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Jean Blaize Grize (in Logique Moderne Fascicule 1 p:28) has been distinguished between two type of implication : implication as relation among sentences and implication as operation, the former is qualified as preorder relation that is reflexive (a→a) , transitive (if a→b and b→c then a→c) and antisymetric (if a→b and b→a then a↔b) , while the later is described through mathematical proprieties namely ; commutativity , associativity … Is this distinction valid or not?

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  • the operation is defined on which set? – miracle173 Aug 23 '17 at 21:16
  • Logical implication is just another operator like AND and OR. Like AND and OR, it has a well known truth table. – Dan Christensen Aug 23 '17 at 21:32
  • Semantic consequences are relations. I don't know if that helps. I know for formulas they are. – W. G. Aug 23 '17 at 21:40
  • It sounds like your author is doing a bad job distinguishing between $\implies$ and $\vdash$. Implication is usually intended to be a symbol that manifests as $\vdash a \implies b$ if and only if $a \vdash b$ (some caveats when there are free variables). If you want to interpret that as a relation, or as a binary combinatorial gate, or as a order, that's up to you. – DanielV Aug 23 '17 at 21:46
  • I could be wrong but I always thought implication could mean two things. Semantic consequences $\models$, and $\rightarrow$. I do not think syntactic consequence, $\vdash$, would be confused with an implication here. Also, I am not a big fan of using the $\implies$ symbol. My logic book is terrible and uses $\implies$ incorrectly half the time. – W. G. Aug 23 '17 at 21:54
  • If I was to guess what your textbook is referring to as a relation among setences deals with $\models$ as referring to formulas, particularly WFFs which are relations. The other would just be $\rightarrow$. I do not know if it this would be a relation or not. The definition of a relation deals has $\rightarrow$ in it and to avoid any circular reasoning. So in that case, I would presume the $\rightarrow$ is not a relation. – W. G. Aug 23 '17 at 22:02
  • I am referring to the $\models$ symbol as a relation here. – W. G. Aug 23 '17 at 22:04

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