Logical implication
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The concept of logical implication encompasses a specific logical function, a specific logical relation, and the various symbols that are used to denote this function and this relation. In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them.
Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following:
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\(\begin{array}{l} p ~\text{implies}~ q. \'"`UNIQ-MathJax1-QINU`"' Regarded as a set, this triadic relation is the same thing as the binary operation: '"`UNIQ-MathJax2-QINU`"' The relationship between \(Cond\!\) and \(L\!\) exemplifies the standard association that exists between any binary operation and its corresponding triadic relation. The conditional sign "\(\rightarrow\!\)" denotes the same formal object as the function name "\(Cond\mbox{ }\!\)", the only difference being that the first is written infix while the second is written prefix. Thus we have the following equation: \[(p \rightarrow q) = Cond (p,\ q)\,.\!\] Consider once again the triadic relation \(L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!\) that is defined in the following equivalent fashion: \[L = \{(p,\ q,\ Cond (p,\ q))\ :\ (p,\ q) \in \mathbb{B} \times \mathbb{B} \}\,.\!\] Associated with the triadic relation \(L\!\) is a binary relation \(L_{..T} \subseteq \mathbb{B} \times \mathbb{B}\!\) that is called the fiber of \(L\!\) with \(T\!\) in the third place. This object is defined as follows: \[L_{..T} = \{ (p,\ q) \in \mathbb{B} \times \mathbb{B}\ :\ (p,\ q,\ T) \in L \}\,.\!\] The same object is achieved in the following way. Begin with the binary operation: \[Cond : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\!\] Form the binary relation that is called the fiber of \(Cond\!\) at \(T\!\), notated as follows: \[Cond^{-1}(T) \subseteq \mathbb{B} \times \mathbb{B}\,.\!\] This object is defined as follows: \[Cond^{-1}(T) = \{ (p,\ q) \in \mathbb{B} \times \mathbb{B}\ :\ Cond (p,\ q) = T \}\,.\!\] The implication sign "\(\Rightarrow\!\)" denotes the same formal object as the relation names "\(L_{..T}\mbox{ }\!\)" and "\(Cond^{-1}(T)\mbox{ }\!\)", the only differences being purely syntactic. Thus we have the following logical equivalence: \[(p \Rightarrow q) \iff (p,\ q) \in L_{..T} \iff (p,\ q) \in Cond^{-1}(T)\,.\!\] This completes the derivation of the mathematical objects that are denoted by the signs "\(\rightarrow\!\)" and "\(\Rightarrow\!\)" in this discussion. It needs to be remembered, though, that not all writers observe this distinction in every context. Especially in mathematics, where the single arrow sign "\(\rightarrow\!\)" is reserved for function notation, it is common to see the double arrow sign "\(\Rightarrow\!\)" being used for both concepts. References
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