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If I say "If pigs fly, unicorns exist", I am implying it is possible that pigs fly and unicorns exist (but impossible that pigs fly and unicorns do not exist). If that is a correct analysis, then it follows that if I say "If Paris is in Germany, Paris is not in Germany", I am implying it is possible that Paris is in Germany and Paris is not in Germany (but impossible that Paris is in Germany and Paris is (sic) in Germany).
So it appears that the 'Paris' statement not only allows a contradiction, but also disallows a tautology! Where have I gone wrong?
EDIT - I have not read posts #37-39 yet.
But if the second one is true for anyad hoc to say we can't, just to avoid a contradiction?
I agree on the principle that contradictions cannot be allowed. I am just puzzled about statements of the form "If P, not-P" where P is false. If such statements are not asserting the logical possibility of (P and not-P), what are they asserting? Are they totally empty?
If I say "If pigs fly, unicorns exist", I am implying it is possible that pigs fly and unicorns exist (but impossible that pigs fly and unicorns do not exist). If that is a correct analysis, then it follows that if I say "If Paris is in Germany, Paris is not in Germany", I am implying it is possible that Paris is in Germany and Paris is not in Germany (but impossible that Paris is in Germany and Paris is (sic) in Germany).
So it appears that the 'Paris' statement not only allows a contradiction, but also disallows a tautology! Where have I gone wrong?
EDIT - I have not read posts #37-39 yet.
But if the second one is true for anyad hoc to say we can't, just to avoid a contradiction?
I agree on the principle that contradictions cannot be allowed. I am just puzzled about statements of the form "If P, not-P" where P is false. If such statements are not asserting the logical possibility of (P and not-P), what are they asserting? Are they totally empty?
How much logic do you know? You need to know a fair bit of logic for me to explain that one. (About wffs, substitution rules, the specfictivity of forms, sentence forms, etc.) I don't want to explain it all here, and I'm not sure that I can do it properly either. Why don't you read a textbook?
OK, fair enough. You guys clearly know a lot more logic than I do. This thread began in a simple, non-technical fashion, based on the ordinary meanings of words. I became involved in it on that basis, but it has now become highly technical and I do not feel I can usefully contribute any further. So I will accept that you are correct, and withdraw from this discussion.
:surrender:
Thanks for your efforts.
OK, fair enough. You guys clearly know a lot more logic than I do. This thread began in a simple, non-technical fashion, based on the ordinary meanings of words. I became involved in it on that basis, but it has now become highly technical and I do not feel I can usefully contribute any further. So I will accept that you are correct, and withdraw from this discussion.
:surrender:
Thanks for your efforts.
One cannot via a truth table (or worlds diagram) prove that a proposition is logically possible.[/I]
If a proposition is not self-contradictory, isn't it logically possible? And you can show that a proposition is not self-contradictory on a truth table.
If I say "If pigs fly, unicorns exist", I am implying it is possible that pigs fly and unicorns exist (but impossible that pigs fly and unicorns do not exist).
Could you explain why you think the above? How do you make the move to statements about logical possibility?
Let me try to explain my reasoning step by step, to help you pinpoint where I may have gone wrong.
1. When I say "If the day is cloudless, the sun shines", I imply (allow) the following possibilities:
(a) The day is cloudy; or
(b) The day is cloudless and the sun shines
and I rule out the following possibility:
(c) The day is cloudless and the sun does not shine.
2. Similarly, when I say "If pigs fly, unicorns exist", I imply the possibilities
(a) Pigs do not fly; or
(b) Pigs fly and unicorns exist
and I rule out
(c) Pigs fly and unicorns do not exist.
3. "If pigs fly, unicorns exist" (IPFUE) is true, since the antecedent is false.
4. Therefore, anything that IPFUE says or implies is also true.
5. IPFUE allows the possibility that pigs fly and unicorns exist - see 2(b) above. [This seems to me implicit in the meaning of the word "if". If you dispute this, please explain why.]
6. Therefore (from 4 and 5) "Pigs fly and unicorns exist" is a true possibility.
7. If it is a true possibility, it must be logically possible, even though it does not describe an actual state of affairs.
8. What about "Pigs fly and unicorns do not exist" (2(c))? Well, IPFUE says it is impossible; IPFUE is necessarily true; hence 2(c) is necessarily (logically) impossible.
9. Applying the above reasoning to "If Paris is in Germany, Paris is not in Germany", we get a true contradiction and a false tautology! (Which is obviously absurd, and a sign of error.)
10. And of course, if you started off with the statement "If pigs fly, unicorns do not exist", you could (by my reasoning) prove exactly the opposite of 7 and 8 above! (Again, absurd.)
I appreciate that my reasoning is faulty somewhere. But since you asked, I have laid it out for your perusal.
ACB,
Maybe my layman verbiage will help us get to the bottom of this (haha)!
Example: "If pigs fly, unicorns exist"
Are you sure that these are the only two possibilities:
- Pigs do not fly; or
- Pigs fly and unicorns exist
Isn't it possible that even if pigs don't fly, unicorns can still exist? All "If pigs fly, exists unicorns" tells us is that if pigs do fly it's necessary that unicorns exist; it does not tell us that if pigs don't fly it's necessary that unicorns don't exist - unicorns could still exist due to something else, right? Therefore, another possibility would be:
- Pigs don't fly and unicorns exist
In order for there to only be two possibilities I think you would have to say:
"Iff pigs fly, unicorns exist"
Which means if and only if pigs fly, unicorns exist.
Example: "If pigs fly, unicorns exist"
Are you sure that these are the only two possibilities:
Isn't it possible that even if pigs don't fly, unicorns can still exist? All "If pigs fly, exists unicorns" tells us is that if pigs do fly it's necessary that unicorns exist; it does not tell us that if pigs don't fly it's necessary that unicorns don't exist - unicorns could still exist due to something else, right? Therefore, another possibility would be:
- Pigs do not fly; or
- Pigs fly and unicorns exist
- Pigs don't fly and unicorns exist
But although the material conditional has definite truth values for the other three permutations of truth values, I don't think anything like that is true of the ordinary language conditional. For instance, it is not at all clear what the truth value of the conditional is when the antecedent is false.
I think you misunderstood me. By "pigs do not fly" I meant both the possibility "and unicorns exist" and the possibility "and unicorns do not exist". Sorry if I didn't make myself clear.
---------- Post added 10-02-2009 at 09:22 PM ----------
I think we are getting to the heart of the matter here. Logic seems to use the word 'if' differently from ordinary language. For example, it would not make sense in an ordinary language conditional to use a self-contradictory antecedent or consequent. Nor could the antecedent be in the present tense when it is definitely false. (We might say "France is not a monarchy, but if it were...", but we would never say "France is not a monarchy, but if it is...") But these things are permissible in logic. This was no doubt the reason for my confusion.
By the way, I think that ordinary language conditionals do imply possibilities where the antecedent is in the present tense and the consequent is in the present or future tense.
Let me try to explain my reasoning step by step, to help you pinpoint where I may have gone wrong.
1. When I say "If the day is cloudless, the sun shines", I imply (allow) the following possibilities:
(a) The day is cloudy; or
(b) The day is cloudless and the sun shines
and I rule out the following possibility:
(c) The day is cloudless and the sun does not shine.
ACB,
Maybe my layman verbiage will help us get to the bottom of this (haha)!
Example: "If pigs fly, unicorns exist"
Are you sure that these are the only two possibilities:
- Pigs do not fly; or
- Pigs fly and unicorns exist
Isn't it possible that even if pigs don't fly, unicorns can still exist? All "If pigs fly, exists unicorns" tells us is that if pigs do fly it's necessary that unicorns exist; it does not tell us that if pigs don't fly it's necessary that unicorns don't exist - unicorns could still exist due to something else, right? Therefore, another possibility would be:
- Pigs don't fly and unicorns exist
In order for there to only be two possibilities I think you would have to say:
"Iff pigs fly, unicorns exist"
Which means if and only if pigs fly, unicorns exist.
I think you misunderstood me. By "pigs do not fly" I meant both the possibility "and unicorns exist" and the possibility "and unicorns do not exist". Sorry if I didn't make myself clear.
---------- Post added 10-02-2009 at 09:22 PM ----------
I think we are getting to the heart of the matter here. Logic seems to use the word 'if' differently from ordinary language. For example, it would not make sense in an ordinary language conditional to use a self-contradictory antecedent or consequent. Nor could the antecedent be in the present tense when it is definitely false. (We might say "France is not a monarchy, but if it were...", but we would never say "France is not a monarchy, but if it is...") But these things are permissible in logic. This was no doubt the reason for my confusion.
By the way, I think that ordinary language conditionals do imply possibilities where the antecedent is in the present tense and the consequent is in the present or future tense.
Thanks Emil, kennethamy, and ACB.
This is a lot to take in, especially since I've never really studied formal logic at all; it appears to me that I have to start at the basics and work my way up before I just jump right into the material presented here!
Bottom line: I have a lot to learn, and I appreciate the enlightening posts even in the midst of people like me who probably sound like a complete idiot (I'm talking to you, Emil, especially, and I'm sure you were pulling your hair out even responding to me!).
Thanks again,
Zeth
---------- Post added 10-03-2009 at 12:51 AM ----------
the modal fallacy since it is so common. (So don't feel bad if you did make it. I made it too before I learned about it. )
There is a very good explanation here about the modal fallacy and related issues.
'The' Modal Fallacy
Lecture Notes on Free Will and Determinism
---------- Post added 10-03-2009 at 12:57 AM ----------
Do you think there is a coherent meaning of "ordinary language conditionals"?
Sure, "if, then" statements in normal language do not correspond completely to material implication or logical implication. Many textbooks mention this and many fail to do so. (I read in a book about the history of modern logic earlier today, and it said that any argument that has an invalid form is invalid. That is not true as I showed earlier.) It may also be used to mean causation and temporal ordering and probably some more things.