Logic is man’s method of reaching conclusions objectively by deriving them without contradiction from the facts of reality—ultimately, from the evidence provided by man’s senses. If men reject logic, then the tie between their mental processes and reality is severed; all cognitive standards are repudiated, and anything goes.
On the back of Morton Hunt’s excellent account of the discoveries of cognitive science can be found the following provocative blurb:
How logical are we? Not very. How much does it matter? Less than you think. The human mind ordinarily reasons by natural processes that are illogical—but that generally lead to correct conclusions.
That, in any case, is what cognitive scientists have discovered in their researches. Since this is obviously a touchy subject for Objectivists, it might be useful to take a glimpse at how cognitive science reached its astonishing conclusion about the limits of logic.
Cognitive scientists used some of the following syllogisms to test how well people think logically. Try them yourself and see if you can figure them out:
Syllogism A:
No Gox box when in purple socks.
Jocks is a Gox wearing purple socks.
Therefore Jocks does not now box.
Syllogism B:
Those who believe in democracy believe in free speech.
Fascists do not believe in democracy.
Therefore Fascists do not believe in free speech.
Syllogism C:
Whatever makes for full employment is socially beneficial.
Being in a state of war tends to make for full employment.
Therefore war is socially beneficial.
Syllogism D:
If it’s raining, the streets are wet.
The streets are wet.
Therefore it's raining.
Syllogism E:
Disease X is known to produce various symptoms, including A, B, and C.
This patient has symptoms A, B, and C.
Therefore this patient has disease X.
Finally, we conclude with an incomplete syllogism. See if you can figure out what logically proceeds from the following two premises:
Syllogism F:
Some of the beekeepers are artists.
None of the chemists are beekeepers.
Therefore…
The point of these examples is to demonstrate how difficult it is to think logically. Logical reasoning is not a skill that comes easy. The human mind takes it up only with great difficulty.
If you have any doubts on this score, try to figure out the syllogisms presented above. Unless you are very well versed in logic, you will likely find it hard going. I will provide the answers in my next post and explain what can be inferred from it as to the nature and function of logic in human life.
22 comments:
I'll bite.
Syllogism A: Valid.
B: Invalid.
C: Valid (but false, in my opinion, because the first premise is false).
D: Valid.
E: Invalid.
F: No conclusion can be drawn.
Very good, Michael! You only missed two. I won't say which ones beyond noting that the syllogism does in fact yield a valid conclusion!
I would also note that any professor logic who used these examples on an exam should probably be taken out and whipped. They are designed with difficulty in mind, to gain insight on how people actually think.
In the last comment, I meant, of course, that the last</em syllogism, syllogism F, yields a valid conclusion.
A. Valid
B. Invalid
C. If you changed, "Therefore war is socially beneficial." to "Therefore war tends to be socially beneficial."
,then I would say, "Valid".
D. If you changed, "The streets aren’t we." to "The streets aren’t wet."
,then I would say, "Valid".
E. Neither necessarily valid nor necessarily invalid
F. None of the beekeepers are chemists.
Either some or all of the artists are beekeepers.
Of course, my answers in terms of logic based on premise you have provided.
I agree with Red's answers, though
for E, I would say:
Invalid (affirming the consequent does not confirm the precedent unless you have an "if and only if")
for F, I would say:
"Some of the artists are not chemists".
http://en.wikipedia.org/wiki/Venn_Diagram
The invalidity of syllogism B hinges on the presentation of the first premise. Had it been less ambiguously worded (of course, this would change the meaning of the sentence), for example: "Only those who believe in democracy believe in free speech," we could then draw the conclusion that fascists do not believe in free speech.
Likewise, with syllogism E we are short of essential information in one and perhaps two respects: First, we don't know what other conditions (if any) produce symptoms A, B, and C. Second, we don't know for sure whether the symptom-combination ABC is sufficient for diagnosis X. ABC appear to be necessary, based on the premise presented, but it is not clear that they are sufficient.
The only conclusion I can find in syllogism F is that there are no beekeeping chemist/artists.
Congratulations Laj: you got them all right. And my apologies to Michael: he only missed one. I goofed on syllogism D: I copied and pasted the wrong syllogism. I've corrected it (and the syllogism is now invalid).
Of course, the validity or invalidty of a syllogism does not necessarily bear on the truth of the conclusion. So some of the quibbles about premises B and C miss the point. B is invalid as it stands, C valid (even if, as is highly likely, the conclusion is false).
"my apologies to Michael: he only missed one."
Woo-hoo!
For the life of me, I couldn't figure out which syllogism (other than F) I got wrong. Last night I was actually lying awake thinking about it!
... Which just goes to show I need better things to occupy my mind.
Greg,
Now I know that time that I spent studying for the LSAT (which I never took) wasn't totally wasted :).
I remember that in logic, the statements:
"some fish are animals"
and
"some animals are fish"
are logically equivalent, though we tend to think in common sense English that the first statement doesn't make sense. That's how I got the conclusion.
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Syllogism E:
Disease X is known to produce various symptoms, including A, B, and C.
This patient has symptoms A, B, and C.
Therefore this patient has disease X.
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1. Therefore this patient does not have disease X.
Would 1. be necessarily valid or invalid?
2. This patient may have disease X, but not necessarily so.
Would 2. be necessarily valid or invalid?
3. Therefore this patient has disease X.
Would 3. be necessarily invalid?
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Syllogism F:
Some of the beekeepers are artists.
None of the chemists are beekeepers.
Therefore…
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Greg, your premise F did not state whether all or only some artists are beekeepers.
"Some of the artists are not chemists"
would be necessarily valid only if there had been information that not all artists are beekeepers in the premise F.
Sorry, Laj, please don't think I'm being a sore loser, I'm just pointing out the faulty premise.
After all, "Paper will tolerate anything written on it." - Stalin
"Some of the artists are not chemists"
would be necessarily valid only if there had been information that not all artists are beekeepers in the premise F.
No, I think Greg's conclusion follows (even though I didn't get it). If all the artists are chemists then some of them would be beekeepers. But none of the chemists are beekeepers; therefore, not all the artists are chemists.
Finally, nice to get a response out of you, Michael!
Michael, I disagree with your conclusion and I'll prove it to you logically.
Let's recap:
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Syllogism F:
Some of the beekeepers are artists.
None of the chemists are beekeepers.
Therefore…
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"Some of the beekeepers are artists." offers at least two possiblities.
1. Some artists are beekeepers.
2. All artists are beekeepers.
When 1. is applied to "None of the chemists are beekeepers."
It offers logically valid certainty that some artists are not chemists.
But when 2. is applied to "None of the chemists are beekeepers.", the logical conclusion would be "No artist is a chemist."
Therefore, "Some of the artists are not chemists" would be logically no longer necessarily valid.
But when 2. is applied to "None of the chemists are beekeepers.", the logical conclusion would be "No artist is a chemist."
Yes, but "some artists are not chemists" doesn't contradict "all artists are not chemists." For instance, "some bricks are not pancakes" doesn't imply that some bricks are pancakes. It says nothing about all bricks. It's only a statement about some bricks, and the statement is true (as far as it goes).
Finally, nice to get a response out of you, Michael!
To be honest, I usually don't read your comments because I find the formatting hard to follow. Sorry.
Yes, but "some artists are not chemists" doesn't contradict "all artists are not chemists." For instance, "some bricks are not pancakes" doesn't imply that some bricks are pancakes. It says nothing about all bricks. It's only a statement about some bricks, and the statement is true (as far as it goes).Exactly, Mike. Red is still caught with confusing the logical "some" with the common sense "some". The logical "some" only means "at least one".
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Yes, but "some artists are not chemists" doesn't contradict "all artists are not chemists."
For instance, "some bricks are not pancakes" doesn't imply that some bricks are pancakes.
It says nothing about all bricks. It's only a statement about some bricks, and the statement is true (as far as it goes).
- Michael Prescott
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Sorry, Michael, when possiblity #2 "All artists are beekeepers."
was applied to "None of the chemists are beekeepers." from F, the logically necessarily valid conclusion involving artists and chemists is, "No artist is a chemist."
Are you saying,
"Some artists are not chemists." and "No artist is a chemist."
are logically equivalent statements?
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Exactly, Mike. Red is still caught with confusing the logical "some" with the common sense "some". The logical "some" only means "at least one". - Laj
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Sorry, Laj, I am fully aware of the meaning of logical "some".
I know the meaning of "some" in logical sense, meaning at least one.
Your statement was, "Some artists are not chemists.", meaning, just like you said, "At least one artist is not a chemist."
However, just as I explained above, when possiblity #2, "All artists are beekeepers." is applied to "None of the chemists are beekeepers." from F., the logically necessarily valid conclusion involving artist and chemists is "No artist is a chemist."
Are you saying, "No artist is a chemist." and "At least one artist is not a chemist." are logically equivalent statements?
Are you saying, "No artist is a chemist." and "At least one artist is not a chemist." are logically equivalent statements?
I wouldn't call the two statements logically equivalent, since "No artist is a chemist" gives us more information than "At least one artist is not a chemist." But the two statements can happily coexist. They don't contradict each other. If no artist is a chemist, then at least one artist is not a chemist.
"Some artists are not chemists" doesn't tell us anything, one way or the other, about some other artists. They may or may not be chemists.
Anyway, this whole discussion nicely illustrates Greg's point - logic can be tricky even for those who've studied it - which supports his further point that formal logic isn't essential for survival. I would guess that most people would get at least half these problems wrong, yet they can still function and survive, and some of them make a very good living.
(Incidentally, this suggests that there are many more types of intelligence than can be measured on an IQ test. Somone may have a relatively low IQ and yet be very intelligent in areas that the test doesn't cover. He might be a great chef, a skilled decorator, a fine dancer, a great negotiator, a wise counselor, etc., etc. Intelligence comes in many flavors, and a standardized test is severely limited in what it can measure.)
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Anyway, this whole discussion nicely illustrates Greg's point - logic can be tricky even for those who've studied it - which supports his further point that formal logic isn't essential for survival. I would guess that most people would get at least half these problems wrong, yet they can still function and survive, and some of them make a very good living. - Michael Prescott
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Indeed.
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(Incidentally, this suggests that there are many more types of intelligence than can be measured on an IQ test. Somone may have a relatively low IQ and yet be very intelligent in areas that the test doesn't cover. He might be a great chef, a skilled decorator, a fine dancer, a great negotiator, a wise counselor, etc., etc. Intelligence comes in many flavors, and a standardized test is severely limited in what it can measure.) - Miachel Prescott
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Incidentally, there have been different IQ tests, not only that there are different versions of the same IQ test.
Furthermore, the IQ test, which is now used is primarily based on relative standing of the taker within a given population, rather than an absolute measure based on what I have read so far.
So a person who has an extremely high IQ today may not necessarily be as "smart" as someone with the exactly IQ score at different time frame and different population base.
Btw. about "logically valid conclusion" regarding F. I am primarily interested in what I consider to be faulty premise of F.
There could be multiple variation of "logically valid conclusion" involving the very same actors, which are not logically equivalent statements.
One could have said, "At least two artists are not chemists.", which also would not have necessarily contradicted, "No artist is a chemist.", depending on the assumption regarding the number of artists in F.
or "At least, three, or four, or five and on and on.......".
You get my drift, why I think F was based on faulty premise.
There should have been more relevant information to avoid this problem in F.
Btw. Where is Herb?
Damn! I wish I saw this post before everyone commented and the answers were revealed, it would've been fun. (And good practice if I wind up taking the LSATs.)
Red Grant: There could be multiple variation of "logically valid conclusion" involving the very same actors, which are not logically equivalent statements.
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I think you could make this claim for just about any argument. After all, if we say "all men are mortal", we could them say "some men are mortal" follows.
That's part of the reason why tests like the LSAT, SAT etc. are multiple choice. The generation of the insight can be less important than the ability to tell whether the insight strictly follows or doesn't follow when it comes to logic.
I don't admit to any genius in deriving the correct answer - I just remembered a test-taking insight from my LSAT prep.
I only read Greg's opening post and the first comment. It's easy to see that A is valid, C, D, and E, are invalid, while no conclusion can be drawn from F.
Why was it so easy? To begin with, it is necessary to know what a syllogism is, and some of these arguments do not even take the form of syllogisms.
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