". Also, the quantifiers
Empty_domain
Logical implication Logical negation Logical NOR Majority function Material conditional Minimal axioms for Boolean algebra Peirce arrow Read-once function
List of Boolean algebra topics
List_of_Boolean_algebra_topics
Apparent logical paradox
important to the paradox is that the conditional in classical (and intuitionistic) logic is the material conditional. It has the property that A → B {\displaystyle
Drinker_paradox
Symbolic logic system
Assuming any C {\displaystyle C} , the introduction rule of the material conditional gives B → C {\displaystyle B\to C} , also when B {\displaystyle B}
Minimal_logic
Formal proof
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to
Conditional_proof
Australian logician
is particularly noted for his views on the interpretation of the material conditional. David Lewis, a frequent visitor to Australian departments of logic
Vic_Dudman
Uniform restraint of the change in functions
|f(x)-f(y)|<\varepsilon } (where A ⟹ B {\displaystyle A\implies B} is a material conditional statement saying "if A {\displaystyle A} , then B {\displaystyle
Uniform_continuity
Concept in mathematical logic
∨ {\displaystyle \lor } ); negation ( ¬ {\displaystyle \neg } ); material conditional ( → {\displaystyle \to } ); and possibly the biconditional ( ↔ {\displaystyle
Functional_completeness
System used to prevent non-paying customers from accessing content that requires payment
Conditional access (CA) is a term commonly used in relation to software and to digital television systems. Conditional access is an evaluation to ensure
Conditional_access
Paradox arising from the question of what constitutes evidence for a statement
{\overline {Rx}}} where → {\displaystyle \rightarrow } indicates the material conditional, according to which "If A {\displaystyle A} then B {\displaystyle
Raven_paradox
Overview of and topical guide to logic
Logical implication Logical truth Logical consequence Name Necessity Material conditional Meaning (linguistic) Meaning (non-linguistic) Paradox (list) Possible
Outline_of_logic
Critical experiment
(disambiguation) Pierre Duhem § Philosophy of science Falsifiability Material conditional Q.E.D. Scientific method Smoking gun Therefore sign Schwartz, Daniel
Experimentum_crucis
Type of formal logic
for the material conditional of LP. For this reason, proponents of LP usually advocate expanding the system to include a stronger conditional connective
Paraconsistent_logic
Type of binary relation
Rational choice theory Hypothetical syllogism – transitivity of the material conditional Smith, Eggen & St. Andre 2006, p. 145 However, the class of von Neumann
Transitive_relation
Polish Dominican and philosopher (1902–1995)
of modal sentences. Bocheński's understanding of the supernatural and material spheres was greatly influenced by his attachment to the logical method
Józef_Maria_Bocheński
Newspaper in Brighton and Hove, England
found that "the newspaper's decision to make its offer to remove the material conditional on the complainant agreeing not to pursue his complaint further,
The_Argus_(Brighton)
Mathematics notation with operators preceding operands
ϕ {\displaystyle \neg \phi } N ϕ {\displaystyle N\phi } negacja Material conditional ϕ → ψ {\displaystyle \phi \to \psi } C ϕ ψ {\displaystyle C\phi \psi
Polish_notation
Branch of logic
"and" (conjunction), "or" (disjunction), "not" (negation), "if" (material conditional), and "if and only if" (biconditional). Examples of such compound
Propositional_logic
Ongoing armed conflict in West Asia
be included as part of a ceasefire deal, thereby making a ceasefire conditional on an end to the 2026 Lebanon war against Hezbollah. On 24 March, Israel
2026_Iran_war
Knowledge representation computer programming language
\mathbb {B} } denotes the booleans, ¬ denotes negation, and ⇒ denotes material conditional. Predicates < and ≥ are built-in and have their usual meaning. Such
FO(.)
Software development process
require lifecycle management beyond the conditional statements used to implement them. A study of practitioner material and developers from 38 companies identified
Feature_toggle
Sentence that resists simple formalization
substituting any non-donkey object for y {\displaystyle y} makes the material conditional true (since its antecedent is false), and so existential clause is
Donkey_sentence
Study of the scope and nature of logic
conceptions: an inference from the premises to a conclusion is valid if the material conditional from the premises to the conclusion is logically true. For example
Philosophy_of_logic
German chemical company
second half of 2025. In November 2025, the EU competition regulator conditionally approved the acquisition after ADNOC offered to ensure the firm aligned
Covestro
English scholastic philosopher (1270s–1340s)
to have taken the position that, in line with modern views on the material conditional, the principle that "from the impossible anything follows" ("ex impossibili
Walter_Burley
Business that provides packing and moving services for relocation
their belongings. The moving process also involves finding or buying materials such as boxes, paper, tape, and bubble wrap with which to pack boxable
Moving_company
115–334 (text) (PDF) 115-335 December 20, 2018 Nicaraguan Investment Conditionality Act (NICA) of 2017 To oppose loans at international financial institutions
List of acts of the 115th United States Congress
List_of_acts_of_the_115th_United_States_Congress
Logical paradox
reductio ad absurdum, discharging assumption 8. 11. p → Kp from line 10 by a classical tautology about the material conditional (negated conditionals)
Fitch's paradox of knowability
Fitch's_paradox_of_knowability
Digital circuit implementation method
previous stage carry equal. A conditional sum adder is a recursive structure based on the carry-select adder. In the conditional sum adder, the MUX level chooses
Carry-select_adder
Design method and manifesto by Luna Maurer, Jonathan Puckey, Roel Wouters and Edo Paulus
identifies external material, including nature, society and human interaction, as the material that activates the process. Conditional design is based on
Conditional_Design_Manifesto
In probability theory, a conditional event algebra (CEA) is an alternative to a standard, Boolean algebra of possible events (a set of possible events
Conditional_event_algebra
Scientific study of digital information
The conditional entropy or conditional uncertainty of X given random variable Y (also called the equivocation of X about Y) is the average conditional entropy
Information_theory
Austrian philosopher (1879–1944)
binary relation A requires B, i.e. A materially implies !B. (All entailment in the axioms is material conditional.) It is defined by axiom III, whereas
Ernst_Mally
Syllogism with conditional premise(s)
hypothetical syllogism is a valid argument form, a deductive syllogism with a conditional statement for one or both of its premises. Ancient references point to
Hypothetical_syllogism
school were (i) their account of modality, (ii) their theory of the material conditional, and (iii) their account of meaning and truth. Modality. According
History_of_logic
American singer-songwriter (1946–2026)
unconditional cash support may be more beneficial for disaster relief than conditional financial support. The report mentioned the impact of the monthly financial
Dolly_Parton
Control flow construct for executing code repeatedly
program contains an infinite loop is known as the halting problem. A conditional loop (also known as an indeterminate loop) is a loop that determines
Loop_(statement)
American semiotician (1925–2022)
today (dot Logical conjunction, vee Logical disjunction, horseshoe Material conditional representing and, or, if) is a lingering, overly abstract, unsystematically
Shea_Zellweger
English singer-songwriter (born 1989)
(2016), A Brief Inquiry into Online Relationships (2018), Notes on a Conditional Form (2020) and Being Funny in a Foreign Language (2022). Each of their
Matty_Healy
Table in statistics
statistics, the conditional probability table (CPT) is defined for a set of discrete and mutually dependent random variables to display conditional probabilities
Conditional_probability_table
1997 murder of a teenager in Canada
death; six involved in the beating received sentences ranging from 60-day conditional sentences to one year in jail, while the two who killed Virk were convicted
Murder_of_Reena_Virk
for the amount paid. There are typically four types of lien waivers: Conditional waiver on progress payment – The safest waiver for claimants, this waiver
Lien_waiver
Average uncertainty in variable's states
{\displaystyle \lim _{p\to 0^{+}}p\log(p)=0.} One may also define the conditional entropy of two variables X {\displaystyle X} and Y {\displaystyle Y}
Entropy_(information_theory)
Form of conditionals in computer programming
alternative to conditional transfer of control, as implemented by conditional branch machine instructions. Predication works by having conditional (predicated)
Predication (computer architecture)
Predication_(computer_architecture)
Intelligence in machines
expensive. For inference to be tractable, most observations must be conditionally independent of one another. AdSense uses a Bayesian network with over
Artificial_intelligence
Class of polymers with a silicon-nitrogen backbone
polysilazanes. Polysilazanes are colorless to pale yellow liquids or solid materials. Conditional of manufacturing, the liquids often contain dissolved ammonia that
Polysilazane
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