Difference between revisions of "Logical implication"
Jon Awbrey (talk | contribs) (→Discussion: TeX) |
Jon Awbrey (talk | contribs) (→Discussion: TeX) |
||
| Line 111: | Line 111: | ||
|} | |} | ||
| − | Associated with the triadic relation <math>L\!</math> is a binary relation <math>L_{\underline{~} | + | Associated with the triadic relation <math>L\!</math> is a binary relation <math>L_{\underline{~} \, \underline{~} \, \operatorname{T}} \subseteq \mathbb{B} \times \mathbb{B}</math> that is called the ''fiber'' of <math>L\!</math> with <math>\operatorname{T}</math> in the third place. This object is defined as follows: |
{| align="center" cellspacing="10" width="90%" | {| align="center" cellspacing="10" width="90%" | ||
| − | | <math>L_{\underline{~} | + | | <math>L_{\underline{~} \, \underline{~} \, \operatorname{T}} = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : (p, q, \operatorname{T}) \in L \}.</math> |
|} | |} | ||
| Line 121: | Line 121: | ||
: <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\!</math> | : <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\!</math> | ||
| − | Form the binary relation that is called the ''fiber'' of <math>\operatorname{Cond}</math> at <math>T | + | Form the binary relation that is called the ''fiber'' of <math>\operatorname{Cond}</math> at <math>\operatorname{T}</math>, notated as follows: |
| − | : <math>\operatorname{Cond}^{-1}(T) \subseteq \mathbb{B} \times \mathbb{B} | + | : <math>\operatorname{Cond}^{-1}(\operatorname{T}) \subseteq \mathbb{B} \times \mathbb{B}.</math> |
This object is defined as follows: | This object is defined as follows: | ||
| − | : <math>\operatorname{Cond}^{-1}(T) = \{ (p, | + | : <math>\operatorname{Cond}^{-1}(\operatorname{T}) = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : \operatorname{Cond} (p, q) = \operatorname{T} \}.</math> |
The implication sign "<math>\Rightarrow\!</math>" denotes the same formal object as the relation names "<math>L_{..T}\mbox{ }\!</math>" and "<math>\operatorname{Cond}^{-1}(T)\mbox{ }\!</math>", the only differences being purely syntactic. Thus we have the following logical equivalence: | The implication sign "<math>\Rightarrow\!</math>" denotes the same formal object as the relation names "<math>L_{..T}\mbox{ }\!</math>" and "<math>\operatorname{Cond}^{-1}(T)\mbox{ }\!</math>", the only differences being purely syntactic. Thus we have the following logical equivalence: | ||
Revision as of 00:24, 13 May 2012
☞ This page belongs to resource collections on Logic and Inquiry.
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
SyllabusFocal nodesTemplate:Col-breakTemplate:Col-breakTemplate:Col-endPeer nodes
Logical operatorsRelated topics
Relational conceptsInformation, InquiryRelated articlesDocument historyPortions 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.
<sharethis /> | |
