Question about Kenny’s critiques on Mill’s logic

Question

Recently, I have read a book called New History of Western Philosophy by Anthony Kenny.

In this book, he starts chapter 4 Logic with a section called Mill’s Empiricist Logic, which contains an introduction to his logic and Kenny’s critiques on him.

My question is about the meaning of the term ‘syllogism’ and ‘non-syllogistic’ used by Kenny there.

What follows is his explanation and critique to it, where he use those terms.

Mill distinguished real inference, which is informative, and verbal inference, which brings no new knowledge about the world.

“He accepted that all reasoning was syllogistic, and he claimed that in every syllogism the conclusion is actually contained and implied in the premisses.”

Since he considered the major premisses in every syllogism as general proposition collected by induction, those were informative, but not justified.

This is why he thought that the syllogism was not a genuine inference.

However, Kenny thinks that “Mill’s criticism of deductive argument involves a confusion between logic and epistemology.”

Because “syllogism is not the only form of inference, and there are many valid non-syllogistic argument (e.g. arguments of the form ‘A=B’, ‘B=C’, therefore ‘A=C’) which are quite capable of conveying information.”

Why is an argument of the form (‘A=B’, ‘B=C’, therefore ‘A=C’) non-syllogistic?

This is my first question.

And second one is what exactly the confusion between logic and epistemology means.

I’m not sure whether I give you enough context to understand my questions, because I don’t understand!

Thanks!

Answer 1

For the first question, it depends exactly how one defines syllogism; see this other question for variations. Presumably Kenny means to exclude reasoning "modulo equational theories", e.g. using first-order logic with equality from his idea/definition of syllogism. This is fair as Aristotle, whose ideas still dominated logic at the time, didn't envisage such a thing. Aristotelian syllogisms were all using monadic, i.e. one-place predicates, so equality--which is a binary predicate--couldn't be formally included. (And Mill didn't really "see" beyond Aristotle.)

For the 2nd question, the distinction between logic and epistemology is spelled out by Kenny:

Mill’s criticism of deductive argument involves a confusion between logic and epistemology. An inference may be, as he says, deductively valid without being informative: validity is a necessary but not a sufficient condition for an argument to produce true information.

Basically Mill was concerned that saying universally quantified utterances like "all men are mortal" already meant that we knew all there is to know on that matter and particularizing that bit of knowledge to individuals was a trivial matter, epistemologically. Mill's point seems to have been than the only way to discover such universally quantified truths was through induction. Obviously he was wrong, but mathematics wasn't put on firm logical basis back then, so he probably didn't see the connection how one could prove "interesting enough" universally quantified statements by deductive means. (Quantification was also not exactly formalized in Mill's time.)

Mill wrote on logic in the early 1840s, by the way, before Boole, Schroder etc. SEP has this to say about Mill's critique of logic (as that was mostly Mill's only dance with the topic):

Deductive or a priori reasoning, Mill thinks, is similarly empty. Predating the revolution in logic that the late nineteenth-century ushered in, Mill thinks of deductive reasoning primarily in terms of the [Aristotelian] syllogism. Syllogistic reasoning, he argues can elicit no new truths about how the world is: “nothing ever was, or can be proved by syllogism which was not known, or assumed to be known, before” (System, VII: 183).

Also, Mill thought that mathematics was actually an inductive way of thinking (unrelated to [Aristotelian] logic):

Mill holds that where we do gain genuinely new knowledge—in cases of mathematics and geometry, for instance—we must, at some level, be reasoning inductively. Mill, that is to say, attempts to account for the genuine informativeness of mathematical and geometric reasoning by denying that they are in any real sense a priori.

There's more on SEP on Mill's conception of mathematics, but that's a bit besides the point here, so I won't detail that further.

Answer 2

Regarding your:

Why is an argument of the form (‘A=B’, ‘B=C’, therefore ‘A=C’) non-syllogistic?

This inference is from basic math arithmetic properties, or more strictly speaking from set-theoretic definition of Equivalence class, which transitivity in this definition plays a central to arrive at your conclusion A=C. So apparently, this is not exactly the classical syllogism popularized by Aristotle.

However, underlying above process, we still implicitly employs the classic syllogism since "=" in your example is a specific instance of Equivalence class acting as the 2nd premise in syllogism, while the axiomatic-like definition of above Equivalence class acts as the 1st premise. So classic syllogism is actually everywhere in arithmetic and math generally, most people just don't pay attention to it since employing syllogism is such a natural mental behavior for all mildly educated people and thus becomes an instinct...

Now regarding your:

And second one is what exactly the confusion between logic and epistemology means.

Historically logic was a branch of epistemology, just like physics broke from metaphysics due to technical reason, modern Positivism (such as the Vienna Circle) moved most formal logic to math realm under the branch called Mathematical Logic. Thus logic becomes math-like formal science which is generally regarded as certain (analytic a priori), while the remaining epistemology is generally regarded as speculative (synthetic posteriori). Under this context, Mill's argument "the major premises in every syllogism as general proposition collected by induction, those were informative, but not justified" is really nothing but claiming in most application of syllogism, the 1st premise is usually based on some empirical inductive reasoning (not certain), so Kenny correctly pointed out this issue is only an epistemic issue, but the validity of syllogistic logical form holds tightly on its own. In a word there's no formal fallacy in syllogism itself as logic, only there may be some epistemic informal fallacy when applying syllogism, but this is a rather common conclusion...