Logical Implication Relation

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.
  • …
• 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.
  • 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

---