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.