For other uses of the term see Material implication ( disambiguation ).

* Material implication

Material implication allows a true consequent to follow from a false antecedent.

* Material conditional, in propositional calculus, or logical calculus in mathematics.

* Material conditional also joins two simpler propositions, and we write, which is read " if then ".

Three significant contributions of the Stoic school were ( i ) their account of modality, ( ii ) their theory of the Material conditional, and ( iii ) their account of meaning and truth.

* Material conditional

* Material conditional

* Material conditional

* Material conditional

:* Material Principle -- " a synopsis and summary of the Christian truth " that people are justified by God's grace through faith in Christ alone.

Material taken from postholes from an enclosure at Douglasmuir, near Friockheim, about five miles north of Arbroath have been radiocarbon dated to around 3500 BC The function of the enclosure is unknown, but may have been for agriculture or for ceremonial purposes.

The 1980 1st-Edition AD & D module Queen of the Demonweb Pits describes the Demonweb Pits as an environment quite unlike the Prime Material Plane, stating that many magical spells and effects do not function as expected here.

B attempts to falsify A's conditional statement (" if Republican then against gun control ") by providing evidence he believes would contradict its implication.

Some logicians ( Kenneth Ross, Charles Wright ) draw a firm distinction between the conditional connective ( the syntactic sign "→"), and the implication relation ( the formal object denoted by the double arrow symbol "⇒").

These logicians use the phrase not p or q for the conditional connective and the term implies for the implication relation.

Some explain the difference by saying that the conditional is the contemplated relation while the implication is the asserted relation.

Relevance logic aims to capture aspects of implication that are ignored by the " material implication " operator in classical truth-functional logic, namely the notion of relevance between antecedent and conditional of a true implication.

When an implication is translated by a hypothetical ( or conditional ) judgment the antecedent is called the hypothesis ( or the condition ) and the consequent is called the thesis.

In propositional logic, material implication is a valid rule of replacement that allows for a conditional statement to be replaced by a disjunction if and only if the antecedent is negated.

Satisfiability -- Scholastic logic -- Second-order predicate -- Self-reference -- Sequent -- Sequent calculus -- Sequential logic -- Singular term -- Soundness -- Square of opposition -- Strict conditional -- Strict implication -- Strict logic -- Structural rule -- Sufficient condition -- Syllogism -- Syllogistic fallacy

The material conditional, ( also known as material implication ) is a logical binary compound using a logical connective often symbolized by a forward arrow "→", and often simply called a conditional.

The truth table associated with the material conditional not p or q ( symbolized as p → q ) and the logical implication p implies q ( symbolized as p → q, or Cpq ) is as follows:

The rule of inference for sufficient condition is modus ponens, which is an argument for conditional implication:

At this stage in the algorithm, it is required that terms with zero-valued coefficients are dropped, so that only binary coefficients equal to one are counted, thus the problem of multiplication or division by zero is not an issue, despite this implication in the factored equation:

In Kripke semantics for relevant logic, the implication operator is a binary modal operator, and negation is usually taken to be a unary modal operator.

In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice ( with join and meet operations written ∧ and ∨ and with least element 0 and greatest element 1 ) equipped with a binary operation a → b of implication such that ( a → b )∧ a ≤ b, and moreover a → b is the greatest such in the sense that if c ∧ a ≤ b then c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound.

The definition of Heyting algebras crucially involves the existence of right adjoints to the binary meet operation, which together define an additional implication operation ⇒.

However, it differs sharply from the constructivist implication that humans arbitrarily construct mathematical principles that have no inherent truth but which instead are created on a conveniency basis.

Other common logical propositions are disjunction (), negation (), implication (), and the logical constants truth () and falsehood ().

This is a complex question and, for simplicity of exposition, we will here consider only vacuous truth as concerns logical implication, i. e., the case when has the form, and is false.

Demonstrating the logical implication simply translates into verifying that the compound truth table is producing a tautology.

Due to the lack of whole works by Xenophanes a lot of meaning is lost and a large amount of guessing is at hand, so that the implication of knowing being something deeper (" a clearer truth ") may have special implications, or it may mean that you can't know something just by looking at it.

It is a poem that, without ever stating it, inevitably suggests the truth of ' ripeness is all ' by developing, with a richness of profundity of implication, the simple perception that ripeness is fall.

" In the language of the Church ’ s social teachings, development communication policies should lead to “ establishing new relationships in human society, under the mastery and guidance of truth, justice, charity and freedom — relations between individual citizens, between citizens and their respective States, between States, and finally between individuals, families, intermediate associations and States on the one hand, and the world community on the other .” In this way, the formulation of such communication policies could be performed based mainly on the developing world rhetoric of nation-building, but at the same time taking serious cognizance of its universal implication and global impact.

As, for example, the teaching on papal infallibility was infallibly taught for a long time de fide tenenda by the universal magisterium, but not de fide credenda until the extraordinary definition at Vatican I, because there was disagreement on whether it was a specifically revealed truth from the deposit of faith, or merely the logical implication of other things in the deposit of faith ( as, for example, the authority of Peter in the college of apostles, the constitution of the Church, her unity, her episcopal structure, etc.

The expression fig leaf has a pejorative metaphorical sense meaning a flimsy or minimal cover for any thing or behaviour that might be considered shameful, with the implication that the cover is only a token gesture and the truth is obvious to all who choose to see it.

Under the Curry – Howard correspondence, the existence of currying and uncurrying is equivalent to the logical theorem, as tuples ( product type ) corresponds to conjunction in logic, and function type corresponds to implication.

The implication behind the title ' President of Ireland ' was that the President would function as the head of all Ireland.

Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's matrix mechanics, immediately understood that these conditions could be met if A, B, C and D are matrices, with the implication that the wave function has multiple components.

The statistic s is said to be complete for the distribution of X if for every measurable function g ( which must be independent of θ ) the following implication holds:

Using the Curry – Howard isomorphism-types also serve to model implication and universal quantification: e. g. a term inhabiting is a function which assigns to any pair of natural numbers a proof that addition is commutative for that pair and hence can be considered as a proof that addition is commutative for all natural numbers.

If the implication of limit preservation is inverted, such that the existence of limits in the range of a function implies the existence of limits in the domain, then one obtains functions that are limit-reflecting.

One implication of the curse of dimensionality is that some methods for numerical solution of the Bellman equation require vastly more computer time when there are more state variables in the value function.

The strong implication being that the mind of the observer ( technically, a conceptually designating consciousness ) provides the name, identity, function, and conventional discreteness to that object.

The structure of a fuzzy system is expressed by the input and output variables and the rule base, while the parameters of a fuzzy system are the rule parameters ( defining the membership functions, the aggregation operator and the implication function ) and the mapping parameters related to the mapping of a crisp set to a fuzzy set, and vice versa.