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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:
\(\begin{array}{l} p ~\text{implies}~ q. \\[6pt] \text{if}~ p ~\text{then}~ q. \end{array}\) 
Here \(p\!\) and \(q\!\) are propositional variables that stand for any propositions in a given language. In a statement of the form \({}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime},\) the first term, \(p,\!\) is called the antecedent and the second term, \(q,\!\) is called the consequent, while the statement as a whole is called either the conditional or the consequence. Assuming that the conditional statement is true, then the truth of the antecedent is a sufficient condition for the truth of the consequent, while the truth of the consequent is a necessary condition for the truth of the antecedent.
Note. Many writers draw a technical distinction between the form \({}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}\) and the form \({}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}.\) In this usage, writing \({}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}\) asserts the existence of a certain relation between the logical value of \(p\!\) and the logical value of \(q,\!\) whereas writing \({}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}\) merely forms a compound statement whose logical value is a function of the logical values of \(p\!\) and \(q.\!\) This will be discussed in detail below.
Contents
Definition
The concept of logical implication is 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.
In the interpretation where \(0 = \operatorname{false}\) and \(1 = \operatorname{true}\), the truth table associated with the statement \({}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime},\) symbolized as \({}^{\backprime\backprime} p \Rightarrow q {}^{\prime\prime},\) appears below:
\(p\!\)  \(q\!\)  \(p \Rightarrow q\!\) 
\(0\!\)  \(0\!\)  \(1\!\) 
\(0\!\)  \(1\!\)  \(1\!\) 
\(1\!\)  \(0\!\)  \(0\!\) 
\(1\!\)  \(1\!\)  \(1\!\) 
Discussion
The usage of the terms logical implication and material conditional varies from field to field and even across different contexts of discussion. One way to minimize the potential confusion is to begin with a focus on the various types of formal objects that are being discussed, of which there are only a few, taking up the variations in language as a secondary matter.
The main formal object under discussion is a logical operation on two logical values, typically the values of two propositions, that produces a value of \(\operatorname{false}\) just in case the first operand is true and the second operand is false. By way of a temporary name, the logical operation in question may be written as \(\operatorname{Cond}(p, q),\) where \(p\!\) and \(q\!\) are logical values. The truth table associated with this operation appears below:
\(p\!\)  \(q\!\)  \(\operatorname{Cond}(p, q)\) 
\(\operatorname{F}\)  \(\operatorname{F}\)  \(\operatorname{T}\) 
\(\operatorname{F}\)  \(\operatorname{T}\)  \(\operatorname{T}\) 
\(\operatorname{T}\)  \(\operatorname{F}\)  \(\operatorname{F}\) 
\(\operatorname{T}\)  \(\operatorname{T}\)  \(\operatorname{T}\) 
Some logicians draw a firm distinction between the conditional connective, the symbol \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime},\) and the implication relation, the object denoted by the symbol \({}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}.\) These logicians use the phrase if–then for the conditional connective and the term implies for the implication relation. Some explain the difference by saying that the conditional is the contemplated relation while the implication is the asserted relation. In most fields of mathematics, it is treated as a variation in the usage of the single sign \({}^{\backprime\backprime} \Rightarrow {}^{\prime\prime},\) not requiring two separate signs. Not all of those who use the sign \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a socalled syncategorematic sign, that is, a sign with a purely syntactic function. For the sake of clarity and simplicity in the present introduction, it is convenient to use the twosign notation, but allow the sign \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) to denote the boolean function that is associated with the truth table of the material conditional. These considerations result in the following scheme of notation.
\(\begin{matrix} p \rightarrow q & \quad & \quad & p \Rightarrow q \\ \text{if}~ p ~\text{then}~ q & \quad & \quad & p ~\text{implies}~ q \end{matrix}\) 
Let \(\mathbb{B} = \{ \operatorname{F}, \operatorname{T} \}\) be the boolean domain consisting of two logical values. The truth table shows the ordered triples of a triadic relation \(L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!\) that is defined as follows:
\(L = \{ (p, q, r) \in \mathbb{B} \times \mathbb{B} \times \mathbb{B} : \operatorname{Cond}(p, q) = r \}.\) 
Regarded as a set, this triadic relation is the same thing as the binary operation:
\(\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.\) 
The relationship between \(\operatorname{Cond}\) and \(L\!\) exemplifies the standard association that exists between any binary operation and its corresponding triadic relation.
The conditional sign \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) denotes the same formal object as the function name \({}^{\backprime\backprime} \operatorname{Cond} {}^{\prime\prime},\) the only difference being that the first is written infix while the second is written prefix. Thus we have the following equation:
\((p \rightarrow q) = \operatorname{Cond}(p, q).\) 
Consider once again the triadic relation \(L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!\) that is defined in the following equivalent fashion:
\(L = \{ (p, q, \operatorname{Cond}(p, q) ) : (p, q) \in \mathbb{B} \times \mathbb{B} \}.\) 
Associated with the triadic relation \(L\!\) is a binary relation \(L_{\underline{~} \, \underline{~} \, \operatorname{T}} \subseteq \mathbb{B} \times \mathbb{B}\) that is called the fiber of \(L\!\) with \(\operatorname{T}\) in the third place. This object is defined as follows:
\(L_{\underline{~} \, \underline{~} \, \operatorname{T}} = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : (p, q, \operatorname{T}) \in L \}.\) 
The same object is achieved in the following way. Begin with the binary operation:
\(\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.\) 
Form the binary relation that is called the fiber of \(\operatorname{Cond}\) at \(\operatorname{T},\) notated as follows:
\(\operatorname{Cond}^{1}(\operatorname{T}) \subseteq \mathbb{B} \times \mathbb{B}.\) 
This object is defined as follows:
\(\operatorname{Cond}^{1}(\operatorname{T}) = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : \operatorname{Cond}(p, q) = \operatorname{T} \}.\) 
The implication sign \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) denotes the same formal object as the relation names \({}^{\backprime\backprime} L_{\underline{~} \, \underline{~} \, \operatorname{T}} {}^{\prime\prime}\) and \({}^{\backprime\backprime} \operatorname{Cond}^{1}(T) {}^{\prime\prime},\) the only differences being purely syntactic. Thus we have the following logical equivalence:
\((p \Rightarrow q) \iff (p, q) \in L_{\underline{~} \, \underline{~} \, \operatorname{T}} \iff (p, q) \in \operatorname{Cond}^{1}(T).\) 
This completes the derivation of the mathematical objects that are denoted by the signs \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) and \({}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}\) 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 \({}^{\backprime\backprime} \rightarrow {}^{\prime\prime}\) is reserved for function notation, it is common to see the double arrow sign \({}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}\) being used for both concepts.
References
 Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
 Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell.
 Edgington, Dorothy (2006), "Conditionals", in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy, Eprint.
 Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.
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