(This post is only a teaser, a brief introduction to the late philosopher David Stove’s philosophy of logic. I do not intend that today’s article will convince anybody. I do not have time to do more.)
Everybody—especially readers of this site—has had experience with logical arguments. People obviously use logical argumentation continuously, whether or not they are aware that the science of logic has been “made formal” by mathematicians and other such creatures.
Their ignorance of this formalization obviously—I hope it is obvious—does not mean that when these people offer a valid argument it is not made invalid because they are unaware of how to prove it valid.
It was David Stove’s contention that formal logic (to be defined in a moment) is a myth. That is to say, that all attempts to formalize logic were doomed to failure. This might sound like yet another post-modern attempt at skepticism. It is not:
My philosophy of logic is so far from being skeptical that it is if anything indecently affirmative. Not only do I believe, as I have implied, that there are logical truths, true judgments of validity or of invalidity; I believe that every normal human being is, in the extent of his knowledge of such truths, a millionaire. Only, I hold, as I have implied, that almost every logical truth which anyone knows, or could know, is either not purely formal, or is singular or of low generality. [p. 128; The Rationality of Induction]
What did Stove mean by formal?
An argument is formal “if it employs at least one individual variable, or predicate variable, or propositional variable, and places no restriction on the values that that variable can take” (emphasis mine). Stove claims that “few or no such things” can be found.
Here is an example of formality: the rule of transposition. “If p then q” entails “If not-q then not-p” for all p and for all q.
This is formal in the sense that we have the variables p and q for which we can substitute actual instances, but for which there are no restrictions. If Stove is right, then we should be able to find an example of formal transposition that fails.
First a common example that works: let p = “there is fire” and q = “there is oxygen”, then
“If p then q” == “If there is fire there is oxygen”.
And by transposition, not-q = “there is no oxygen” and not-p = “there is no fire” then
“If not-q then not-p” == “If there is no oxygen then there is no fire.”
For an example 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. (Stove credits Vic Dudman with this example.)
So we have found an instance of formal transposition that fails. Which means logic cannot be “formal” in Stove’s sense. It also means that all theorems that use transposition in their proofs will have instances in which those theorems are false if restrictions are not placed on its variables. (It’s worse, because transposition is logically equivalent to several other logical rules; we won’t go into that now.)
It is Stove’s contention that all logical forms will have an example where it goes bad, like with transposition.
Exercise: can we find counterexamples to the two most popular logical forms, modus ponens and modus tollens? I haven’t tried yet, but I rely on the way-above-average intelligence of our readership to provide some.
Modus ponens: “If p then q, p, therefore q, for all p, q”. Example: “If Socrates is a man then he is mortal, Socrates is a man, therefore he is mortal.”
Modus tollens: “If p then q, not-q, therefore not-p, for all p, q”. Example: “If Socrates is a man then he is mortal, Socrates is not mortal (he is immortal), therefore Socrates is not a man.”
Hm. I am not convinced by the examples. The negation of “baby cries”, “baby does not cry”, introduces nuances that “baby is not crying” or “bay cries not” do not. “If we are not beating baby he is not crying” seems perfectly sound to me. I think the “does” formulation introduces notions of causal direction that go beyond the original logic.
Rich,
“If we are not beating Baby he is not crying” is empirically false at the least. Which makes the argument false. I think it might help to write out the transposition with all the nuances you can imagine and see if any of the resulting statements are true.
Hi Briggs,
As an engineer, my vulgar math skills are definitely intuitionist. Thus, I challenge the Law of the Excluded Middle on the basis of having insufficient empirical support.
If the baby cries, we beat it. If we did not beat the baby, it did not cry.
The conclusion may be empiracly false, but that is only because the arguement is built on a flawed assumptiong. Of course, we don’t beat the baby every time it cries. But, if you did. Then by the fact that you had not beaten said child, that child must not have given you a reason to do so.
Stephen Hawking said that he did not like the Shroedinger cat expirement because it was cruel. I finaly understand what he was talking about.
All,
If you think you have found something wrong with the Baby example, try applying your analysis to the Fire example to see if your technique works there, too.
Again, thanks mr Briggs for finding me out such an interesting contrarian :). I’m reading his book and I like it! (goin in for chapter 2 in 3…2…1…)
in the given baby arguement, beating the baby “q” is a consequence of the baby crying “p”, not the reason the baby is crying.
“If NOT q then NOT p” still holds because if you didn’t beat the baby, then it is because the baby didn’t cry.
If the arguement is the baby cries in response to being beaten, beating the baby is “p” and the baby crying is “q”
If the object is on fire, oxygen is present
If there is no oxygen, the object is not on fire.
Genericly true.
If there is oxygen, the object is on fire.
If the object is not on fire, there is no oxygen.
The object here must be extremely combustible.
If we beat the baby, it will cry.
If the baby is not crying, is has not been beaten.
A semi-valid premise. If we assume it is true, the conclusion is follows.
If the baby crying, we will beat it.
If we did not beat the baby, is must not have been crying.
A broken premise with a warped conclusion. But, if we take the assumption as given, the logic is sound.
The Baby example doesn’t work because the wording is ambiguous. While it is true that “This is formal in the sense that we have the variables p and q for which we can substitute actual instances, but for which there are no restrictions,” we are still required to use the same p and the same q in the original proposition and in the transposition (not just the same words – the same interpretation of those words).
“let p = “Baby cries†and q = “we beat himâ€, thus
â€If p then q†== “If Baby cries then we beat himâ€.
So what does if p then q mean:
i) every time the baby cries we start beating him
ii) if the baby is crying then we are currently beating him
iii) if the baby ever cries at all it implies we are baby beating people
iv) the only reason baby every cries is because we have been beating him
You could probably come up with other plausible interpretations. But the key is sticking with one interpretation for the proposition and the transposition. If you do that, then the transposition works fine.
For the fire example, there is one strongly obvious interpretation for the original and the transpostion – the fire and the oxygen must occur at the same time. So there is no problem there either.
Your analogy is correct, but the assumption, If p then q is wrong.
“If the baby cries, we beat him” implies that we ALWAYS beat the baby if he cries. Assuming that preposterous assumption is true, if we don’t beat the baby, he hasn’t cried, is a correct conclusion.
1a) If alcohol is a poison, then drinking it will kill you.
1b) Alcohol is a poison, therefore drinking it is fatal.
2a) If alcohol is a poison, then drinking it will kill you.
2b) Alcohol is not a poison, therefore drinking it will not kill you.
And we all know of instances where the logic breaks down, in both cases.
Mike D your 2a and 2b line of argument is not correct.
“If P then Q” is not equivalent to “If not P then not Q”
I think the easiest way to really understand “If P then Q” is as “not(not Q) and P”. You cannot have “P” and the “negation of Q” both being true at the same time.
In the alcohol example that means: If “alcohol is a poison” then “drinking alcohol will kill you” is equivalent to saying that you cannot have both “Drinking alcohol will NOT kill you” and “Alcohol is a poison” true at the same time.
All,
I think we’re losing a little focus. Let’s recall that if the entire statement “If there is fire there is oxygen” is true, then the entire statement “If there is no oxygen there is no fire” must be true.
And even that transposition, the entire statement, is false, as is the premise, in certain cases. For example, a hydrogen-only star can be said to create fire without oxygen. And of course other fire-making oxidizers besides oxygen exist; flourine, for example. If we claim the wording is ambiguous, we have to find examples where it is not.
Criticisms of Stove’s conclusion that point to the initial statement being false won’t work unless we can find examples of transposition where the initial statement is always true. But recall that the since transposition is symmetric, it is only our choice that decides what the initial statement is. Thus both statements must be true. Can we find an example of transposition where this is the case? Certainly, in mathematics we can.
Nobody has yet had a hand at finding a counter example to modus ponens or modus tollens.
I also have to give stronger arguments: Stove’s chapter is full of them. Some of them are more technical and require more work on my part to type up. But I’ll get to it….
language and word play. An unfortunate word, “beatâ€
The complexity of our language allows more sensitive (in the precise sense) statements to be made.
“cases ruleâ€
This strikes me as playing with verb tenses and trying to oversimplify something which has evolved to a high level of complexity for a good reason.
When substituting or simplifying, subtlety is missed, obviously. This is like Kermit over again.
The word “beat†is often spoken incorrectly from a grammatical point of view although not in this country, so that I have no doubt that certain dictionaries have inserted a whole new rule to flatter the masses into thinking they are speaking correctly.
(he, she beats) I, they, we beat.
I have beaten; he/she has beaten; they, we have beaten.
I was beating; he/she was beating; they, we were beating.
I will beat; he/she will beat; they, we will beat.
I am beating; he/she is beating; they, we are beating.
Maybe this is why Vic Dudman chose this example.
Would the word slap or smack have made the point more or less clear? I think so,
Who would use the words “I beat†to mean that I had “beaten†or “I was beating�
“If P then Q” is logically equivalent to “If not Q then not P.”
That means that those two expressions must have the same truth value. Which means that:
when “If P then is Q” is true, “if not Q then not P” is also true,
and when “If P then Q” is false, “if not Q then not P” is also false.
So in the fire example
“If there is fire there is oxygen†is logically equivalent to “If there is no oxygen there is no fireâ€
Therefore: if “If there is fire there is oxygen” is true then “If there is no oxygen there is no fire” is true.
And
if “If there is fire there is oxygen” is false then “If there is no oxygen there is no fire” is false.
Dr Briggs has produced evidence to suggest that the statement “If there is fire there is oxygen” is false, so if we accept this evidence then we will logically deduce that “If there is no oxygen there is no fire” is also false.
It seems that the traditional rules of logic are working perfectly fine here.
Raymond Smullyan gave an example of the faulty conflation of two different meanings with this syllogism:
Nothing is better than eternal happiness.
A ham sandwich is better than nothing.
Therefore a ham sandwich is better than eternal happiness.
One can logically accept the first two premises and still reject the conclusion because the “nothing” in the first two sentences referred to different “nothings”.
For another example of different meanings, consider the two sentences,
“Fruit flies like a banana”, and
“Fruit flies like a banana”.
The first sentence is referring to entomological gastronomy, while the
second obviously is referring to horticultural aerobics.
In summary, I think the problem was in language. The P and Q in the sentence,
P implies Q, did not have the same meaning as the P and Q in the second sentence, not P implies not Q.
I’ll finish with the two sentences,
“Time flies like an arrow.”, and
“Fruit flies like a banana.”
Modus ponens: “If p then q, p, therefore q, for all p, qâ€.
Let p = I press the accelerator pedal, and let q = car moves.
I press the accelerator pedal, therefore the car moves.
Modus tollens: “If p then q, not-q, therefore not-p, for all p, qâ€.
The car doesn’t move therefore I didn’t press the accelerator pedal.
…now I’m going shopping!
Patrick,
Your are missing what Stove is saying—and restating what I have said. Stove is in no way implying the ordinary rules of logic do not work, that they are not “perfectly fine”. Recall that he has said that every person knows millions of ordinary logical truths. True statements of logic, that is; including transpositions. His only claim is the one made: it is possible to find an example where variable substitution will produce a falsity in a “logical form”, a form which, after all, is a human classification.
I have showed that even the fire example might not work, as I see you agree. My challenge was to then demonstrate a transposition which works. One that does not suffer from the same difficulties some think are in the Baby example.
Alan,
Smullyan’s right. There is nothing like a ham sandwich. But that criticism, excellent though it is, does not apply to the Baby example because the meaning of the words does not undergo any kind of shift during the transposition. Nor do the words in the fire example.
All,
It might help to recall that in mathematics, theorems frequently (always?) place restrictions on the kinds and types of variables. Strict limitations are placed on the variables. Stove is saying that such limitations also need to be present in statements of propositional and first and second order logic. Stated in this fashion, it seems a lot less skeptical.
I do have the feeling people are taking his approach as somehow “invalidating logic”, which is assuredly not his intention. In fact, the exact opposite is true. Stove was as anti-skeptical as they came.
Joy,
Those are good positive examples. Now we need some possible counter examples.
“But that criticism, excellent though it is, does not apply to the Baby example because the meaning of the words does not undergo any kind of shift during the transposition.”
If the meaning of the words isn’t changing, then the claim that the transposition is false needs some careful explaining.
—————————————————————————————-
â€If not-q then not-p†== “If we do not beat Baby then he does not cry.â€
which is obviously false.
—————————————————————————————-
No. That’s not obvious at all.
“If we are not beating Baby he is not crying†is empirically false at the least. You say this because you know that babies, in general, cry for other reasons too. This is your empiricism. In the universe of the argument, when the baby cries we are a beating him, always, so if I treat the argument as formal then it’s valid and if its premsises are true so is its conclusion. Have we actually said any more than that the baby argument is empirically implausible?
I have no trouble at all in accepting that real-world arguments are unlikely to be purely formal because real-world language is rarely tidy. I must get one of Ben Goldacre’s T-shirts: http://badscience2.spreadshirt.net/en/GB/Shop
I think the Smullyan analogy was correct here. From the statement,
“If the baby crys I beat him”, I form a picture of a baby crying, then “I” am either moving towards the baby to beat it, or already in the process of beating the baby every time it cries if p implies q is correct. As an additional factor, “I” will only beat the baby if I HEAR
it crying. If I’m at work, or outside mowing the lawn, I won’t hear the baby and I won’t beat it. So a clearer statement, more in agreement with real life, is “If the baby cries and I hear it, I head towards the baby and proceed to beat it”.
If I’m NOT beating the baby, or heading towards the baby with the intent of beating it, it’s not crying or I don’t hear it would be obviously true, and would be the correct formation of
not p implies not q.
I presume the major reason for disagreements is that people obviously don’t accept
“p implies q” in their opponents syllogisms. After reading this thread, I get the impression that opponents may agree with the premises proposed, but have different interpretations of p and q. I now suspect this may affect a significant minority of disputes.
I think Briggs is right, the discussion is getting bogged down with the baby example. I think the problem is that there is really no “universal set of propositions” with respect to which we can take logical negation. If we think of P and Q as sets, then “P is a subset of Q” is analogous to p implies q. For example, if Q=”the set of situations in which there is oxygen” and P=”the set of situations in which there is fire” then saying that P is a subset of Q is saying “if there is fire then there is oxygen.”
But in order to discuss “not P” and “not Q” in terms of sets, we would need a definition of the complement of a set. In order to do this, there has to be some “universal set” U. So then “not P” would be the set of all elements that are in U but not P. In the fire example, is there some “universal set of situations” in which we can find “not P”? Does that universal set contain situations where physical laws as we know them are violated?
It isn’t clear to me that a general universal set exists in the area of logic. U would have to contain, a priori, all logical propositions. Perhaps such a thing is impossible to define, just as it is impossible to define a “largest set” in math.
Seems to me though that Bayesians might disagree. They often talk about the likelihood, or probability of an event occuring without a well-defined meaning for the set of all possible events!
Patrick, you are absolutely correct. I should have proposed:
2b Drinking alcohol did not kill me, therefore it is not a poison.
Right? Not q implies not p
Or something like that. Too many beers. Possibly some brain cells did die.
I like my example because the set {poisons} is a fuzzy set. It is a lot fuzzier than {fire}. “One man’s meat is another man’s poison” and all like that there.
Logical syllogisms require concretely defined sets. Fuzziness causes leaks. Just ask Grothendieck, if you can find him.
P = baby giggles Q = we tickle baby
Baby giggles therefore we tickle baby
we do not tickle baby therefore baby does not giggle
is false because we all know that, “youth smiles without any reason, it is one of it’s chiefest charmsâ€
All examples that I think of are false, I can’t think of a true one and there are many. I’ve got them going round in my head like a bad pop song.
This is a wicked trick by Briggs.
I must not have all the facts in order that I can believe
I do not believe therefore I must have all the facts
My favorite statement in my syllabus is, “You may take a makeup exam only if you have a legitimate excuse.†(“There is fire only if there is oxygen.”) Note that “P only if Q” is equivalent to “If P then Q.”
“You said that if I have a legitimate excuse, then I may take a makeup exam.†(“If there is oxygen, then there is fire.â€) Nope, I didn’t say so in my syllabus.
Oh come on JH, the only ones who get confused by ordinary language are scientists, philosophers or logicians.
Raymond Smullyan published two book with chess retrograde puzzles which somehow fell in a crack between the subset of chess and puzzle fans. Here is a real beaut:
http://www.chessville.com/chessprints/2004/20041212.htm
( Manchester Guardian, 1957)
All,
We seem to be spinning wheels a bit there. So I think what we’ll do is to push on with another example. Sometime this week, I’ll start another thread which contains a different example. The more examples we have, the better we’ll be able to keep in mind the entire idea.
Harold, I cannot agree with you more! I guess one of the main points of logic is that it allows us to distinguish good reasoning from bad.
It seems easier to understand the formal logic when applied to mathematics. However, when it’s used in natural language arguments, I painfully endure a semantic disadvantage. After a while, I start to believe these arguments have personalities and egos. It would be a painstaking endeavor for me to fight their egos. ^_^
I have yet to see an example here where the transposition turns out to be false for any situation where the original arguement is not also false.
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