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!


  • Syllogism is a form of inference with very restricted forms of premises and conclusions. Most inferences in propositional logic are non-syllogistic, and multi-place predicates (like = in the example) cannot even be expressed by traditional syllogistic means. As for logic vs epistemology, Mill and Kenny are simply operating with different concepts of "logic". Kenny's modern one is much more narrow, see What are the differences between philosophies presupposing one Logic versus many logics?
    – Conifold
    Apr 12, 2021 at 5:33
  • Thanks! Reading answers below, I've got another question. Now it is certain that many of Kenny's critiques are based on modern logic which was made after Mill. If so, however, I think there is no reason for Kenny to show that some deductive arguments are informative, because it is enough for that purpose to make distinction between epistemology and logic. Rather, Kenny's answer makes informativity important for deduction's validity. Apr 12, 2021 at 5:55
  • I think Kenny's criticism largely goes past Mill because it is anachronistic, they use words differently. But he makes a double point from the pedestal of today: some of Mill's remarks about uninformativity should be split off into epistemology, but even aside from that, some deductive arguments are informative, pace Mill. It would be odd to say that Wiles's deductive proof of the Last Fermat theorem was uninformative considering that we did not know if it is true before it, for 300 years. Mill's second sin is that he restricts "deductive proof" to syllogistic.
    – Conifold
    Apr 12, 2021 at 6:17
  • @Conifold: yeah, Mill thought math was inductive. Apr 12, 2021 at 6:23
  • You said "some deductive arguments are informative, pace mill." But, according to Kenny, Mill also thought that some deductive arguments are informative, whether the information from that is justifiable is skeptical for him though. I'm, then, curious about the term "information" is used in the same way by both Kenny and Mill. How could that argument A=B, B=C, therefore A=C be informative? Apr 12, 2021 at 6:41

2 Answers 2


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.

  • Thanks for your detailed explanation! Now I understand the point of Kenny's critiques. If I got it right, Kenny wanted to point it out that Mill didn't know that there were non-syllogistic arguments, which are not only valid but also informative, because of influence of aristotelian logic. However another question arises from this explanation. Mill thought that verbal inference is not informative because it brings no new knowledge about the world. If so, that argument(‘A=B’, ‘B=C’, therefore ‘A=C’) seems non-informative to me. Then, how could it be a solution for Mill's critique? Apr 12, 2021 at 4:43
  • @ElectricEye: frankly, I don't why Kenny thinks that is a non-obvious fact (i.e. something obtained by transitivity of equality). It seems no more "deep" than from A ⊆ B and B ⊆ C infer A ⊆ C, which is actually the set equivalent/formulation of an Aristotelian syllogism, if I recall [that] correctly. Apr 12, 2021 at 5:08
  • Thanks a lot! Though the point you mentioned is not yet answered, this conversation was really helpful for me. Apr 12, 2021 at 5:21
  • @ElectricEye: actually Kenny doesn't say that the equality transitivity example is more non-obvious than a syllogism, just that it's not a syllogism "But syllogism is not the only form of inference, and there are many valid non-syllogistic arguments (e.g. arguments of the form ‘A = B’, ‘B = C’, therefore ‘A = C’) which are quite capable of conveying information." I guess the only unclear part is the strength of "conveying information". Kenny is actually critical of Mill's criticism of syllogisms. Kenny's point is that syllogisms too convey information. [continues[ Apr 12, 2021 at 5:55
  • "Even in the case of syllogism, it is possible to give an account that makes it a real inference if we interpret ‘All men are mortal’ not as saying that ‘mortal’ is a name of every member of the class of men but—in accordance with Mill’s own account of naming—as saying that there is a connection between the attributes connoted by ‘man’ and by ‘mortal’." I guess this is the sense in which a syllogism "conveys information", according to Kenny. So then a syllogism "makes the connection" e.g. between Socrates and mortality. Apr 12, 2021 at 5:57

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...

  • Thanks for your help! By the way I've got another question about Kenny's critiques. Now I understand why he pointed out that it doesn't matter whether the 1st premise of syllogism was epistemologically justifiable. But, then, why does he give an example of informative, valid, non-syllogistic arguments in order to refute Mill's logic? If informativity(whether an inference give a new knowledge about the world) of syllogism does not matter, isn't it enough for that purpose to point out epistemology/logic distinction? Apr 12, 2021 at 5:02
  • Frankly I tend to agree with u, the major issue in Mill's theory is the mixing of logic and epistemic induction premise fallacy. Actually as I've already pointed out in my above answer, from Kenny's own redundant criticism it reveals Kenny's own problem in understanding of the ubiquity of syllogism. Even application of transitivity in ur case requires classical syllogism... Apr 12, 2021 at 6:22

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