# Logical Implication Relation

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A Logical Implication Relation is a Boolean Logic Relation/Propositional Sentence Operation where ...

**AKA:**Implication Connective, Implication Relation, Material Conditional, →, Conditional, ⊃, Implies, Implication, Logical Implication, Logical Implication Operation, Implication Operation.**Context:**- It is a Commutative Relation.
- It is an Idempotent Relation.
- It is an Associative Relation.
- It is a Distributive Relation.
- It is a Monotone Relation.
- It is a Preorder Relation.
- It has Truth Table:
**A****B****A→B**T T T T F F F T T F F T

**Example(s):**- [math]\displaystyle{ A }[/math] →
*B*. - a Horn Clause.
- …

- [math]\displaystyle{ A }[/math] →
**Counter-Example(s):**- [math]\displaystyle{ A }[/math] ⇒ [math]\displaystyle{ B }[/math], an Entailment Relation.

**See:**Bi-implication Relation, Propositional Logic Formula, If-Then Rule.

## References

### 2016

- (Weisstein, 2016) ⇒ Eric W. Weisstein. (1999 - 2016). “Implies." From MathWorld -- A Wolfram Web Resource. ⇒ http://mathworld.wolfram.com/Implies.html
**Implies**is the connective in propositional calculus which has the meaning**"if A is true, then B is also true."**In formal terminology, the term conditional is often used to refer to this connective (Mendelson, 1997, p. 13). The symbol used to denote "implies" is A⇒B, A [math]\displaystyle{ \supset }[/math] B (Carnap, 1958, p. 8; Mendelson, 1997, p. 13), or A[math]\displaystyle{ \rightarrow }[/math]B. The Wolfram Language command Experimental`ImpliesRealQ[ineqs1, ineqs2] can be used to determine if the system of real algebraic equations and inequalities ineqs1 implies the system of real algebraic equations and inequalities ineqs2.

- In classical logic, A⇒B is an abbreviation for [math]\displaystyle{ \rightharpoondown }[/math]A v B, where [math]\displaystyle{ \rightharpoondown }[/math]A denotes NOT and v denoted OR (though this is not the case, for example, in intuitionistic logic). ⇒ is a binary operator that is implemented in the Wolfram Language as Implies[A, B], and can not be extended to more than two arguments.
- A=>B has the following truth table (Carnap, 1958, p. 10; Mendelson, 1997, p. 13).
A B A⇒B T T T T F F F T T F F T

- If A⇒B and B⇒A (i.e., A⇒B ^ B⇒A), then A and B are said to be equivalent, a relationship which is written symbolically as A[math]\displaystyle{ \Leftrightarrow }[/math]B, A[math]\displaystyle{ \leftrightarrow }[/math]B, or A=B (Carnap, 1958, p. 8).

### 2012

- (Weisstein, 2016) ⇒ Rudolf Carnap. (2012). “Introduction to symbolic logic and its applications". Courier Corporation. ⇒ http://goo.gl/vqIpQS

### 2009

- (Mendelson, 2009) ⇒ Elliott Mendelson (2009). “Introduction to mathematical logic". CRC press. http://goo.gl/EWiHJA
- (Wikinary, 2009) http://en.wiktionary.org/wiki/material_conditional
- Noun
- A conditional statement in the indicative mood.
*A implies B is a material conditional.*

- A conditional statement in the indicative mood.
- Synonyms: conditional; if-then statement

- Noun
- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Material_conditional
- 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:
- X \rightarrow Y,
- X \supset Y, and sometimes
- 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 not p or 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