Logic

[Plain Simple]

In the thread Orci on Start Trek, timelines, canon and everything (SPOILERS) in the Star Trek XI forum a quite intensive sub-discussion started about logic, truth, and what not. Since it started to go quite off topic, I start this thread for a continuation of that discussion if wanted.

 

[Neopeius]

You'll have to restate the case for us initiates.

 

[Plain Simple]

It got quite involved and when I try to recap the discussion I risk misinterpreting other peoples words, but I think I can word it as follows.

For some reason the subject of Creationism and Intelligent Design (C/ID) popped up in that thread (don't ask me why...) and the claim was made that they are illogical because they are based on premisses that contradict 'known facts' about the world. This point was opposed by those claiming that, although it might be true that C/ID is based on 'false' premisses (it might also not be true, that's not the point here), that doesn't necessarily make it illogical. It is perfectly well possible for a theory or argument to be completely logical although starting from false premisses. Logic does not equal truth. So to be clear, the discussion was not on the merits of C/ID as a scientific theory, but on whether or not it being based on false premisses is a reason for calling it illogical.

 

[MIB]

Yes, I do believe there is a sound basis for calling it illogical. Arguing from false premises is a logical fallacy. Whatever good logic may follow the fallacy, it will still be based on that fallacy and thus will be bad logic.

 

[Plain Simple]

Denying the antecedent is a non-logical reasoning, not a logical reasoning starting from factually false premisses. At least, if we're talking about the same thing: http://en.wikipedia.org/wiki/Denying_the_antecedent.

Edit: in the time that I wrote my response MIB seems to have edited his post to remove the reference to 'denying the antecedent'.

What was meant in this discussion (again, if I word other peoples statements correctly) was, for example, something of the form

premisses

1. A implies B
2. A

conclusion

3. B

Now, my stand (and other's) is that this is a logical reasoning, no matter the truth value of 1. and 2. in the sense that given the premisses the conclusion follows, even if one of the premisses (for example 2.) states that the earth is only 6000 years old or it's daily raining pigs or this bbs is a Spice Girls fan site.

 

[MIB]

Only in the sense that you are drawing admitedly false conclusions from false premises. If that is the actual goal, one could argue that one is using 'good' logic. Otherwise, that person would be doing little more than trying to make themselves feel better over justfiying their beliefs with bad logic.

 

[Alidar Jarok]

I think that good logic based on false premises could technically be true, although it's a bit of an argument over semantics considering the post that brought up this discussion.

 

[Plain Simple]

It turned out that my try to save the original thread by starting this spin off came too late: it got closed. Which of course doesn't mean the discussion in here cannot continue if people are interested.

 

[Goliath]

The distinction being made here is the distinction between validity and soundness.

A deductive argument is said to be valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. Otherwise, a deductive argument is said to be invalid.

A deductive argument is sound if and only if it is both valid, and all of its premises are actually true. Otherwise, a deductive argument is unsound.

Source: Internet Encyclopedia of Philosophy.

Consider the following argument:

Camelopard is God.
You all should bow down and worship God.
Therefore, you should all bow down and worship Camelopard.

This is a valid argument. It's perfectly logical. If its premises are true, then its conclusion cannot be false.

But it's not a sound argument. At least one of its premises is false: I am not, in fact, God. As a consequence, its conclusion is false as well.

 

[Plain Simple]

The distinction being made here is the distinction between validity and soundness.

Indeed. Thanks for mentioning the term "soundness", I didn't know that was what it is called. Too bad none of the people from the other thread have come to this one (yet), since some fresh explanations by other people might have clarified things.

 

[TrekGuide.com]

Well, I was enjoying the other thread, even if it went wildly off-topic around page 25 or so ...

I was making a philosophical statement about the Mirror Universe having identical people as in the Federation Universe, to which 3D Master replied, "Prove it." (Ironically, that is the sort of response that 3D Master himself would get from a creationist when arguing that there is no God.) Trying to "prove," in a scientific sense, what is "true" in a fictional Mirror Universe as depicted on a fictional TV series, is not "logical" in the Mr. Spock sense.

I think 3D Master and others were arguing from the belief that everything that is logical is true, and everything that is true is logical, and that logic is the same as science, as far as proving what is true or not.

But to avoid the same emotional debates and circular arguments that got the other thread closed, we should define the terms and how they differ from each other.

What is "logic"? What is it used for? How does it differ from "truth" and "science"?

What is "truth"? How does it differ from "logic"?

What is "science"? How does it relate to "logic" and "truth"?

In the other thread I made up the following example:

1. Richmond is a state capital.
2. Richmond is a city in California.
3. Sacramento is the capital of California.
​

All the elements in that statement are "true," but it is not a "logical" statement, since the sentences do not support each other.

Here is another:

1. All Elves have pointed ears.
2. Mr. Spock has pointed ears.
3. Mr. Spock is a Vulcan.
​

Again, these are "true" statements, but they are not "logical."

Now, can anyone provide an example of a statement that is "logical," but is not "true" (first defining what "logical" and "true" mean)?

 

[Shaw]

In the other thread I made up the following example:

1. Richmond is a state capital.
2. Richmond is a city in California.
3. Sacramento is the capital of California.
​

All the elements in that statement are "true," but it is not a "logical" statement, since the sentences do not support each other.

Here is another:

1. All Elves have pointed ears.
2. Mr. Spock has pointed ears.
3. Mr. Spock is a Vulcan.
​

Again, these are "true" statements, but they are not "logical."

That's funny... I'm guessing that you haven't had much set theory.

 

[Plain Simple]

But to avoid the same emotional debates and circular arguments that got the other thread closed, we should define the terms and how they differ from each other.

Very sensible idea. Every in-depth discussion should start with that.

1. All Elves have pointed ears.

I would hardly call that a true statement, unless you do not mean it in the fictional sense, but in a true world sense, in which case the statement is about the elements of a (probably) empty set and thus true. But if you mean in fiction, than you can probably easily find Elves without pointed ears. I don't think for example that it has ever been definitively established whether or not Tolkien's elves have pointed ears. But let's not turn this into a thread about Elves' ears. We can open another spin off thread in SF&F for that.

Now, can anyone provide an example of a statement that is "logical," but is not "true" (first defining what "logical" and "true" mean)?

As logic I take first order (predicate) logic, albeit in a less formalised way for the sake of discussion (so we can discuss in words, not symbols), as truth I take all those things that are accepted by the scientific community as verified by empirical observations (or to put it more specific: theoretical constructs that have never been sufficiently falsified by empirical observations). Note that I assume these to include 'common sense observations' like 'my walls are painted white' even if no scientist ever came in to observe that and write a peer reviewed paper about it. I just assume that it is accepted in the scientific community that if you and your friends and family look at your walls and see them to be white in white light, they are white.

These choices of mine are inspired by, what I think, was the original intention in the previous thread. I do not think however that, when you go to slightly different definitions, that changes the following much.

So, here is a logical untrue string of statements.

1. My head is an orange.
2. Oranges are green.
3. All green things talk sense.
4. My head talks sense.

Note that 1, 2, and 3 are not true, but the line of reasoning follows logic. And 4 is true (sometimes ), but that is not of importance for either the validity or soundness of the argument. I could have just as well written "no sense" instead of "sense" in 3 and 4 and still the argument would have been logical and false.

 

[Plain Simple]

In the other thread I made up the following example:

1. Richmond is a state capital.
2. Richmond is a city in California.
3. Sacramento is the capital of California.
​

All the elements in that statement are "true," but it is not a "logical" statement, since the sentences do not support each other.

Here is another:

1. All Elves have pointed ears.
2. Mr. Spock has pointed ears.
3. Mr. Spock is a Vulcan.
​

Again, these are "true" statements, but they are not "logical."

That's funny... I'm guessing that you haven't had much set theory.

Care to elaborate on that? The above examples do not give constructs that defy logic, they are just independent sentences that are not connected, not be valid logic, nor by an invalid argument. An example of the latter would be "All Elves have pointed ears, Mr. Spock has pointed ears, Mr. Spock is an Elf", or, if you want a non-logical one with true statements: "All humans are mammals, I am a mammal, I am human".

Edit: actually, now I'm doubting whether there is a way to make explicit when an argument is not logical and when logic just doesn't enter into it at all (whether validly or abused).

 

[Shaw]

Care to elaborate on that?

Well, in the first case the grouping is an attempt to make an association that isn't valid. He is doing this by avoiding the definition of Richmond.

Richmond(1) is an element of the set of state capitals.
Richmond(2) is an element of the set of cities in California.
Both Richmond(1) and Richmond(2) are elements of the set of cities called Richmond.
Richmond(1) ≠ Richmond(2).​

You guys want linear arguments, but set theory doesn't have to be linear... but the elements should be well defined (and in this case they are not).

The second example deals with subsets... Elves are a subset of the set of beings with pointed ears, Spock is an element of the set of beings with pointed ears but not an element of the subset made up of Elves.

If you guys don't know how to make these arguments, then you're going no where fast.

And Wiki isn't a substitute for an education... most community colleges offer courses on Logic (and I believe they include elements of set theory in those too). If you want to learn this stuff, don't attempt to learn it on the fly in a thread on some forum... really learn it.

 

[Goliath]

What is "logic"? What is it used for? How does it differ from "truth" and "science"?

Logic has many meanings. For the purposes of this discussion, it probably means just "valid reasoning." Valid reasoning is "logical." Invalid reasoning is "illogical."

Logic is used to make inferences. If all men are mortal, and Socrates is a man, we can infer from these two facts that Socrates is mortal: if A=B, and B=C, then A=C.

"Science" also has many meanings, but again, for the purposes of this discussion, it probably means the systematic investigation of the physical world, and the knowledge that results from this investigation.

"Truth" is surprisingly difficult to define. There are many different theories of truth, but I personally find "minimalist" or "deflationary" theories the most convincing.

Instead of trying to define "truth," minimalists focus on what we mean when we say something "is true." And what we mean when we say that something "is true" is simply that we affirm that thing.

If you say, "it's raining," and I say, "that's true," I am merely signalling my agreement with your statement. It's equivalent to nodding your head, or saying "yes, it is raining."

Thus, "truth" is merely every statement to which we should nod our heads, or which we should affirm.

Now, can anyone provide an example of a statement that is "logical," but is not "true" (first defining what "logical" and "true" mean)?

"Statements" are not logical. Logic is the technique of making correct inferences from statements.

Consider the example I gave above:

Camelopard is God.
You should all bow down and worship God.
Therefore, you should all bow down and worship Camelopard.

None of these statements, by themselves, is logical or illogical. The logic in this example can be found in the inference I made from these statements.

In this case, while my inference was valid, it was unsound. I made the correct inference from my premises: if I am God, and you all should bow down and worship God, then you should all bow down and worship me--QED.

But at least one of premises was obviously untrue: nobody is going to nod their heads and say, "yes, Camelopard is God"--not even me.

And since my premise was untrue, it follows that my conclusion was also untrue: you should not all bow down and worship me.

 

[Plain Simple]

Care to elaborate on that?

Well, in the first case the grouping is an attempt to make an association that isn't valid. He is doing this by avoiding the definition of Richmond.

Richmond(1) is an element of the set of state capitals.
Richmond(2) is an element of the set of cities in California.
Both Richmond(1) and Richmond(2) are elements of the set of cities called Richmond.
Richmond(1) ≠ Richmond(2).​

You guys want linear arguments, but set theory doesn't have to be linear... but the elements should be well defined (and in this case they are not).

The second example deals with subsets... Elves are a subset of the set of beings with pointed ears, Spock is an element of the set of beings with pointed ears but not an element of the subset made up of Elves.

If you guys don't know how to make these arguments, then you're going no where fast.

And Wiki isn't a substitute for an education... most community colleges offer courses on Logic (and I believe they include elements of set theory in those too). If you want to learn this stuff, don't attempt to learn it on the fly in a thread on some forum... really learn it.

I can assure you that as a professional mathematician I'm quite familiar with set theory and logic. Now I don't work in the area of (mathematical) logic or set theory specifically, so I'm not an expert on all the intricacies, but I do know (I hope, and so does my employer I suppose) a bit about logic. Which is of course not to say that I cannot be mistaken on these issues or that when writing things down fairly quickly for this thread I cannot mess up details.

Now, I choose 'linear' examples as you call them because they are usually quite easy to understand and still illustrate the point in this case.

And as for your two Richmonds, I understood the example to mean that some pre-defined Richmond was assumed to belong to both the set of state capitals as well as cities in California. You're right, it should be defined beforehand. So probably the correct interpretation of the example as it is written down (perhaps not as it was intended) is the following: Let Richmond be an element in the set of state capitals. Premise: Richmond is an element of the set of cities in California.

Of course I agree whole heartedly that Wikipedia is not a substitute for good education, but I can tell you from personal experience that the mathematics entries on Wikipedia are often very good and informative. And you can hardly expect people to start an education in logic just to understand or participate in this thread. That's why I linked to Wikipedia, so the people who are not familiar with the terms I used (because I was too lazy to write out in more detail what I meant) to look them up.

 

[Plain Simple]

And since my premise was untrue, it follows that my conclusion was also untrue: you should not all bow down and worship me.

Actually that is not so. It is perfectly well possible to arrive at a true conclusion from a false premise via logic. Your conclusion that we should all bow down and worship you could still be true (in the minimalist interpretation: it could still be possible that we nod our heads and say "yes indeed, we should bow down and worship Camelopard") even if you are not God. It just does not follow logically from the previous statements.

 

[Shaw]

I can assure you that as a professional mathematician I'm quite familiar with set theory and logic...

In what field of study?

My background is in differential geometry and differential topology, but I was still expected to have an adequate background in things like logic, set theory, group theory, etc.

How in the world can you do proofs without knowing logic? How did you get through Modern Algebra or Real Analysis without an understanding of logic?

What is your definition of a professional mathematician? Granted, I was a professional mathematician when I was being paid by the National Science Foundation and the Department of Energy to do math (I was doing research in tight immersions of surfaces, both smooth and polyhedral)... but I still hold myself (and others) to a pretty high standard of what counts as a mathematician (and I fall short of that currently).

Sorry... but I'm not seeing it here from you in this topic. Here is a link so that you have some idea about my background (for full disclosure).

 

[Goliath]

Care to elaborate on that?

Well, in the first case the grouping is an attempt to make an association that isn't valid.

Actually, no, it isn't. You didn't read what he said closely enough.

What he said was:

Richmond is a state capital.
Richmond is a city in California.
Sacramento is the capital of California.

And as he pointed out--correctly--there is nothing logical about this. That third sentence isn't a conclusion. No inference was made. It's just a series of three sentences.

What you say would be true only if he had said something like this:

Richmond is a state capital.
Richmond is a city in California.
Therefore, Richmond is the capital of California.

But he didn't say that. And even if he had, we wouldn't need set theory to see the problem here: this syllogism commits the fallacy of equivocation, by confusing two different senses of the word 'Richmond'.

Knowledge of set theory is no substitute for attentive reading.