Yes, of course J.S. Mill oversold his famous "canons of induction" in his book A System of Logic (1843). No doubt Francis Herbert Bradley in his Principles of Logic (1883 version) had some right to play the curmudgeon in scorning the canons as worthless insofar as logical proofs go.
A useful Wikipedia exposition of the canons:
https://en.wikipedia.org/wiki/Mill%27s_Methods
Here are the canons (from Bradley):
A useful Wikipedia exposition of the canons:
https://en.wikipedia.org/wiki/Mill%27s_Methods
Here are the canons (from Bradley):
1. If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree is, is the cause (or effect) of the given phenomenon.
2. If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every instance in common but one, that one occurring only in the former: the circumstance in which alone the two instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon.
3. If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common but the absence of that circumstance; the circumstance in which alone the two sets of instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon.
4. Subduct from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents.
5. Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation.
That Mill did not see these canons as hard and fast rules is even noted by Bradley who pointed out that Mill had observed that Method 1, in particular, along with other methods, had on occasion given rise to false results.
Still, Bradley claims that scientific men such as Rudolph Hermann Lotze, Christoph von Sigwart, William Whewell and William Stanley Jevons "had taken a view of the process of scientific discovery that was not favorable" to the canons.
No doubt not only Mill, but Bradley, would have benefited from the advances of modern statistics -- in particular the methods of quantitative correlation. Further, a solid course in the logic of set theory would have been enlightening to these gentlemen. But these developments were yet some years off.
On the other hand, though Mill may have been imprudent in his zeal for his canons, I think it is quite obvious that any empirical approach to discovery is subject to the criticism that an airtight logical proof is impossible. I expect that Bradley would have been familiar with Hume.
The "inductive logic" of Mill and his followers was fundamentally flawed, Bradley says. Because Mill's canons "presuppose universal truths, therefore they are not the only way of proving them. But if they are the only way of proving them, then every universal truth is unproved."
Again, any reader of Hume already is aware of this issue. (Further, one might question Bradley's reliance on universals [which came to replace Plato's Forms]. Bertrand Russell was one philosopher who questioned whether universals can be said to exist.)
Bradley, while generally praising Jevons, devoted a chapter to errors he had uncovered in Jevons' Principles of Science (1874), which outlined Jevons' equational logic, a precursor of modern symbolic logic that supplanted Booles' pioneering work. Bradley objects to Jevons' use of the sign "=" to indicate equality. Rather, says Bradley, the sign indicates identity. But in that case, if two propositions are identical, then to say that "A = B" is to say that "A = A" is to say nothing. (Russell later addressed this point; see posts on this blog.)
Bradley goes on to caution that, though Jevons' "logical machine" (a "logic piano" he invented) could calculate and do a form of reasoning, it fell short of what the human mind can do.
There may have been some mathematical insecurity showing here.
Readers, notes Bradley, may wonder why his critique of Jevons didn't include a mathematical analysis. Bradley is "compelled to throw the burden of the answer on those who had charge of my education, and who failed to give me the requisite instruction." Even so, the "mathematical logician" failed to impress Bradley, considering that "so long as he fails to treat (for example) such simple arguments as 'A before B and B with C, therefore A before C,' he has no strict right to demand a hearing."
In any case, it seems likely that when Mill used the word logic he had in mind processes of thought that were somewhat larger than those entertained by the scholastic philosophers and their modern formalist descendants.
On ground that Mill's methods are not truly inductive, Bradley thumps Mill's logic as a fiasco. "And if I am told that these flaws, or most of them, are already admitted by Inductive Logicians, I will not retract the word I have used. But to satisfy the objector I will gave way so far as to write for fiasco, confessed fiasco."
Admittedly, Mill's writings in general show a not brilliant -- though very intelligent -- mind at work. Yet, I can only admire the brilliance of his analysis of "induction." Do we not have here a precise portrayal of the scientific method of investigation? He gives us rules for weeding out probable causes and for reduction of error. Though his rules are useless for absolute proof, as in a mathematical theorem, they would certainly be helpful if studied by potential jurors, judges and officers of the court.
A case adjudicated by Mill's standards would result in a verdict in which jurors usually were convinced beyond a reasonable doubt. In that sense, Mill's canons do indeed undergird what is accepted as proof.
Scientists and engineers similarly use Millsian standards -- perhaps bolstered by modern statistical methods which in fact stand upon Millsian assumptions -- and accept "proofs" that they realize are subject to revocation and revision.
So, though I am quite enjoying Bradley's quirky and bombastic book on logic (a subject he eventually dropped in favor of Hegelian metaphysics), I cannot quite accept his pedantical bruising of Mill. Allowances, I think, should be given for Mill's rather loquacious and often impressionistic style, even when he is honing in on fine points.
Bradley's Logic, for all its irascibility, is a fascinating critique of the various assumptions of 19th century logicians. He had grounds for questioning the law of the excluded middle[1] long before Ludwig Brouwer and the intuitionists. And his off-beat analyses demonstrate that there is more to the subject of logic than most of us have been taught.
Take this tidbit, for example:
Still, Bradley claims that scientific men such as Rudolph Hermann Lotze, Christoph von Sigwart, William Whewell and William Stanley Jevons "had taken a view of the process of scientific discovery that was not favorable" to the canons.
No doubt not only Mill, but Bradley, would have benefited from the advances of modern statistics -- in particular the methods of quantitative correlation. Further, a solid course in the logic of set theory would have been enlightening to these gentlemen. But these developments were yet some years off.
On the other hand, though Mill may have been imprudent in his zeal for his canons, I think it is quite obvious that any empirical approach to discovery is subject to the criticism that an airtight logical proof is impossible. I expect that Bradley would have been familiar with Hume.
The "inductive logic" of Mill and his followers was fundamentally flawed, Bradley says. Because Mill's canons "presuppose universal truths, therefore they are not the only way of proving them. But if they are the only way of proving them, then every universal truth is unproved."
Again, any reader of Hume already is aware of this issue. (Further, one might question Bradley's reliance on universals [which came to replace Plato's Forms]. Bertrand Russell was one philosopher who questioned whether universals can be said to exist.)
Bradley, while generally praising Jevons, devoted a chapter to errors he had uncovered in Jevons' Principles of Science (1874), which outlined Jevons' equational logic, a precursor of modern symbolic logic that supplanted Booles' pioneering work. Bradley objects to Jevons' use of the sign "=" to indicate equality. Rather, says Bradley, the sign indicates identity. But in that case, if two propositions are identical, then to say that "A = B" is to say that "A = A" is to say nothing. (Russell later addressed this point; see posts on this blog.)
Bradley goes on to caution that, though Jevons' "logical machine" (a "logic piano" he invented) could calculate and do a form of reasoning, it fell short of what the human mind can do.
There may have been some mathematical insecurity showing here.
Readers, notes Bradley, may wonder why his critique of Jevons didn't include a mathematical analysis. Bradley is "compelled to throw the burden of the answer on those who had charge of my education, and who failed to give me the requisite instruction." Even so, the "mathematical logician" failed to impress Bradley, considering that "so long as he fails to treat (for example) such simple arguments as 'A before B and B with C, therefore A before C,' he has no strict right to demand a hearing."
In any case, it seems likely that when Mill used the word logic he had in mind processes of thought that were somewhat larger than those entertained by the scholastic philosophers and their modern formalist descendants.
On ground that Mill's methods are not truly inductive, Bradley thumps Mill's logic as a fiasco. "And if I am told that these flaws, or most of them, are already admitted by Inductive Logicians, I will not retract the word I have used. But to satisfy the objector I will gave way so far as to write for fiasco, confessed fiasco."
Admittedly, Mill's writings in general show a not brilliant -- though very intelligent -- mind at work. Yet, I can only admire the brilliance of his analysis of "induction." Do we not have here a precise portrayal of the scientific method of investigation? He gives us rules for weeding out probable causes and for reduction of error. Though his rules are useless for absolute proof, as in a mathematical theorem, they would certainly be helpful if studied by potential jurors, judges and officers of the court.
A case adjudicated by Mill's standards would result in a verdict in which jurors usually were convinced beyond a reasonable doubt. In that sense, Mill's canons do indeed undergird what is accepted as proof.
Scientists and engineers similarly use Millsian standards -- perhaps bolstered by modern statistical methods which in fact stand upon Millsian assumptions -- and accept "proofs" that they realize are subject to revocation and revision.
So, though I am quite enjoying Bradley's quirky and bombastic book on logic (a subject he eventually dropped in favor of Hegelian metaphysics), I cannot quite accept his pedantical bruising of Mill. Allowances, I think, should be given for Mill's rather loquacious and often impressionistic style, even when he is honing in on fine points.
Bradley's Logic, for all its irascibility, is a fascinating critique of the various assumptions of 19th century logicians. He had grounds for questioning the law of the excluded middle[1] long before Ludwig Brouwer and the intuitionists. And his off-beat analyses demonstrate that there is more to the subject of logic than most of us have been taught.
Take this tidbit, for example:
... if we refuse to isolate a relation within [a conceptual] whole, if we prefer to treat the entire compound synthesis as the conclusion we want, are we logically wrong? Is there any law which orders us to eliminate, and, where we cannot eliminate, forbids us to argue?...
... for the conclusion is not always a new relation of the extremes; it may be merely the relation of the whole which does not permit the ideal separation of a new relation. And, having gone so far, we are led to go farther. If, the synthesis being made, we do not always go on to get from that the fresh relation, if we sometimes rest in the whole we have constructed, why not sometimes again do something else? Why not try a new exit? There are other things in the world besides relations; we all know there are qualities, and a whole put together may surely, if not always at least sometimes, develop new qualities. If then by construction we can get to a quality, and not to a relation, once more we shall have passed from the limit of our formula.
Earlier in his book, Bradley had taken a dim view of the whole notion of relation as it applies to logic.
Another interesting thought: Does comparison count as a form of inference? Bradley asks. The suggestion that whenever we compare, we are reasoning runs counter to our established ideas. But how can we repulse it? he adds. "We start from data, we subject these data to an ideal process, and we get a new truth about these data."
We are liable to compare ABC and DBF and observe that they are alike in B, notes Bradley. "No doubt we may question the validity of this inference, but I do not see how we can deny its existence."
In modern parlance, we would say X = {A,B,C} and Y = {D,B,F}. Hence X ∩ Y = {B}. One might even contend that the set theoretic mode helps us pin down better what we mean by likeness.
I suppose that when logic is restricted to meaning-free formulas, many of Bradley's insights and concerns are bypassed. That is to say, for example, if P is held true and Q is held true, the formula "P implies Q" is deemed to be true.
But, one is uncomfortable with the proposition, "If Socrates is a man, then Mars is a planet." We demand more. The proposition "If Socrates is a man and all men are mortal, then Socrates is mortal" is usually accepted. The Socrates syllogism of course dovetails nicely with naive set theory, as in
(all x ∈ M)(x has property of mortality) & xo ∈ M. ∴ xo has the property of mortality
(where xo stands for Socrates).
By accepting parts of naive set theory as axiomatic, we avoid Russell's troubled efforts to define sets in terms of propositions.
In this regard, some logicians make a distinction between adjunctive operations and connective operations. If adjunctive reasoning is used, a false proposition implies every proposition and a true proposition is implied by every proposition. Connective reasoning is more common in routine discourse. The distinction works out to mean that a truth table may be read in both directions in the adjunctive case but in only one direction in the connective case.
The adjunctive camp has been warring against the connective camp since antiquity, according to Hans Reichenbach, U1 who is comfortable with both approaches. In any case, those who believe that "implication" should mean formal derivability, as in the Socrates set theoretic formulation, would favor the connective interpretation. (Reichenbach notes that he prefers the term "adjunctive implication" to Russell's "material implication.")
Returning to the issue of syllogisms, Bradley insists that the concept of major and minor premise is delusional. Though he may have a point as to how these terms were generally used, it seems to me that in a set theoretic context, we have the set which contains the members. A particular member then might be seen as equivalent to a minor premise. Of course, something I have not shown is my consideration that the various syllogisms (and their symbolic equivalents) yield Euclidean-style proofs when we can establish set membership (even though we don't generally bother).
Bradley, I assume, wrote his Logic in order to pave the way for his coming foray into metaphysics, Appearance and Reality (1893). His entire Logic is an attack on the assumptions of "popular" logic and reasoning. I will agree that he raises many interesting points, though I would say that at least some have been effectively answered by recent developments, such as information theory.
1. Sometime in the 1930s (I'll have to track down the precise reference at some point), Bertrand Russell pointed out that the assumption that a proposition was either true or false was based on a metaphysical notion that the cosmos is completely describable with facts (propositions with a true or false value). But this notion is assuredly unprovable. In the thirties there was no way to know whether there were any mountains on the dark side of the moon. It was presumed that the assertion or its negation must be true. Yet, one could as well have said the proposition was undecidable and hence neither true nor false.
It is noteworthy that experiments in quantum mechanics demonstrate that propositions such as "Schroedinger's poor cat has expired" are undecidable until a quantum event has been recorded. And, it is not altogether absurd to surmise that the lunar dark side's mountainous terrain was in superposition with a flat surface prior to the dark side's observation via spaceborne cameras three decades after Russell's comment.
U1. Elements of Symbolic Logic by Hans Reichenbach (Macmillan 1947). Chapter II, Section 7.
Another interesting thought: Does comparison count as a form of inference? Bradley asks. The suggestion that whenever we compare, we are reasoning runs counter to our established ideas. But how can we repulse it? he adds. "We start from data, we subject these data to an ideal process, and we get a new truth about these data."
We are liable to compare ABC and DBF and observe that they are alike in B, notes Bradley. "No doubt we may question the validity of this inference, but I do not see how we can deny its existence."
In modern parlance, we would say X = {A,B,C} and Y = {D,B,F}. Hence X ∩ Y = {B}. One might even contend that the set theoretic mode helps us pin down better what we mean by likeness.
I suppose that when logic is restricted to meaning-free formulas, many of Bradley's insights and concerns are bypassed. That is to say, for example, if P is held true and Q is held true, the formula "P implies Q" is deemed to be true.
But, one is uncomfortable with the proposition, "If Socrates is a man, then Mars is a planet." We demand more. The proposition "If Socrates is a man and all men are mortal, then Socrates is mortal" is usually accepted. The Socrates syllogism of course dovetails nicely with naive set theory, as in
(all x ∈ M)(x has property of mortality) & xo ∈ M. ∴ xo has the property of mortality
(where xo stands for Socrates).
By accepting parts of naive set theory as axiomatic, we avoid Russell's troubled efforts to define sets in terms of propositions.
In this regard, some logicians make a distinction between adjunctive operations and connective operations. If adjunctive reasoning is used, a false proposition implies every proposition and a true proposition is implied by every proposition. Connective reasoning is more common in routine discourse. The distinction works out to mean that a truth table may be read in both directions in the adjunctive case but in only one direction in the connective case.
The adjunctive camp has been warring against the connective camp since antiquity, according to Hans Reichenbach, U1 who is comfortable with both approaches. In any case, those who believe that "implication" should mean formal derivability, as in the Socrates set theoretic formulation, would favor the connective interpretation. (Reichenbach notes that he prefers the term "adjunctive implication" to Russell's "material implication.")
Returning to the issue of syllogisms, Bradley insists that the concept of major and minor premise is delusional. Though he may have a point as to how these terms were generally used, it seems to me that in a set theoretic context, we have the set which contains the members. A particular member then might be seen as equivalent to a minor premise. Of course, something I have not shown is my consideration that the various syllogisms (and their symbolic equivalents) yield Euclidean-style proofs when we can establish set membership (even though we don't generally bother).
Bradley, I assume, wrote his Logic in order to pave the way for his coming foray into metaphysics, Appearance and Reality (1893). His entire Logic is an attack on the assumptions of "popular" logic and reasoning. I will agree that he raises many interesting points, though I would say that at least some have been effectively answered by recent developments, such as information theory.
1. Sometime in the 1930s (I'll have to track down the precise reference at some point), Bertrand Russell pointed out that the assumption that a proposition was either true or false was based on a metaphysical notion that the cosmos is completely describable with facts (propositions with a true or false value). But this notion is assuredly unprovable. In the thirties there was no way to know whether there were any mountains on the dark side of the moon. It was presumed that the assertion or its negation must be true. Yet, one could as well have said the proposition was undecidable and hence neither true nor false.
It is noteworthy that experiments in quantum mechanics demonstrate that propositions such as "Schroedinger's poor cat has expired" are undecidable until a quantum event has been recorded. And, it is not altogether absurd to surmise that the lunar dark side's mountainous terrain was in superposition with a flat surface prior to the dark side's observation via spaceborne cameras three decades after Russell's comment.
U1. Elements of Symbolic Logic by Hans Reichenbach (Macmillan 1947). Chapter II, Section 7.
