Bertrand Russell |

I said:

Traditional logic does not attempt to reduce logic to a quantitative calculus, largely because it views logic as a linguistic and metaphysical art, not a technical mathematical calculus.McPike responds:

Surely this is just wrong? 'Traditional logic' also views logic as a formal tool, one which it is necessary to master "before" attempting something like metaphysics. It can be treated (taught and learned) just as abstractly and formally as modern logic.But to say that traditional logic is formal (or at least has a formal branch—the old traditional logic included material logic, which was not formal) is not the same thing as saying that all rational discourse can be reduced to "a kind of mathematical calculus," which was the point of my post. I think the latter statement is more specific than the mere issue of formality.

For one thing, I think it would be fair to say that the system of traditional logic recognizes that there are what I would call "material leakages" in the system which defy exclusively formal treatment. The conditional statements I pointed to are just one example of this. Oblique syllogisms (syllogisms in which there is a relational term playing an essential role in the inference—"John is

*the son of*Mary) would be another. In both these cases the formal clothing we try to fit our rational expression into doesn't perfectly fit. There is some material relation that inserts itself into the otherwise formal structure of the reasoning and that recognition is built into the formal system of traditional logic.

For another, traditional logicians have traditionally disputed the idea that logic—even in its formal aspect—is purely quantitative in nature, in the way in which the kind of logic fathered by the

*Principia Mathematica*seems to be. Most traditional manuals on logic begin with the distinction between

*comprehension*and

*extension*. Comprehension has to do with the

*intellectual content*of logical terms, whereas extension has to do with their referents in the world. The comprehension of the term 'man', for example, would be a "rational, sentient, living, material substance." The extension of the term 'man' would be "all the men who are, were, or will be."

Comprehension is

*qualitative*in nature because it asks questions involving the kind of things to which terms refer, whereas extension is

*quantitative*, since it asks how much or how many things a term refers to. I'm willing to be disproven here, but it seems to me that modern systems of logic (at least the propositional and predicate calculus) are all extension and no comprehension. That is reflected in the title modern logicians have affixed to their system: propositional and predicate

*calculus*. I'm no expert in set theory, a fixture of much of the modern logic that traces itself to Russell and Whitehead, but from what I know of it, it seems to be one bit of evidence for my claim here.

And it doesn't seem to me a complete coincidence that those who developed modern logic were almost exclusively mathematicians (Frege, Boole, Russell, Whitehead, et al.).

My point was simply that although the formal branch of traditional logic is the treatment of reasoning in a formal way, there is a recognition that, in doing so, there are material (and qualitative) considerations that affect the course and conduct of the reasoning, a recognition that modern systems do not seem to allow for in their attempt to cram all rational discourse into a purely formal system. The modern system of logic not only does not allow for material considerations in its formal system, it doesn't, as traditional logic does, recognize a material (or "major") branch of logic at all, any material considerations having been relegated to the dust heap of rejected Aristotelian metaphysics (e.g. Russell), or to the field of rhetoric (as seems to be the case with what is now called "informal logic").

*I will post my answer to McPike's challenge to my use of conditional statements as examples of the differences between the two systems tomorrow.*

## 5 comments:

Thanks for the response, Martin. This is a fascinating subject. I'll look forward to reading the rest of your response tomorrow before offering any comments.

Thanks again, Martin, but I find this unconvincing. Traditional 'logic' may have included more than its formal branch (the study of valid forms of reasoning) but if the usage of modern philosophy restricts the meaning of 'logic' to a subset of what was traditionally called 'logic,' then you have to recognize the equivocation that is thus created if you want to avoid reasoning fallaciously.

As for the alleged modern claim that "all rational discourse can be reduced to a kind of mathematical calculus," it seems to me that the goal of modern logic was to be able to *express* all rational discourse using a kind of mathematical calculus, not to reduce it to being nothing more than that! An analogy: it's like saying the goal of producing electronic versions of all print-media is to reduce all print-media to those versions. Or if someone writes a book on the elements of style, you accuse him of wanting to reduce all language to its stylistic elements. That just doesn't follow.

In regard to what you call 'leakage,' Frege's original project in logic was a particular project that aimed to be 'leak-free.' But 'modern logic' is surely well aware (thanks to Russell, and related developments in GĂ¶del and Tarski) that that project failed - Frege himself fully recognized this! Can you perhaps explain more clearly what you mean when you attribute this project to Russell and Whitehead?

Regarding comprehension, modern logic and traditional logic both presuppose comprehension, since questions of extension presuppose comprehension (at least enough for recognition) of the extensionally treated elements (i.e., the subjects being logically quantified). But logic is not the discipline which settles questions of comprehension - higher sciences are required for that.

"it seems to me that modern systems of logic (at least the propositional and predicate calculus) are all extension and no comprehension. That is reflected in the title modern logicians have affixed to their system: propositional and predicate calculus."

I think this is clearly a very weak argument. In any case, I wonder if you are familiar with Fred Sommers and his 'term calculus' (inspired by both Aristotelian logic and Leibniz)? He would certainly seem to be a crucial figure to look at in this context.

David McPike said...

Thanks again, Martin, but I find this unconvincing. Traditional 'logic' may have included more than its formal branch (the study of valid forms of reasoning) but if the usage of modern philosophy restricts the meaning of 'logic' to a subset of what was traditionally called 'logic,' then you have to recognize the equivocation that is thus created if you want to avoid reasoning fallaciously.I think I adequately explained that my main point was to establish that modern logicians tend to recognize no material considerations in their system of formal logic. I added the point about their failure to recognize any material branch to underscore their exclusion of material influences from their logical system. Equivocal usages of the term 'logic' do not affect my point at all.

I think you are trying to create an equivocation here through your assumption that the modern system only treats the excludes the material as a matter of usage. I simply don't accept that. If you're saying this is just a matter of usage, I think you are just wrong.I'm sorry you don't accept that, but I, on the other hand, do not accept that the difference is simply a matter of usage--and consequently I don't accept your characterization of my point as an equivocation.

The whole point of the these two posts is that the difference between the two systems and they way they construct their logic is fundamentally tied to the differences in their metaphysical assumptions. The "usage" is a reflection of these assumptions. The equivocation you are arguing for is not implicit in my argument, it is rather created by your assumption that it is a mere matter of usage, why, as should be evident from my argument here.

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