# Re: [peirce-l] Frege against the Booleans

```Sorry, corrections in bold:

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`Jon,`
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The way I learned it, (formal) implication is not the /assertion/ but the /validity/ of the (material) conditional, so it's a difference between 1st-order and 2nd-order logic, a difference that Peirce recognized in some form. If the schemata involving "p" and "q" are considered to expose all relevant logical structure (as usually in propositional logic), then a claim like "p formally implies q" is false. On the other hand, a proposition /à la/ "if p then q" (or "p materially implies q") is contingent, neither automatically true nor automatically false. I agree that you can see it as the same relationship on two different levels. That seems the natural way to look at it.
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Another kind of implication is expressed by rewriting a proposition like "Ax(Gx-->Hx)" as "G=>H". In other words "All G is H" gets expressed "G implies H". In first-order logic, at least, it actually comes down to a material conditional compound of two terms in a universal proposition.
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If in addition to logical rules one has postulated or generally granted other rules, say scientific or mathematical rules, then these lead to scientific or mathematical implications, the associated conditionals being true by the scientific or mathematical rules, not just contingently on a case-by-case basis. Anyway, all these kinds of implication do seem like the same thing in various forms.
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It's not clear to me how any of this figures into the concept-vs.-judgment question. The only connection that I've been able to make out in my haze is that when we say something like "p formally implies p", we're thinking of the proposition p as if it were a concept rather than a judgment; our concern is limited to validity *as of an argument* "p ergo p". If we *_/say/_* 'p, ergo p' or, in a kindred sense, "p proves p," we're thinking of p as a judgment, and our concern includes the soundness as well as validity *of the argument "p ergo p"*.
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Best, Ben

On 5/11/2012 2:25 PM, Jon Awbrey wrote:

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```Ben,

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Just to give a prototypical example, one of the ways that the distinction between concepts and judgments worked its way through analytic philosophy and into the logic textbooks that I knew in the 60s was in the distinction between a "conditional" ( → or -> ) and an "implication" ( ⇒ or => ). The first was conceived as a function (from a pair of truth values to a single truth value) and the second was conceived as a relation (between two truth values). The relationship between them was Just So Storied by saying that asserting the conditional or judging it to be true gave you the implication.
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I think it took me a decade or more to clear my head of the dogmatic slumbers that this sort of doctrine laid on my mind, mostly because the investiture of two distinct symbols for what is really one and the same notion viewed in two different ways so obscured the natural unity of the function and the relation.
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Cf. http://mywikibiz.com/Logical_implication

Regards,

Jon

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