Information about Material Implication
The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain conditionals in logic. In propositional logic, it expresses a binary truth function ⊃ from truth-values to truth-values. In predicate logic, it can be viewed as a subset relation between the extension of (possibly complex) predicates. In symbols, a material conditional is written as one of the following:
The material conditional is false when X is true and Y is false - otherwise, it is true. (Here, X and Y are variables ranging over formulæ of a formal theory.) We call X the antecedent, and Y the consequent. The material conditional is also commonly referred to as material implication with the understanding that the antecedent (X) materially implies the consequent (Y).
A distant approximation to the material conditional is the English construction 'if...then...', where the ellipses are to be filled with English sentences. However, this is the most common reading of the material conditional in English. A closer approximation to X → Y is 'it's false that X be true while Y false'—i.e., in symbols,
. Arguably this is more intuitive than its logically equivalent disjunction ¬X ∨ Y.
Other properties of implication:
There are various kinds of conditionals in English; e.g., there is the indicative conditional and the subjunctive or counterfactual conditional. The latter do not have the same truth conditions as the material conditional. For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below.
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The material conditional is false when X is true and Y is false - otherwise, it is true. (Here, X and Y are variables ranging over formulæ of a formal theory.) We call X the antecedent, and Y the consequent. The material conditional is also commonly referred to as material implication with the understanding that the antecedent (X) materially implies the consequent (Y).
A distant approximation to the material conditional is the English construction 'if...then...', where the ellipses are to be filled with English sentences. However, this is the most common reading of the material conditional in English. A closer approximation to X → Y is 'it's false that X be true while Y false'—i.e., in symbols,
. Arguably this is more intuitive than its logically equivalent disjunction ¬X ∨ Y.
Definition
Logical implication and the material conditional are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of false just in case the first operand is true and the second operand is false.Truth table
The truth table associated with the material conditional if p then q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:| p | q | ? |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Johnston diagram
The Johnston diagram of "If A then B")
If A then B
Formal properties
The material conditional is not to be confused with the entailment relation ⊨ (which is used here as a name for itself). But there is a close relationship between the two in most logics, including classical logic which we only consider here. For example, the following principles hold:- If
then 
for some 
. (This is a particular form of the deduction theorem.)
- The converse of the above
- Both ⊃ and ⊨ are monotonic; i.e., if then
, and if 
then 
for any α, Δ. (In terms of structural rules, this is often referred to as weakening or thinning.)
Other properties of implication:
- ::
- transitivity: (
- commutativity: (
- idempotency:
- 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.
Philosophical problems with material conditional
The truth function ⊃ does not correspond exactly to the English 'if...then...' construction. For example, any material conditional statement with a false antecedent is true. 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 Pigs fly 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.There are various kinds of conditionals in English; e.g., there is the indicative conditional and the subjunctive or counterfactual conditional. The latter do not have the same truth conditions as the material conditional. For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below.
References
- Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
- Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell.
- Edgington, Dorothy (2006), "Conditionals", in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Eprint.
- Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.
See also
Conditionals
Related topics
Conditional may refer to:
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- Causal conditional, if X then Y, where X is a cause of Y
- Conditional mood, a verb form in many languages
- Conditional probability, the probability of an event A given that another event B has occurred
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Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration.
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In logic and mathematics, a propositional calculus (or a sentential calculus) is a formal system in which formulas representing propositions can be formed by combining atomic propositions using logical connectives, and a system of formal proof rules
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predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulas contain variables which can be quantified.
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formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities.
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In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems.
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In logic and mathematics, logical implication is a logical relation that holds between a set T of formulae and a formula B when every model (or interpretation or valuation) of T is also a model of B.
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In logic and mathematics, a logical value, also called a truth value, is a value indicating the extent to which a proposition is true.
In classical logic, the only possible truth values are true and false.
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In classical logic, the only possible truth values are true and false.
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proposition is the content of an assertion, that is, it is true-or-false and defined by the meaning of a particular piece of language. The proposition is independent of the of communication.
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Johnston diagrams, which look similar to Euler or Venn diagrams, illustrate formal propositional logic in a visual manner. Logically they are equivalent to truth tables; some may find them easier to understand at a glance.
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Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. They are characterised by a number of properties[1]; non-classical logics are those that lack one or more of these properties, which are:
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In mathematical logic, the deduction theorem states that if a formula F is deducible from E then the implication E → F is demonstrable (i.e. it is "deducible" from the empty set).
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monotonic function (or monotone function) is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
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Monotonicity of entailment is a property of many logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In sequent calculi this property can be captured by an inference rule called weakening, or sometimes
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A non-monotonic logic is a formal logic whose consequence relation is not monotonic. Most studied formal logics have a monotonic consequence relation, meaning that adding a formula to a theory never produces a reduction of its set of consequences.
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Relevance logic, also called relevant logic, is any of a family of non-classical substructural logics that impose certain restrictions on implication. (It is generally, but not universally, called relevant logic by Australian logicians, and relevance logic
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associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed.
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In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example:
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- 4 • (2 + 3) = (4 • 2) + (4 • 3).
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The term transitivity may refer to:
In grammar
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In grammar
- Transitivity (grammatical category)
- transitive verb, when a verb takes an object
- Transitive relation, a binary relation
- Intransitivity
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Commutativity is a widely used mathematical term that refers to the ability to change the order of something without changing the end result. It is a fundamental property in most branches of mathematics and many proofs depend on it.
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Idempotence IPA: /ˌaɪdɨmˈpoʊtənts/ describes the property of operations in mathematics and computer science that yield the same result after the operation is applied multiple times.
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The indicative conditional is the logical operation given by statements of the form "If A then B" in ordinary English (or similar natural languages). The indicative conditional, unlike the material conditional, does not have a stipulated definition.
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A counterfactual conditional, subjunctive conditional, or remote conditional, is a conditional (or "if-then") statement indicating what would be the case if its antecedent were true.
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Willard Van Orman Quine (June 25, 1908 – December 25, 2000), usually cited as W.V. Quine or W.V.O. Quine but known to his friends as Van, was one of the most influential philosophers and logicians of the 20th century.
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A counterfactual conditional, subjunctive conditional, or remote conditional, is a conditional (or "if-then") statement indicating what would be the case if its antecedent were true.
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The indicative conditional is the logical operation given by statements of the form "If A then B" in ordinary English (or similar natural languages). The indicative conditional, unlike the material conditional, does not have a stipulated definition.
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In logic a corresponding conditional is a statement whose principal connective is the material implication symbol, and whose antecedent is the conjunction of the premises or an argument and whose consequent is the conclusion of that argument.
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In logic, a strict conditional is a material conditional that is acted upon by the necessity operator from modal logic. For any two propositions and , the formula says that materially implies while says that strictly implies .
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