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 Template:Col-break Inquiry Live Template:Col-break Logic Live Template:Col-end Peer nodes Template:Col-break Logical Implication @ MyWikiBiz Logical Implication @ MathWeb Wiki Logical Implication @ NetKnowledge Template:Col-break Logical Implication @ OER Commons Logical Implication @ P2P Foundation Logical Implication @ SemanticWeb Template:Col-end Logical operators Template:Col-break Exclusive disjunction Logical conjunction Logical disjunction Logical equality Template:Col-break Logical implication Logical NAND Logical NNOR Negation Template:Col-end Related topics Template:Col-break Ampheck Boolean domain Boolean function Boolean-valued function Differential logic Template:Col-break Logical graph Minimal negation operator Multigrade operator Parametric operator Peirce's law Template:Col-break Propositional calculus Sole sufficient operator Truth table Universe of discourse Zeroth order logic Template:Col-end Relational concepts Template:Col-break Continuous predicate Hypostatic abstraction Logic of relatives Logical matrix Template:Col-break Relation Relation composition Relation construction Relation reduction Template:Col-break Relation theory Relative term Sign relation Triadic relation Template:Col-end Information, Inquiry Template:Col-break Inquiry Dynamics of inquiry Template:Col-break Semeiotic Logic of information Template:Col-break Descriptive science Normative science Template:Col-break Pragmatic maxim Truth theory Template:Col-end Related articles Jon Awbrey, “Semiotic Information” Jon Awbrey, “Introduction To Inquiry Driven Systems” Jon Awbrey, “Prospects For Inquiry Driven Systems” Jon Awbrey, “Inquiry Driven Systems : Inquiry Into Inquiry” Jon Awbrey, “Propositional Equation Reasoning Systems” Jon Awbrey, “Differential Logic : Introduction” Jon Awbrey, “Differential Propositional Calculus” Jon Awbrey, “Differential Logic and Dynamic Systems” Document history Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders. Template:Col-break Logical Implication, MyWikiBiz Logical Implication, Wikiversity Beta Logical Implication, PlanetMath Template:Col-break Logical Implication, GetWiki Logical Implication, Wikinfo Logical Implication, Textop Wiki Template:Col-break Logical Implication, Wikipedia Template:Col-end <sharethis />