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. \'"`UNIQ-MathJax1-QINU`"' 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 "\(\Rightarrow\!\)" denotes the same formal object as the relation names "\(L_{..T}\mbox{ }\!\)" and "\(\operatorname{Cond}^{-1}(T)\mbox{ }\!\)", the only differences being purely syntactic. Thus we have the following logical equivalence:

\[(p \Rightarrow q) \iff (p,\ q) \in L_{..T} \iff (p,\ q) \in \operatorname{Cond}^{-1}(T)\,.\!\]

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

  • 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.

Syllabus

Focal nodes

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Peer nodes

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Logical operators

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Related topics

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Relational concepts

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Information, Inquiry

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