# A Challenge to Frege's predicate logic

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Challenge: solve this logic puzzle using only the predicate calculus.

Here are the premises -

1. The only animals in this house are cats

2. Every animal that loves to gaze at the moon is suitable for a pet

3. Animals that I detest are animals that I avoid

4. Only animals that prowl at night are carnivorous

5. No cat fails to catch mice

6. Only the animals in this house ever take to me

7. Kangeroos are not suitable for pets

8. None but carnivores kill mice

9. I detest animals that do not take to me

10. Animals that prowl at night always love to gaze at the moon

What is the conclusion?

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Davy,

I don't mind getting the dunce award. Conclusion: I love cats.

?

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Without working through the details I see that all the premises are conditionals. If you don't have at least one non-compound premise to kick off the series of inferences, I don't see how you could get any but trivial conclusions - repeat the premises, string them together with AND or OR, prefix them with double negations, etc.

Corrrection: #5 and #6 will give you two non-compound statements each, so you're off an running. Get to it later.

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Having symbolized it I return to my original position. You can't formally infer much from these premises. If the conclusion you have in mind is that translation from ordinary language into symbols is, like all translation, ambiguous, you're right, but that isn't news.

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Seems like a very long-winded way of saying: "I avoid kangaroos." I used predicates and logic. Does that count? ;-). REB

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Correct, Roger. I used Sommer's cancellation technique given in Kelley's "The Art of Reasoning". It's one of Lewis Carroll's 'Sillygisms', you can find a load more in his 'Symbolic Logic'.

The point I was trying to make was that it's impossible to solve this problem, or in fact, any similar problem where it's required to deduce a conclusion from premises, using the predicate calculus. Sure, you can test for validity and consistency, but in order to do that you have to get the conclusion first, by some means other than PL. You could try guessing and then testing for validity, but that's a pretty inefficient way of going about it.

Predicate calculus may well be the best tool for exploring the foundations of mathematics (which is what it was invented for), but for philosophy, science and general reasoning, it really sucks (my opinion).

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Correct, Roger. I used Sommer's cancellation technique given in Kelley's "The Art of Reasoning". It's one of Lewis Carroll's 'Sillygisms', you can find a load more in his 'Symbolic Logic'.

The point I was trying to make was that it's impossible to solve this problem, or in fact, any similar problem where it's required to deduce a conclusion from premises, using the predicate calculus. Sure, you can test for validity and consistency, but in order to do that you have to get the conclusion first, by some means other than PL. You could try guessing and then testing for validity, but that's a pretty inefficient way of going about it.

Predicate calculus may well be the best tool for exploring the foundations of mathematics (which is what it was invented for), but for philosophy, science and general reasoning, it really sucks (my opinion).

It is well known (proved first by Goedel) that First Order Logic is undecidable. There is no finite algorithm for deciding whether a closed well formed formula (one with no free variables) is True or False (it has to be one or the other). However, if a closed well formed formula IS True, then it is provable from the Hilbert-Ackerman axioms. So First Order logic is complete. Provablew coincides with True. However if the axioms for arithmetic are added (thus permitting arithmetic induction) then the formal system is incomplete. This is the famous Goedel block-buster theorem that doomed Hilbert's program of reducing all of mathematics of a decidable formalism.

Logic not the tool of discovery. It is the tool of justification.

Ba'al Chatzaf

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Logic not the tool of discovery. It is the tool of justification.

Seems to me like that's a very narrow view of logic. What about inference? surely both are needed for a full account?

Inference is the act or process of deriving logical conclusions from premises known or assumed to be true.[1] The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.
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Seems to me like that's a very narrow view of logic. What about inference? surely both are needed for a full account?

Logic is the science or discipline of valid inference. It does not supply the basic assumptions. That one gets through induction, abduction and lucky guesses.

Ba'al Chatzaf

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Seems to me like that's a very narrow view of logic. What about inference? surely both are needed for a full account?

Logic is the science or discipline of valid inference. It does not supply the basic assumptions. That one gets through induction, abduction and lucky guesses.

Ba'al Chatzaf

What about ~inductive~ inference? What about all the Aristotelian logicians who treated induction as part of logic, not from the "enumerative" induction perspective of Hume, but as an actual logical process of deriving a generalization from premises, which must avoid certain inductive fallacies in order to be valid?

Have you read Louis Groarke's book An Aristotelian Account of Induction? (McGill-Queens University Press, 2009) I especially would like to see what you make of his third chapter: A "Deductive" Account of Induction. I'll quote from the conclusion of that chapter:

Aristotle defines an inductive argument as an uninterrupted inference that moves from particulars to a universal. He conceives of it formally as a special kind of syllogism that uses convertible or interchangeable terms to connect a necessary property to a natural kind. In propositional logic, we might conceive of induction as an argument form appealing to some version of the principle of shared identity, that things of the same kind must possess the same properties. These definitions are more or less equivalent. Every instance of convertibility is one of shared identity.

On the traditional view, to reason inductively would be to move from particular to general; deductively, from general to particular. Clearly, deductive reasoning depends on induction. We need to arrive at a universal before we can reason back down to the particular case. This dependence of deductive on inductive reasoning tends to be obscured by the contemporary focus on formal logic. Formal logicians can pluck generalizations, so to speak, out of thin air. Real-world generalizations, by contrast, depend on a prior inductive inference.

It is remarkable to hear repeated again and again the pious empiricist dogma: induction is not a valid form of argument. Hume and his heirs did not refute earlier views, however; rather, they never made any serious attempt to understand them. Earlier historical authors had more logical dexterity than we imagine. Careful examination of their work shows that they could make logical sense of induction as a valid argument form. Instead of summarily rejecting the tradition, it behoves [sic] contemporary authors to spend some time trying to understand what earlier authors were about.

REB

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Induction has a problem. It is possible for an assertion be be true of a large number of instances and yet be false for the entire class from which the instances are taken. In short, inductive inference is not logically valid. True premises do not guarantee a true conclusion.

The classical example. One billion swans lately seen are white therefore all swans are white. Whoops. A black swan was spotted in Australia. Do you see the problem.

In the case of deduction with a valid deduction true premises MUST yield a true conclusion. The rules of -deductive- logic are truth value preserving.

Inductive logic lacks necessity. There is no guarantee that the next item observed will conform to the inductively generated generalization.

Ba'al Chatzaf

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Induction has a problem. It is possible for an assertion be be true of a large number of instances and yet be false for the entire class from which the instances are taken. In short, inductive inference is not logically valid. True premises do not guarantee a true conclusion.

The classical example. One billion swans lately seen are white therefore all swans are white. Whoops. A black swan was spotted in Australia. Do you see the problem.

In the case of deduction with a valid deduction true premises MUST yield a true conclusion. The rules of -deductive- logic are truth value preserving.

Inductive logic lacks necessity. There is no guarantee that the next item observed will conform to the inductively generated generalization.

Ba'al Chatzaf

Ba'al, if you're going to persist in this, you really need to read Groarke's book. I guarantee he will not only challenge your perspective, but will give you a very interesting read. He is a very clear thinker, considering the difficulty of the issue.

In the meantime, here is how he addresses the "black swan" issue. He devotes two different passages to it, which I'll quote in full. But please, no more "yes but's." I'm not interested in addressing all your objections, which would dissolve away if you read Groarke's book.

Excerpt 1:

Seen from an Aristotelian perspective, the problem with bad examples of induction is not invalidity but the falsehood of premises. Consider the ubiquitous (and admittedly tiresome) textbook example of white and black swans, used to show that inductive arguments must be invalid. This trope is often accompanied by a story. Everyone (in Europe) assumed swans are white; that is, they induced the general conclusion "all swans are white." But then black swans were discovered in Australia. So induction is unreliable.

But is this induction truly an invalid inference? As modern-day deductivists point out, natural-language arguments usually contain hidden elements. We need to fill in the blanks to understand what is going on. In the present instance, what Europeans were (allegedly) assuming seems clear. They were assuming that all swans possess the same colour. They reasoned, informally: These birds are white; these birds are swans; all swans are the same colour; therefore, all swans must be white. Note, however, that this is a valid argument. If the premises are all true, then the conclusion must be true. Of course, the premises are not all true. All swans are not the same colour. But that has no bearing on the issue of validity.

Note that the hidden premise "all swans have the same colour" is not a mere repetition of the other premises. It makes a different kind of claim. It assumes, in effect, that the term "these particular birds" and the term "swans" are, with respect to colour, convertible. This identification of the two terms is more of an assumption than an enference. (It does not require the kind of insight Aristotle associates with induction.) Still, we can try to formalize the argument in Aristotelian terms. Define our terms: S, these particular birds; P, white birds; and M, swans. The Europeans (allegedly) reasoned, "These particular birds are white birds. These particular birds are swans, convertible to all swans are (equivalent to) these particular birds. Therefore all swans must be the same colour as these birds; i.e., all swans must be white." Despite the awkward phrasing, this is a valid argument. It fits the following form. Major premise: All S is P. Minor premise: (All S is M, convertible to) all M is S. Conclusion: Therefore, all M is P. As it turns out, the subject and middle terms are not convertible. At least when it comes to colour, the nature of these individuals birds is not interchangeable with the nature of all swans. Whiteness is not a necessary property of swans. The argument goes astray then, because the hidden premise about convertibility is false, not because of something inherently wrong with the logical form.

The sceptic may object that we can never know whether the claim about convertibility is true. But even if we cannot know whether these white swans are interchangeable with other swans (in terms of colour), this would still be a valid argument. The argument posits convertibility. It assumes that these white swans are interchangeable or representative (in the relevant sense) with all other swans. If we can never know whether this is true, we will never know if the premises in the argument are true. But this does not attract from the validity of the argument. The argument only tells us that the conclusion must follow, if convertibility holds. Whether this is, in fact, the case is another issue.

Excerpt 2:

The ubiquitous counter-example of black swans swims through modern textbooks. We all know the refrain: gullible people once thought that whiteness was a necessary property of "swanness." Then they discovered that there are black swans in Australia. So this proves that induction is not reliable. Or is it? The test case deserves a second look.

As it turns out, the commonsense intuition that familiar white swans are a natural kind is basically correct. Nothing about this cognitive leap should make us doubt induction. Biologists, to this day, distinguish between diverse species of swan, largely on the basis of the colour of their plumage. There are, as it turns out, various species of swans. Some are pure white (the mute swan, the trumpeter swan, the whooper swan, the whistling swan, etc.), some white and black (the South American black-necked swan), the coscoroba swan (with black wing tips) and some almost entirely black (the Australian black swan, with white flight feathers). When people ordinarily declare that "all swans are white," they are not making a rigorous scientific claim. What they mean presumably is that the kind of bird we call a "swan" (most likely, the mute swan or the polish mute swan) is white. And they are right. Indeed, that kind of bird is white. We might be surprised to learn of the existence of black swans in Australia, but outside of Australia people are not ordinarily talking about those kinds of birds; they are talking about the birds they know, about the birds they refer to when they use the term "swan."

The black-swan example seems more a rhetorical trope than anything else. Plain parlance is too loose to stand up to precise scrutiny. On being told that there are black swans in Australia, we would, in all likelihood, ordinarily conclude that they must be a different species of bird. And we would be right. Black swans are a different species of bird. They do not provide a counter-example to the carefully worded inductive claim, "the species of swans we have here in Europe are white." We could move rigorously from the particular claim, "this Polish mute swan and that Polish mute swan are white," to the universal generalization that "all mute swans are white." This would be sound inductive reasoning.

White plumage is a necessary feature of European swans (in fact, of swans in the Northern Hemisphere), but it is not a necessary feature of the genus swan. Are people who claim that all swans are white referring to the species or to the genus? We cannot really know, but it seems more sensible to suggest that they are referring to the species. They are referring to "the kind" of bird they know. They are not claiming anything about unborn [i think he mean to say: unknown...reb] birds living in habitats half a world away. If they are, they are guilty of lazy thinking. But this is to take an uncharitable view of what is actually happening.

Aristotle does not claim that human beings never make mistakes. People are often, for example, inattentive, but that should not destroy confidence in inductive science. Whatever philosophical vocabulary we settle into -- notions such as genus, species, naure, essence, necessary or accidental property -- will be indispensable tools for making sense of the world...

I don't know what the rest of you reading this think of it, but to me, it is correct, awesomely well stated, and something Objectivists should embrace whole-heartedly, whether or not dyed-in-the-wool empiricists or modern logicians see its merits.

REB

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Nice post Roger.

I never had the problem that a lot of folks have about inference, induction and the other form of instant argument, or insight, analogical proofs.

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Induction is how we get from a finite set of particulars to an open ended general (universally quantified) proposition.

Abduction is how we get from effects to possible causes.

Deduction is how we get from premises to conclusions in such a way that a true premise must necessarily yield a true conclusion.

Deduction is driven by necessity. Induction and abduction are not. Which is why we have erroneous inductions (the road of science is littered with the corpses of dead theories - phlogiston, aether, caloric, vital essence....).

If B is validly deduced from A then it is impossible for A to be true and B not to be true.

Whereas if generalization G is inferred from a corpus of fact F then it is possible that a future discovered fact f might falsify G. This possibility does not exist with deduction. All inductions must be taken as provisional with the possibility that a fact discovered in the future could falsify the induction.

Inductions could be wrong and abductions are not absolute. Example: Newton abduced from observing celestial bodies and falling objects that there is a force that masses exert on each other. That hypothetical force he took to be gravitation. Einstein abduced that apparent gravitational force is the result of curvature of the space-time manifold produced by mass and or energy. For Newton, gravitation was force. For Einstein it was curvature, the geometry of the manifold. Einstein's theory is better supported by fact than is Newton's.

Ba'al Chatzaf

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Which is why we have erroneous inductions (the road of science is littered with the corpses of dead theories - phlogiston, aether, caloric, vital essence....). Ba'al Chatzaf

This is very likely true, a minority made the cut. We only see the tip of the iceberg.

But do you know how many correct inductions led to true theories? (After a subsequent deductive process.)

Which would prove once and for all the significance of induction.

Can you have such insight into all induction applied by all scientists? I doubt that.

Tony

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Which is why we have erroneous inductions (the road of science is littered with the corpses of dead theories - phlogiston, aether, caloric, vital essence....). Ba'al Chatzaf

This is very likely true, a minority made the cut. We only see the tip of the iceberg.

But do you know how many correct inductions led to true theories? (After a subsequent deductive process.)

Which would prove once and for all the significance of induction.

Can you have such insight into all induction applied by all scientists? I doubt that.

Tony

No one is denying the -significance- of induction. Induction is how we go from particulars to generalities. In a way, all learning is a kind of induction. From a finite set of experiences we generalize and deal with the many situations we have not yet encountered. Without induction we could not survive., Even so, induction does not guarantee correctness. It is possible to induce from a set of facts (describable by a finite set of true assertions) to a false conclusion. We then have to go back and make a better generalization.

Induction's cousin abduction is how we come up with possible causes of what we see. Mill's method of finding correlations is a method of determining possible cause. It is one of the special forms of abduction. Even so, abduction does not guarantee correctness. The example I have, Newton abducting from falling bodies and celestial objects moving through the heavens did not lead to a correct cause of gravitation. Newton's force law does not fully account for the motion of the planets, for example. Einstein's abduction, the general theory of relativity, does a better job. Einstein finds the "causes" of gravitation in the curvature of the space-time manifold, not in forces acting at a distance.

In any case, without abduction your local automobile mechanic would not be able to find out why your engine is misfiring or why smoke is coming out of your tail-pipe.

Ba'al Chatzaf

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I really ought to say something nice about induction. I will give two examples of inductions that came out right on the mark.

Example 1: The Periodic table of the elements which was first constructed by Dmitri Mendele'ev. Medele'ev studied the 63 elements known at the time he worked. He ordered them by atomic weight and by their properties (gas, liquid, solid, metal, non-metal, color, hardness etc). He came up with a two dimensional matrix ordered along on axis by weight and the other by property. In a brilliant feat of induction he realized that there were gaps in his table and he made predictions about what the unknown elements would be like, when they were discovered. As time passed the gaps were filled, and sure as can be, he predictions about atomic weight and properties were right on the mark. Be aware that Medele'ev had no idea what the atoms themselves were like. He was working before Thompson discovered that atoms were made of smaller parts (Thompson discovered the electron). However, Medele'ev was able to guess from the external attributes, weight and various properties (see above) what the general ordering of the elements would be.

It was empirical inductive science at its best. It far exceeded guessing the color of swans, for example. It was inductive because there was no known a priori principle to account for the grouping of properties in octaves. Just a note. In the early 20th century Mosely found a way of counting the protons in the nucleus and the correct way of ordering the elements was by proton count (atomic number) rather than atomic weight. Even so, Mendele'ev was very close.

Example 2: Charles Darwin. During his 5 years aboard HMS Beagle Darwin studied living animals and plants. He found fossils of long departed extinct animals etc. He carefully measured, weight and described his findings. When he got to the Gallapigos Islands he saw a variety of birds. Alike and yet not alike. Each kind of bird seemed very well adapted to the conditions on the island on which it lived. Darwin put together his empirical findings and combined them with a principle put forth by Malthus, to wit, the fecundity of life must necessarily lead to a struggle to acquire the means of livelihood. Some kinds of animals and plants succeeded, some did. Later on he compared nature to what animal and plant breeders had been doing for hundreds of years. From this he derived the principle of Natural selection. It was a brilliant induction. Darwin had no idea what the mechanism was by which plants and animals endowed their off-spring with the characteristics that enabled them to survive in their environments. It is ironic that in the very year Darwin and Wallace presented their ideas on natural selection that Mendel published his findings about heredity in a very obscure Czeck journnal. Mendel had hit, by induction on a combinatorial principle of inheritance. Mendel's work was not widely published and it had to be re-discovered some 40 years later by a Dutch biologist. Pay attention: Darwin did not have the foggiest idea how characteristics were inherited by the offspring from the parent, yet he intuited a correct principle which could explain how animals adapted and changed to fit the environment. Darwin's empirical approach and the biochemistry of genetics were joined in the late 1930s to give rise to the modern theory of genetic inheritance which showed the machinery of inheritance. The rest, as they say, is history.

Here are two brilliant examples of induction which produced sound results. Neither were the trivial kind of enumeration and correlation that Francis Bacon loved so well.

Induction triumphed! There is no way the machinery of elements or of biological inheritance could have be found by a priori means. Aristotle on steroids or Plato at his best could not have come up with the right answers a priori.

Ba'al Chatzaf

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Induction is how we get from a finite set of particulars to an open ended general (universally quantified) proposition.

Abduction is how we get from effects to possible causes.

Deduction is how we get from premises to conclusions in such a way that a true premise must necessarily yield a true conclusion.

Deduction is driven by necessity. Induction and abduction are not.

Bob,

So how do we get to true premises before we start deducing?

Michael

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Induction is how we get from a finite set of particulars to an open ended general (universally quantified) proposition.

Abduction is how we get from effects to possible causes.

Deduction is how we get from premises to conclusions in such a way that a true premise must necessarily yield a true conclusion.

Deduction is driven by necessity. Induction and abduction are not.

Bob,

So how do we get to true premises before we start deducing?

Michael

You look at the world and see if the premise is true, if you have a premise to check. If you don't you look around and see what is true. If you like what you see state the true thing you see and you have a premise. In any case the only way to come up with true particulars to start the argument chain is to look at the world. Otherwise you use particulars that were previously proved and use them to crank out further chains leading to yet more conclusions.

The primordial statements, those that are not conclusions of prior arguments are gotten the old fashioned way; looking, measuring, comparing.

Reality cannot be deduced a priori. The general statements that are tautologically true tell us nothing specific about the world. The purpose of such general statements is to trap illogical conclusions or assumptions. For example if you start off with the premise that it is both raining and not raining on your street at the same time, the violation of the general law of non-contradiction tells you that you have a bogus assertion. If you make an assumption and it leads by inference to an outright contradiction (guaranteed to be false) then your assumption is false.

To get anywhere one needs both induction or empirical methodology AND deduction. Induction is the engine of discovery. Deduction is the engine of justification.

Ba'al Chatzaf

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Ok but sticking with deductive logic, it may be true that any conclusions derivable from a set of premises are implied by (or 'contained' in) them, so in that sense the conclusions are necessary, but it doesn't mean that the conclusions are necessarily obvious. If that were the case, they wouldn't be any need for any kind of deductive machinery at all. It isn't at all obvious that, for example, Pythagoras' theorem (or any other theorem of geometry) is merely a long winded way of stating the axioms of Euclid! The discovery of new theorems in mathematics may involve induction, but much of the reasoning is purely deductive, and no-one would deny that new theorems are discovered, not merely justified after they've been found.

'Forward chaining', as used in the 'traditional' Aristotelian logic (and Boolean algebra) is not possible in predicate logic because it is unguided and (as Ba'al pointed out) not guaranteed to terminate. However, in many (maybe most) applications, a mathematical or logical problem is not formulated as a theorem to be proved. Consequents are not given for verification; the task is to determine consequents.

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Ok but sticking with deductive logic, it may be true that any conclusions derivable from a set of premises are implied by (or 'contained' in) them, so in that sense the conclusions are necessary, but it doesn't mean that the conclusions are necessarily obvious. If that were the case, they wouldn't be any need for any kind of deductive machinery at all. It isn't at all obvious that, for example, Pythagoras' theorem (or any other theorem of geometry) is merely a long winded way of stating the axioms of Euclid! The discovery of new theorems in mathematics may involve induction, but much of the reasoning is purely deductive, and no-one would deny that new theorems are discovered, not merely justified after they've been found.

'Forward chaining', as used in the 'traditional' Aristotelian logic (and Boolean algebra) is not possible in predicate logic because it is unguided and (as Ba'al pointed out) not guaranteed to terminate. However, in many (maybe most) applications, a mathematical or logical problem is not formulated as a theorem to be proved. Consequents are not given for verification; the task is to determine consequences

Most important theorems are -discovered- by a combination of intuition and fiddling with already know results. Finding the proofs involves another kind of mental work. Part of it is working backwards from the desired result (the theorem) toward the basic postulates of the system. The key question is: what would imply this theorem is true?

While it is true that the Pythagorean theorem is implicit in the postulates teasing out the proof involves proving dozens of other theorems and lemmas. The right triangle theorem does not tumble out trivially from the postulates. Look at the numbering of the theorems. The right triangle theorem is number 47 in Book I of Euclid. Dozens of theorems pertaining to congruences had to be proven first.

Knowing what to prove is an art. An infinite number of theorems follow from the postulates but very few of them are important, meaningful or interesting. Knowin what to prove (and ultimately how to prove it) is as much an art as it is a technical discipline.

Ba'al Chatzaf

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• 3 years later...

Induction has a problem. It is possible for an assertion be be true of a large number of instances and yet be false for the entire class from which the instances are taken. In short, inductive inference is not logically valid. True premises do not guarantee a true conclusion.

The classical example. One billion swans lately seen are white therefore all swans are white. Whoops. A black swan was spotted in Australia. Do you see the problem.

In the case of deduction with a valid deduction true premises MUST yield a true conclusion. The rules of -deductive- logic are truth value preserving.

Inductive logic lacks necessity. There is no guarantee that the next item observed will conform to the inductively generated generalization.

Ba'al Chatzaf

Ba'al, if you're going to persist in this, you really need to read Groarke's book. I guarantee he will not only challenge your perspective, but will give you a very interesting read. He is a very clear thinker, considering the difficulty of the issue.

In the meantime, here is how he addresses the "black swan" issue. He devotes two different passages to it, which I'll quote in full. But please, no more "yes but's." I'm not interested in addressing all your objections, which would dissolve away if you read Groarke's book.

Excerpt 1:

Seen from an Aristotelian perspective, the problem with bad examples of induction is not invalidity but the falsehood of premises. Consider the ubiquitous (and admittedly tiresome) textbook example of white and black swans, used to show that inductive arguments must be invalid. This trope is often accompanied by a story. Everyone (in Europe) assumed swans are white; that is, they induced the general conclusion "all swans are white." But then black swans were discovered in Australia. So induction is unreliable.

But is this induction truly an invalid inference? As modern-day deductivists point out, natural-language arguments usually contain hidden elements. We need to fill in the blanks to understand what is going on. In the present instance, what Europeans were (allegedly) assuming seems clear. They were assuming that all swans possess the same colour. They reasoned, informally: These birds are white; these birds are swans; all swans are the same colour; therefore, all swans must be white. Note, however, that this is a valid argument. If the premises are all true, then the conclusion must be true. Of course, the premises are not all true. All swans are not the same colour. But that has no bearing on the issue of validity.

Note that the hidden premise "all swans have the same colour" is not a mere repetition of the other premises. It makes a different kind of claim. It assumes, in effect, that the term "these particular birds" and the term "swans" are, with respect to colour, convertible. This identification of the two terms is more of an assumption than an enference. (It does not require the kind of insight Aristotle associates with induction.) Still, we can try to formalize the argument in Aristotelian terms. Define our terms: S, these particular birds; P, white birds; and M, swans. The Europeans (allegedly) reasoned, "These particular birds are white birds. These particular birds are swans, convertible to all swans are (equivalent to) these particular birds. Therefore all swans must be the same colour as these birds; i.e., all swans must be white." Despite the awkward phrasing, this is a valid argument. It fits the following form. Major premise: All S is P. Minor premise: (All S is M, convertible to) all M is S. Conclusion: Therefore, all M is P. As it turns out, the subject and middle terms are not convertible. At least when it comes to colour, the nature of these individuals birds is not interchangeable with the nature of all swans. Whiteness is not a necessary property of swans. The argument goes astray then, because the hidden premise about convertibility is false, not because of something inherently wrong with the logical form.

The sceptic may object that we can never know whether the claim about convertibility is true. But even if we cannot know whether these white swans are interchangeable with other swans (in terms of colour), this would still be a valid argument. The argument posits convertibility. It assumes that these white swans are interchangeable or representative (in the relevant sense) with all other swans. If we can never know whether this is true, we will never know if the premises in the argument are true. But this does not attract from the validity of the argument. The argument only tells us that the conclusion must follow, if convertibility holds. Whether this is, in fact, the case is another issue.

Excerpt 2:

The ubiquitous counter-example of black swans swims through modern textbooks. We all know the refrain: gullible people once thought that whiteness was a necessary property of "swanness." Then they discovered that there are black swans in Australia. So this proves that induction is not reliable. Or is it? The test case deserves a second look.

As it turns out, the commonsense intuition that familiar white swans are a natural kind is basically correct. Nothing about this cognitive leap should make us doubt induction. Biologists, to this day, distinguish between diverse species of swan, largely on the basis of the colour of their plumage. There are, as it turns out, various species of swans. Some are pure white (the mute swan, the trumpeter swan, the whooper swan, the whistling swan, etc.), some white and black (the South American black-necked swan), the coscoroba swan (with black wing tips) and some almost entirely black (the Australian black swan, with white flight feathers). When people ordinarily declare that "all swans are white," they are not making a rigorous scientific claim. What they mean presumably is that the kind of bird we call a "swan" (most likely, the mute swan or the polish mute swan) is white. And they are right. Indeed, that kind of bird is white. We might be surprised to learn of the existence of black swans in Australia, but outside of Australia people are not ordinarily talking about those kinds of birds; they are talking about the birds they know, about the birds they refer to when they use the term "swan."

The black-swan example seems more a rhetorical trope than anything else. Plain parlance is too loose to stand up to precise scrutiny. On being told that there are black swans in Australia, we would, in all likelihood, ordinarily conclude that they must be a different species of bird. And we would be right. Black swans are a different species of bird. They do not provide a counter-example to the carefully worded inductive claim, "the species of swans we have here in Europe are white." We could move rigorously from the particular claim, "this Polish mute swan and that Polish mute swan are white," to the universal generalization that "all mute swans are white." This would be sound inductive reasoning.

White plumage is a necessary feature of European swans (in fact, of swans in the Northern Hemisphere), but it is not a necessary feature of the genus swan. Are people who claim that all swans are white referring to the species or to the genus? We cannot really know, but it seems more sensible to suggest that they are referring to the species. They are referring to "the kind" of bird they know. They are not claiming anything about unborn [i think he mean to say: unknown...reb] birds living in habitats half a world away. If they are, they are guilty of lazy thinking. But this is to take an uncharitable view of what is actually happening.

Aristotle does not claim that human beings never make mistakes. People are often, for example, inattentive, but that should not destroy confidence in inductive science. Whatever philosophical vocabulary we settle into -- notions such as genus, species, naure, essence, necessary or accidental property -- will be indispensable tools for making sense of the world...

I don't know what the rest of you reading this think of it, but to me, it is correct, awesomely well stated, and something Objectivists should embrace whole-heartedly, whether or not dyed-in-the-wool empiricists or modern logicians see its merits.

REB

(Hope no-one minds me reviving this old thread...)

Good quotes, though "Understanding Objectivism" contains a more intuitive (to my eyes) answer to the black swan objection. The answer is simply that we have to make inductions within the context of the rest of our knowledge about the world.

"All swans we have seen are white, therefore all swans are white" is not a valid induction if you keep in mind that colour of feathers is not normally an essential feature of animal species. Many species of animal come in different breeds with different colours of skin, fur, or feathers, so 18th century Europeans could have --and should have -- surmised that different-coloured swans might live on other continents.

A valid induction would be "all swans we have seen are mortal, therefore all swans are mortal". 17th century Europeans observed that death was universal to all animals, so "all swans are mortal" would have been a valid conclusion. Furthermore, aging is observed universally, making it clearer why death is universal (the induction may not have been valid if animals were observed to remain in perpetual youth then collapse suddenly - one may surmise that some curable disease caused death).

Our modern knowledge of the universal principle of entropy further confirms "all swans are mortal" as a valid induction.

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• 2 months later...

Ok but sticking with deductive logic, it may be true that any conclusions derivable from a set of premises are implied by (or 'contained' in) them, so in that sense the conclusions are necessary, but it doesn't mean that the conclusions are necessarily obvious. If that were the case, they wouldn't be any need for any kind of deductive machinery at all. It isn't at all obvious that, for example, Pythagoras' theorem (or any other theorem of geometry) is merely a long winded way of stating the axioms of Euclid! The discovery of new theorems in mathematics may involve induction, but much of the reasoning is purely deductive, and no-one would deny that new theorems are discovered, not merely justified after they've been found.

'Forward chaining', as used in the 'traditional' Aristotelian logic (and Boolean algebra) is not possible in predicate logic because it is unguided and (as Ba'al pointed out) not guaranteed to terminate. However, in many (maybe most) applications, a mathematical or logical problem is not formulated as a theorem to be proved. Consequents are not given for verification; the task is to determine consequents.

Coming with something that might be proved is the creative act in mathematics. The bright idea leaps out of the mathematicians head like Athena from the brow of Zeus. All important advances in mathematics have been driven by artistic creation, not by grinding out propositions mechanically.

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Ok but sticking with deductive logic, it may be true that any conclusions derivable from a set of premises are implied by (or 'contained' in) them, so in that sense the conclusions are necessary, but it doesn't mean that the conclusions are necessarily obvious. If that were the case, they wouldn't be any need for any kind of deductive machinery at all. It isn't at all obvious that, for example, Pythagoras' theorem (or any other theorem of geometry) is merely a long winded way of stating the axioms of Euclid! The discovery of new theorems in mathematics may involve induction, but much of the reasoning is purely deductive, and no-one would deny that new theorems are discovered, not merely justified after they've been found.

'Forward chaining', as used in the 'traditional' Aristotelian logic (and Boolean algebra) is not possible in predicate logic because it is unguided and (as Ba'al pointed out) not guaranteed to terminate. However, in many (maybe most) applications, a mathematical or logical problem is not formulated as a theorem to be proved. Consequents are not given for verification; the task is to determine consequents.

Coming with something that might be proved is the creative act in mathematics. The bright idea leaps out of the mathematicians head like Athena from the brow of Zeus. All important advances in mathematics have been driven by artistic creation, not by grinding out propositions mechanically.

Intuition is not outside the realm of Logic, of deduction or induction. What you're describing is a psycho-epistemological process, of the conscious mind tasking the subconscious, and later at the "moment of intuition", the subconscious is serving up the conclusion to the conscious mind after having made the appropriate connections.