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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)
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Either way, one reads such a rule in the following manner: "From <math>\textit{Expression~1}</math> and <math>\textit{Expression~2}</math> infer <math>\textit{Expression~3}.</math>
Either way, one reads such a rule of inference in the following manner:
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From <math>\textit{Expression~1}</math> and <math>\textit{Expression~2}</math> infer <math>\textit{Expression~3}.</math>
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Looking to Example 1, the rule of inference known as ''modus ponens'' says the following: From the premiss <math>p \Rightarrow q</math> and the premiss <math>p\!</math> one may logically infer the conclusion <math>q.\!</math>
Looking to Example 1, the rule of inference known as ''modus ponens'' says the following: From the premiss <math>p \Rightarrow q</math> and the premiss <math>p\!</math> one may logically infer the conclusion <math>q.\!</math>
Modus ponens is an ''illative'' or ''implicational'' rule. Passage through its turnstile incurs the toll of some information loss, and thus from a fact of ''q'' alone one cannot infer with any degree of certainty that ''p'' ⇒ ''q'' and ''p'' are the reasons why ''q'' happens to be true.
Modus ponens is an ''illative'' or ''implicational'' rule. Passage through its turnstile incurs the toll of some information loss, and thus from a fact of <math>q\!</math> alone one cannot infer with any degree of certainty that <math>p \Rightarrow q</math> and <math>p\!</math> are the reasons why <math>q\!</math> happens to be true.
Further considerations along these lines may lead us to appreciate the difference between ''implicational rules of inference'' (IROI's) and ''equational rules of inference'' (EROI's), the latter indicated by an ''equational line of inference'' (ELOI) or 2-way turnstile "|–|".
Further considerations along these lines may lead us to appreciate the difference between ''implicational rules of inference'' and ''equational rules of inference'', the latter indicated by an ''equational line of inference'' or a 2-way turnstile <math>{}^{\backprime\backprime} \Vdash {}^{\prime\prime}.</math>
==Variations on a theme of transitivity==
==Variations on a theme of transitivity==