# Logical Implication Relation

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## References

### 2016

In classical logic, A⇒B is an abbreviation for $\rightharpoondown$A v B, where $\rightharpoondown$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$\Leftrightarrow$B, A$\leftrightarrow$B, or A=B (Carnap, 1958, p. 8).

### 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.
• Synonyms: conditional; if-then statement
• (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