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Is logic "purely" formal?

According to philosopher David Stove, logic is not formal in the sense that it places no restriction on the value a variable can take, and gives these examples using contraposition:

The "formal" statement of contraposition is 'if p then q' => 'if not-q then not-p'

e.g. let p = 'there is fire' and q = 'there is oxygen'

so,

if p then q = 'if there is fire then there is oxygen'

and applying contraposition,

if not-q then not-p = 'if there is no oxygen then there is no fire'

So far so good, but here is an argument which fails:

let p = 'baby cries' and q = 'we beat him'

so if p then q = 'if baby cries then we beat him'

the contrapositive is:

if not-q then not-p = 'if we do not beat baby then he does not cry'

which is false. Here's the article I got this from : http://wmbriggs.com/post/447/

Comments?

Re: Is logic "purely" formal?

Hi Smith.

David Stove’s contention that “almost every logical truth which anyone knows, or could know, is either not purely formal, or is singular or of low generality” is itself a claim to formality and generality, and therefore self-contradictory and surely false – for a start. Behind that is a mass of misrepresentations of logic and fake examples.

Let us consider the example, given by Bill Briggs: “in which formal transposition fails, let p = “Baby cries” and q = “we beat him”, thus “If p then q” = “If Baby cries then we beat him”. But then by transposition, not-q = “We do not beat Baby”, not-p = “he does not cry”, thus “If not-q then not-p” = “If we do not beat Baby then he does not cry,” which is obviously false.”

Well, the conclusion is false here because the premise is false, not because the inference process is false. “If Baby cries then we beat him”? 1) Is this a generality? Who is “we”, the whole world or a certain family or some families? 2) Is this a real if-then statement? It cannot be, because “we beat him” is a volitional act, so logically it cannot be an inevitable consequence of “baby cries” but at best a probable consequence. That is, at best, “If Baby cries then we are likely to beat him”. So, this is not a true example of a formal and general if-then statement, nor therefore a correct example of if-then contraposition. The example given is, thus, fallacious nonsense.

Furthermore, the contraposite there presented is misleading. The real contraposite of “If Baby cries then we beat him” is not “If we do not beat Baby then he does not cry,” but “If we did not beat Baby it can be assumed that he did not cry,” i.e. “if we did not beat him it is because he did not cry.” In cases like this, when dealing with human actions and reactions, I would recommend use of the form “when-then” rather than “if-then”, i.e. with natural (de re) conditionals rather than logical (de dicta) conditionals (see my book Future Logic (part 4) for a study of de re conditioning: http://www.thelogician.net/FUTURE-LOGIC/De-Re-Conditioning-IV.htm.

It is fashionable to attack logic, but such attacks are all illogical bunk.

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Re: Is logic "purely" formal?

Hi Avi, thanks. I had a feeling there was something fishy about it, but couldn't put my finger on where the error was.

Avi Sion
Hi Smith.
Furthermore, the contraposite there presented is misleading. The real contraposite of “If Baby cries then we beat him” is not “If we do not beat Baby then he does not cry,” but “If we did not beat Baby it can be assumed that he did not cry,” i.e. “if we did not beat him it is because he did not cry.”

Right. In this case the 'if then' is being used in the sense of causality, so the proposition could be expressed as 'baby crying causes us to beat him' in which case the contrapositive would be 'not beating baby "causes" him to not cry'. It doesn't mean that he may not be crying for other reasons, only insofar as 'we are not beating him'.

To be fair to Stove, I don't think he was attacking logic, at least, I assume he wasn't if he's written a book called 'The Rationality of Induction' (although I haven't read it). But then I wonder what his point is. It seems trivial to say that there are restrictions on the value a variable can take, but that doesn't make logic any less formal, any more than the need to restrict values in some mathematical formula makes maths 'informal'.

Anyway, I'm happy to have found your site.

Re: Is logic "purely" formal?

Hi Smith, thanks.

I haven't read Stove's'The Rationality of Induction', so I do not know what he advocates in it. I ought to read it, but I have a pile of books I still want to read waiting for me.

If you're interested in Induction, I strongly recommend my book 'Hume's Problems with Induction', which should inoculate anyone against fallacious claims regarding this subject.

You can read it online here: http://www.thelogician.net/LOGICAL-and-SPIRITUAL-REFLECTIONS/Hume/Hume-Problems-with-Induction-A.htm. Or you can buy it at amazon.com or at lulu.com in softcover or e-book form.

If you leave your e-mail address at www.thelogician.net, I will add you on to my mailing list for future announcements.

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Re: Is logic "purely" formal?

You can download David Stove's 'The Rationality of Induction' from here :

https://drive.google.com/file/d/1vQL9QsQfdimuiBAdhVQfRpOhqx_twP_d/view

That example comes from the chapter 'The Myth of Formal Logic', which has other examples. Here's one :

"All men are mortal entails "If Socrates is a man then Socrates is mortal"; which in turn entails "If Socrates is a man then it is not the case that Socrates is an immortal man". So, entailment being transitive, "All men are mortal" entails "If Socrates is a man then it is not the case that Socrates is an immortal man". Conjoin that conditional with the necessarily true conditional, "If Socrates is an immortal man then Socrates is a man". Hypothetical syllogism then give you, "If Socrates is an immortal man then it is not the case that Socrates is an immortal man". Entailment being transitive, and necessarily true premises being dispensable, we now have that "All men are mortal" entails "If Socrates is an immortal man then it is not the case that Socrates is an immortal man". And so it does: there is nothing untoward so far.

But if hypothetical syllogism and universal instantiation are valid in all cases, it likewise follows from "All men are mortal" that "If Socrates is an immortal man then it is not the case that Socrates is an immortal man". And that is false. "All men are mortal" is contingent. But "If Socrates is an immortal man then it is not the case that Socrates is an immortal man", is necessarily false. And a contingent proposition cannot entail a necessary falsity.

The argument seems to be valid, but is Stove right to say that :

There are no logical forms, above a low level of generality, of which every instance is invalid: every such supposed form has valid cases. There are few or no logical forms, above a low level of generality, of which every instance is valid: nearly every such supposed form has invalid cases or paradoxical cases. The natural conclusion to draw is that formal logic is a myth, and that over validity, as well as over invalidity, forms do not rule: cases do.

Re: Is logic "purely" formal?

Joe,

First, the very idea that one can debunk “formal logic” by means of formal logic (i.e. using symbolic terms) is itself self-contradictory and absurd. Second, the idea that, by allegedly showing with a single example (or even many) that some seemingly logical argument(s) are self-contradictory, one can thereby declare the whole wide enterprise of formal logic to be false and worthless is again absurd – refuting one argument form does not imply refutation of all other (unconnected) argument forms. So, in both these instances already, one can immediately see that your critic of formal logic (David Stove) is a sophist.

Thirdly, consider Stove’s alleged refutation.
1) He says: "All men are mortal entails "If Socrates is a man then Socrates is mortal"; which in turn entails "If Socrates is a man then it is not the case that Socrates is an immortal man". So, entailment being transitive, "All men are mortal" entails "If Socrates is a man then it is not the case that Socrates is an immortal man".”
Yes, If All men are mortal and Socrates is a man, then Socrates is mortal; that is a syllogism: 1/AFF. If Socrates is mortal, then Socrates is not immortal; that is an obversion. As regards “immortal man” – this is a compound statement of the given Socrates is a man and the inferred Socrates is mortal (or not-immortal).
However, it is NOT true that All men are mortal BY ITSELF IMPLIES If Socrates is a man then Socrates is a not-immortal man. All we can say is that All men are mortal and Socrates is a man TOGETHER IMPLY that Socrates is a not-immortal man. Nesting is possible; i.e. one can say: If all men are mortal, then if Socrates is a man it follows that Socrates is a not-immortal man. But the nested hypothetical cannot be dissociated from its condition (that All men are mortals); and treated as an independent truth thereafter.
2) Thus, when Stove then claims: “Conjoin that conditional with the necessarily true conditional, "If Socrates is an immortal man then Socrates is a man". Hypothetical syllogism then gives you, "If Socrates is an immortal man, then it is not the case that Socrates is an immortal man".” – he is engaged in an illicit process, treating a conditional conditional as a categorical conditional.
If all men are mortal, then if Socrates is a man it follows that Socrates is a not-immortal man.
Plus: If Socrates is an immortal man then Socrates is a man.
Does NOT according to formal logic yield the conclusion: If Socrates is an immortal man, then Socrates is a not-immortal man.
To repeat, that is because the hypothetical syllogism is not as Stove depicts it, but a conjunction of the major premise “If all men are mortal AND Socrates is a man, then Socrates is a not-immortal man” and the minor premise “If Socrates is an immortal man, then Socrates is a man”. The middle term in the major premise (mortal AND man) is thus MORE SPECIFIC than the middle term in the minor premise (just man), and NO valid conclusion can be drawn.

Stove commits THE FALLACY OF FOUR TERMS, and thus shows his utter ignorance and stupidity, if not his utter dishonesty (“there is nothing untoward so far”!). Instead of proving that (as he puts it) “formal logic is a myth“, he demonstrates how important it is to study it truly and not get misled by sophistry.
Many evil people try hard to debunk logic, because they wish for the triumph of irrationality. This is the agenda of the Woke crowd, and of their sick predecessors since the so-called Enlightenment. It is fashionable – but all wrong.
Avi

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Re: Is logic "purely" formal?

As regards the argument that 'if baby cries, then we beat him' implies the contraposite 'if we do not beat baby, then he does not cry', which is false.
Here you should ask the question, why is the inference claimed false?
1) Baby cries – we then beat him - presumably invariably?. But this is a volitional act, and the beater may not always be present anyway when Baby cries. So, is the premise always true, for a start?
2) Is it correct to say that if Baby is not beaten, he does not cry? Surely, when baby starts crying he was not yet being beaten, which is why the beater then started beating him! If he was already being beaten, then one can understand why he cried, and also one may wonder whether the continued beating was related to the crying!
Etc. Etc. As you can see, there are many difficulties in these “if-then” statements – they are not really clear and consistent if-then statements, and so one can understand that they lead to doubt.
Truly this “philosopher” David Stove is an idiot, who is looking for ways to attack logic while lacking an understanding of logic.

(Joe, I wrote this thinking you had written the related question - but now realize it is an old post by Smith, not you. Still I leave the comment for others to see.)

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Re: Is logic "purely" formal?

Hi Avi, I take your points about trying to 'disprove' logic using logic, and of course you are right to say that using one or two examples doesn't show that there is no such thing as formal logic (although to be fair to Stove, he does acknowledge that there are many 'logical truths').

Following the argument closely, I see that you are correct. As you say, where it goes wrong is using the compound term 'immortal & man'. It seems reasonable that 'if Socrates is a man then it is not the case that Socrates is an immortal man' is implied by 'All men are mortal', but when you take the contrapositive of the conditional you end up with 'if Socrates is an immortal man then Socrates is not a man', which is contradictory.

To be honest, I don't know what to make of Stove or his 'agenda'. He's not one of the woke liberals for sure, because he was a conservative and banned from Sydney university for being politically incorrect. He wrote an essay called 'The Intellectual Capacity of Women', which didn't go down too well. He seemed to take great delight in offending as many people as possible, but could be very funny sometimes, usually at the expense of famous philosophers. I particularly like his comment on Plato and universals :

It is possible for something to be a certain way and for something else to be the same way.

So,

There are universals.

(Tumultuous applause, which lasts, despite occasional subsidences, 2,400 years.)

This is an outrageous misrepresentation, but funny nevertheless.

You can find links to some of his work here

By the way, Stove uses the term 'entails' and 'entailment' in his argument, and I've never been able to find a precise definition of what entailment actually means. It's not the same as implication, is it? Do you have a definition in one of your books?

Re: Is logic "purely" formal?

Hi Joe.
For the record, in more formal terms, Stove's error can be expressed as follows:

If Q and R, then S (= if Q, then (if R, then S)
If P, then R, "therefore" (according to Stove) If P, then S

This is of course an invalid syllogism, because the middle term is different in the major premise (Q and R) and in the minor premise (R): and the former is narrower in scope than the latter.

When I accused Stove of being Woke, I deliberately used a currently fashionable word, even knowing that it is not exactly the right word to use in this context (since it has a much wider application). My point was that this contemporary fashion is not that new. Today the focus is, for instance, on creating confusion between the male and female genders and claiming that one can go from one to the other at will or that there are many gradations in between or beyond them. Not long ago, reason was assailed by means of 'deconstruction'. A few centuries back, Hume and after him Kant attacked reason is various subtle ways, denying causation or soul for instances. In ancient Greece, the Sophists did their best to confuse people. My point is that this is a clear trend. The assault on reason is not an accident. Rather it is the work of people who have a definite, deep-seated will to find fault with reason (making unreason seem logical).

Clearly, Stove's argument was an instance of this evil intent. Had he wished to find the truth of the matter, he would have thought about the validity of his argument a bit more carefully, and found out for himself its fallaciousness. Any high-school kid who has studied logic knows the fallacy of four terms. But Stove was evidently eager to stick his sword in logic's back. So, even if Stove is a political conservative, he is in relation to the art and science of logic clearly on the side of the bad guys (the anti-logic crowd).

As regards his statement about Plato and universals, this is a bit facile - and indicative of Stove's superficiality. In truth, our ability to perceive or conceive, or merely intuit, similarities and differences between concrete or abstract things is a great wonder, a marvelous mystery. Most people don't realize it. That Plato was one of the first, if not the first, to truly realize this, and to try and find some answer to how it comes about, is one of the reasons why he is rightly regarded since antiquity as one of the greatest philosophers that ever was.

As for the use of the term entailment rather than implication, I generally avoid the former and use the latter. Some people use the former for the conclusions of arguments and the latter for straight if-then statements - but that to my mind is silly hair-splitting, intellectual snobbery. In truth, the relation of premises and conclusions is one of implication - there is no difference which justifies using a distinct term for it.

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Re: Is logic "purely" formal?

As for the use of the term entailment rather than implication, I generally avoid the former and use the latter. Some people use the former for the conclusions of arguments and the latter for straight if-then statements - but that to my mind is silly hair-splitting, intellectual snobbery. In truth, the relation of premises and conclusions is one of implication - there is no difference which justifies using a distinct term for it.

Thanks. I wasn't sure that entailment meant the same thing as implication, especially as logicians (some, anyway) use the double turnstyle symbol for it. And according to Wikipedia, it means different things in different contexts. I don't know why modern logicians are so obsessed with symbols; logic is not mathematics, and it just puts people off who may have otherwise been interested in learning the subject. Not only are symbols not really necessary most of the time, but there seems to be no set of symbols which are used consistently.

As regards his statement about Plato and universals, this is a bit facile - and indicative of Stove's superficiality. In truth, our ability to perceive or conceive, or merely intuit, similarities and differences between concrete or abstract things is a great wonder, a marvelous mystery. Most people don't realize it. That Plato was one of the first, if not the first, to truly realize this, and to try and find some answer to how it comes about, is one of the reasons why he is rightly regarded since antiquity as one of the greatest philosophers that ever was.

Agreed. Abstraction is an amazing ability and is the one thing which sets us apart from other animals. Looking at Stove's articles and books, he is a very negative philosopher who seems to be content to criticize others; as far as I can see he hasn't produced any original work.

The other example in 'The Myth of Formal Logic' is a so-called paradox, similar to the liar paradox:

... (the) syllogistic rule: '"All F are G and x is F" entails "x is G", for all x, all F, all G.'
That is a purely formal judgement of validity with as good a claim on our belief as any.
Here I offer, not a counterexample, but a counterexample-or-paradox, the paradox being an obvious relative of the Liar. ...

(d)
All arguments with true premisses and false conclusion are invalid.
(d) is an argument with true premisses and false conclusion.
______________________________
(d) is invalid.

If (d) is invalid then our syllogistic rule is false straight off. If (d) is valid, then its conclusion is false, and so one of its premisses must be false. Then the problem is to find the false premiss. The first premiss is true. So is the second part (we are supposing) of the second premiss. The falsity must therefore be in the first part of the second premiss: but where? Indeed, since the conclusion, if false, is necessarily false, and since the first premiss is necessarily true,and the second part of the second premiss is necessarily true: please find the necessary falsity which is asserted by the first part of the second premiss, (the part which says both premisses are true).

This is a trick, because (d) refers to the whole argument, but the second premise is a PART of the argument which refers to the WHOLE argument, which is not yet complete. I've seen this kind of 'paradox' before, and it isn't really a true paradox. Arguments which refer to themselves are just sophistry.

Re: Is logic "purely" formal?

Hi Joe,

Regarding the Stove argument you quote, I read the symbol (d) as referring to the major premise only, thus getting:

“If A and B are true, and C is claimed to follow from them jointly but is false, then C does not follow from A and B.” This principle is a universally true statement for all deductive arguments, according to formal logic. This is therefore a true major premise (called (d) by Stove).
“The preceding principle (i.e. (d)) is false” is therefore surely FALSE by formal logic. This is therefore a false minor premise by Stove.
It follows that we cannot even suppose the “The preceding principle (i.e. (d)) is false” to be true, and no paradox or logical problem of any sort arises. Stove’s ‘conclusion’ is thus worthless.

(Note in passing that Stove uses the terms valid/invalid instead of true/false, showing that the does not even clearly know the distinction between these concepts.)

But reading your analysis, it is evident that you probably have pinned down more precisely what he was up to - sowing confusion between the whole argument and the mere major premise thereof, making his minor premise a sort of self-reference. I suspect that Stove is not consciously engaging in sophistry, but is merely too unintelligent to see the errors he commits.

I agree with your assessment of him: "Looking at Stove's articles and books, he is a very negative philosopher who seems to be content to criticize others; as far as I can see he hasn't produced any original work". I wonder if he deserves to be called a "philosopher" - he is obviously a hater rather than a lover of wisdom.

As for the word entailment, I admit that I use it occasionally as a weaker term than implication. Perhaps we should say entailment when dealing with inductive (probabilistic) 'implication', so as to distinguish this from deductive (100% firm) implication.

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