Logical Implication Relation

From GM-RKB
(Redirected from Logical Implication)
Jump to: navigation, search

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



References

2016

In classical logic, A⇒B is an abbreviation for [math]\rightharpoondown[/math]A v B, where [math]\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).
ABA⇒B
TTT
TFF
FTT
FFT
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]\Leftrightarrow[/math]B, A[math]\leftrightarrow[/math]B, or A=B (Carnap, 1958, p. 8).

2012

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:
      • pq
        TT
        T
        TF
        F
        FT
        T
        FF
        T