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A material conditional (also known as “material implication”, “material consequence”, or simply “implication”) is a logical connective often symbolized by a forward arrow “→”. A single statement formed from the connection of two, for instance “pq” (called a conditional statement), is typically interpreted as “If p, then q” or “q if p”. The material implication between two sentences p, q is typically symbolized as

1. $p \rightarrow q$;
2. $p \supset q$ (Although this symbol is confused with the superset symbol used by Algebra of sets.);
3. $p \Rightarrow q$ (Although this symbol is often used for logical consequence (i.e. logical implication) rather than for material implication.)

As placed within the material conditionals above, p is known as the antecedent, and q as the consequent, of the conditional. One can also use compounds as components, for example pq → (rs). There, the compound pq (short for “p and q”) is the antecedent, and the compound rs is the consequent, of the larger conditional of which those compounds are components.

Implication is one form of logical consequence. In an argument such as “Fred is Mike’s brother’s son", the statement “Fred is Mike’s nephew” is a material consequence of “Fred is Mike’s brother’s son”, but not a formal consequence. This is because the validity of the argument depends on the content of the premise and conclusion (including the meanings of the words “brother”, “son”, and “nephew”) rather than on the logical form or structure of the argument.[1]

In classical logic, the compound pq is equivalent to the negative compound: not (both p and not q). In particular, it is considered true if p is false, whether or not q is true. The only case in which the statement pq is false is when p is true and q is false. In everyday English, saying “It is false that if p then q” is not often taken as flatly equivalent to saying “p is true and q is false” but, when used within classical logic, those phrasings are taken as logically equivalent. (Other senses of English “if…then…” require other logical forms.)

## Definitions of the material conditional

Logicians have many different views on the nature of material implication and approaches to explain its sense.[2]

### As a truth function

In classical logic, the compound pq is logically equivalent to the negative compound: not both p and not q. Thus the compound pq is false if and only if both p is true and q is false. By the same stroke, pq is true if and only if either p is false or q is true (or both). Thus → is a function from pairs of truth values of the components p, q to truth values of the compound pq, whose truth value is entirely a function of the truth values of the components. Hence, the compound pq is called truth-functional. The compound pq is logically equivalent also to ¬pq (either not p, or q (or both)), and to ¬q → ¬p (if not q then not p). But it is not equivalent to ¬p → ¬q, which is equivalent to qp.

#### Truth table

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:

$p$ $q$ $p \rightarrow q$
T T T
T F F
F T T
F F T

### As a formal connective

The material conditional can be considered as a symbol of a formal theory, taken as a set of sentences, satisfying all the classical inferences involving →, in particular the following characteristic rules:

Unlike the truth-functional one, this approach to logical connectives permits the examination of structurally identical propositional forms in various logical systems, where somewhat different properties may be demonstrated. For example, in intuitionistic logic which rejects proofs by contraposition as valid rules of inference, (p → q) ⇒ ¬p ∨ q is not a propositional theorem, but the material conditional is used to define negation.

## Formal properties

When studying logic formally, the material conditional is distinguished from the semantic consequence relation $\models$. We say $A \models B$ if every interpretation that makes A true also makes B true. However, there is a close relationship between the two in most logics, including classical logic. For example, the following principles hold:

• If $\Gamma\models\psi$ then $\varnothing\models(\varphi_1\land\dots\land\varphi_n\rightarrow\psi)$ for some $\varphi_1,\dots,\varphi_n\in\Gamma$. (This is a particular form of the deduction theorem. In words, it says that if Γ models ψ this means that ψ can be deduced just from some subset of the theorems in Γ.)
• The converse of the above
• Both $\rightarrow$ and $\models$ are monotonic; i.e., if $\Gamma\models\psi$ then $\Delta\cup\Gamma\models\psi$, and if $\varphi\rightarrow\psi$ then $(\varphi\land\alpha)\rightarrow\psi$ for any α, Δ. (In terms of structural rules, this is often referred to as weakening or thinning.)

These principles do not hold in all logics, however. Obviously they do not hold in non-monotonic logics, nor do they hold in relevance logics.

Other properties of implication (the following expressions are always true, for any logical values of variables):

• distributivity: $(s \rightarrow (p \rightarrow q)) \rightarrow ((s \rightarrow p) \rightarrow (s \rightarrow q))$
• transitivity: $(a \rightarrow b) \rightarrow ((b \rightarrow c) \rightarrow (a \rightarrow c))$
• reflexivity: $a \rightarrow a$
• totality: $(a \rightarrow b) \vee (b \rightarrow a)$
• truth preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of material implication.
• commutativity of antecedents: $(a \rightarrow (b \rightarrow c)) \equiv (b \rightarrow (a \rightarrow c))$

Note that $a \rightarrow (b \rightarrow c)$ is logically equivalent to $(a \and b) \rightarrow c$; this property is sometimes called currying. Because of these properties, it is convenient to adopt a right-associative notation for → where $a \rightarrow b \rightarrow c$ denotes $a \rightarrow (b \rightarrow c)$.

Note also that comparison of the truth table shows that $a \rightarrow b$ is equivalent to $\neg a \or b$, and it is sometimes convenient to replace one by the other in proofs. Such a replacement can be viewed as a rule of inference.

## Philosophical problems with material conditional

Outside of mathematics, it is a matter of some controversy as to whether the truth function for material implication provides an adequate treatment of conditional statements in English (a sentence in the indicative mood with a conditional clause attached, i.e., an indicative conditional, or false-to-fact sentences in the subjunctive mood, i.e., a counterfactual conditional).[3] That is to say, critics argue that in some non-mathematical cases, the truth value of a compound statement, "if p then q", is not adequately determined by the truth values of p and q.[3] Examples of non-truth-functional statements include: "p because q", "p before q" and "it is possible that p".[3] “[Of] the sixteen possible truth-functions of A and B, material implication is the only serious candidate. First, it is uncontroversial that when A is true and B is false, "If A, B" is false. A basic rule of inference is modus ponens: from "If A, B" and A, we can infer B. If it were possible to have A true, B false and "If A, B" true, this inference would be invalid. Second, it is uncontroversial that "If A, B" is sometimes true when A and B are respectively (true, true), or (false, true), or (false, false)… Non-truth-functional accounts agree that "If A, B" is false when A is true and B is false; and they agree that the conditional is sometimes true for the other three combinations of truth-values for the components; but they deny that the conditional is always true in each of these three cases. Some agree with the truth-functionalist that when A and B are both true, "If A, B" must be true. Some do not, demanding a further relation between the facts that A and that B.”[3]

The truth-functional theory of the conditional was integral to Frege's new logic (1879). It was taken up enthusiastically by Russell (who called it "material implication"), Wittgenstein in the Tractatus, and the logical positivists, and it is now found in every logic text. It is the first theory of conditionals which students encounter. Typically, it does not strike students as obviously correct. It is logic's first surprise. Yet, as the textbooks testify, it does a creditable job in many circumstances. And it has many defenders. It is a strikingly simple theory: "If A, B" is false when A is true and B is false. In all other cases, "If A, B" is true. It is thus equivalent to "~(A&~B)" and to "~A or B". "AB" has, by stipulation, these truth conditions.

Dorothy Edgington, The Stanford Encyclopedia of Philosophy, “Conditionals”[3]

The meaning of the material conditional can sometimes be used in the natural language English "if condition then consequence" construction (a kind of conditional sentence), where condition and consequence are to be filled with English sentences. However, this construction also implies a "reasonable" connection between the condition (protasis) and consequence (apodosis) (see Connexive logic).[citation needed]

The material conditional can yield some unexpected truths when expressed in natural language. For example, any material conditional statement with a false antecedent is true (see vacuous truth). So the statement "if 2 is odd then 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement, "if my password is Swordfish then Paris is in France" is true. These problems are known as the paradoxes of material implication, though they are not really paradoxes in the strict sense; that is, they do not elicit logical contradictions. These unexpected truths arise because speakers of English (and other natural languages) are tempted to equivocate between the material conditional and the indicative conditional, or other conditional statements, like the counterfactual conditional and the material biconditional. It is not surprising that a rigorously defined truth-functional operator does not correspond exactly to all notions of implication or otherwise expressed by 'if...then...' sentences in English (or their equivalents in other natural languages). For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below.

## Further reading

 6519 videos foundNext >
 019a 14Rules: Material Implication and Tautologyexercises in 019b send questions or comments to: ulogiq@gmail.com. The cause for material conditional lifeSrimad Bhagavatam Lecture given by Sri Amitasana Dasa. For more lectures on Srimad Bhagavatam log on to http://www.iskconbangalore.org/pravachan. Practice Problems - Rules of ImplicationProfessor Thorsby goes over some practice problems using the rules of implication. Philosophical Issues in Propositional LogicA discussion of the truth table for material implication, the paradox of material implication, and the paradox of entailment. Christopher Gauker on LanguageIn this interview, Christopher Gauker discusses his conception of language. He also explains why universal instantiation is an invalid rule of inference, why... Antinatalism and Depression CXV: Meaning vs HarmMaterial Conditional: If harm has value, then meaning has value. Are we in agreement? If not, why not? Disclaimer: This video contains terms like "Brahman" a... Natural Deduction Short Lesson: How to turn any proposition into a conditional statementIn this video I demonstrate the process of turning any proposition into a conditional statement using the rules of natural deduction including addition, comm... Environmental Sociology 3 (4/5): Macrotheories: The Origins of the Human-Environmental WorldThe outline of the five split parts of this session is below the introduction] WHO: I am Dr. Mark D. Whitaker (Ph.D., Sociology; University of Wisconsin-Mad... 11.Conditional format and Materials Tracking.mp4you are welcome in our free specialized website in construction planning and scheduling http://www.planningengineer.net 11.Conditional format and Materials T... T.R.R | Storm the office of Interior Minister Egypt Habib el-AdliHabib Ibrahim El Adly was born in Cairo in March 1, 1938. Received a Bachelor of Laws and Diploma of material conditional from the Police College in 1961. Re...
 6519 videos foundNext >

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