Falsity implies anything?!?!?

Emil

Thu 1 Oct, 2009 08:29 am
@ACB,
ACB;94580 wrote:
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.

ACB

Thu 1 Oct, 2009 09:07 am
@Emil,
Emil;94581 wrote:

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?

kennethamy

Thu 1 Oct, 2009 11:05 am
@ACB,
ACB;94588 wrote:
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?

I guess I am missing your point. Let P=F. If P then not-P is, If F then not-F. Which comes to, If F then T (double negation). Which comes to, T.

If P then not P, does not assert that it is possible that P and not-P. Are you confusing the conditional sign (if... then...) with conjunction? (and)? As I already said, If P then not P asserts, it is not the case that (P and not, not P). Because a conditional statement asserts that it is false that the antecedent is true, and the consequent is false.

I hope this helps, but I think that there is some confusion going on. But I cannot put my finger on it.

---------- Post added 10-01-2009 at 01:14 PM ----------

ACB;94580 wrote:
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.

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?

Emil

Thu 1 Oct, 2009 11:55 am
@ACB,
ACB;94588 wrote:
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?

ACB

Thu 1 Oct, 2009 05:12 pm
@Emil,
Emil;94607 wrote:
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:

Emil

Thu 1 Oct, 2009 05:37 pm
@ACB,
ACB;94661 wrote:
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:

Not a problem. Recall that even though my knowledge is more numerous than yours, it does not follow that I do not learn something from talking with you. Teaching teaches the teacher too. (Yes I tried making up some funny way of saying it. )

But some more points.

One cannot via a truth table (or worlds diagram) prove that a proposition is logically possible. One can prove that an argument form is valid, and thus that any argument of that form is valid. One can sometimes prove that an argument form is invalid. (Generally easy in propositional logic and harder or impossible in predicate logic.) That does not implythat all arguments of that form are invalid. The form of the argument may not have been captured in the formalization chosen. When I claimed that your implication was false I assumed that it was perfectly formalized.

Also, we did not make a lot of important distinctions here: between wff's, propositions, sentences, argument forms, etc. You'd need to read a textbook about that. I recommend Possible Worlds if you want a highly technical textbook. I read at least two other textbooks before reading that one. It may be too tough for a newbie in logic. Kennethamy usually recommends some textbook by a Copi. I don't know it.

Since it is very hard to prove invalidity, the onus is typically placed upon the person making the argument to show that it is valid. That is quite easy if it has a valid form in propositional logic. It is sometimes very hard to prove in predicate logic.

kennethamy

Thu 1 Oct, 2009 05:38 pm
@ACB,
ACB;94661 wrote:
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:

But I would like it if you tried to explain what you were getting at. I do not dismiss what you said. But I don't understand your reason for saying it. Or, I would like to explain to you why I think you are wrong. Co

---------- Post added 10-01-2009 at 07:43 PM ----------

Emil;94666 But some more points. [I wrote:
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.

Emil

Fri 2 Oct, 2009 04:53 am
@kennethamy,
kennethamy;94667 wrote:

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.

kennethamy

Fri 2 Oct, 2009 06:01 am
@Emil,
Emil;94739 wrote:

I was assuming, of course, that the formalization was the specific one. I could formalize some bachelors are not bachelors as P, and that formalization would not show that proposition as being self-contradictory.

ACB

Fri 2 Oct, 2009 06:39 am
@kennethamy,
kennethamy;94597 wrote:
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.

kennethamy

Fri 2 Oct, 2009 07:00 am
@ACB,
ACB;94747 wrote:
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.

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.

I don't see why you say this. I can state the conditional, if dogs are not dogs, then cats are not cats. Now that conditional is true, since the antecedent is false. But neither antecedent, nor consequent imply possibilities, since both are self- contradictory. A conditional statement does not imply that either the antecedent or consequent is true, nor does it imply that either are possibly true.

There may be a problem with the notion of possibility here. By "possibility" is meant, logical possibility. I.e. consistency. And maybe with the notion of implication, too. P implies Q just means that if P is true, then Q is true (and no more). P implies Q, does not imply either P nor Q since P implies Q can be true, while P is false, and Q is false. And, the same goes for possible P, and possible Q, as I showed in my example above.

Zetherin

Fri 2 Oct, 2009 09:36 am
@JeffD2,
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.

kennethamy

Fri 2 Oct, 2009 10:33 am
@Zetherin,
Zetherin;94776 wrote:
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 I have lost what is at issue here. I thought that the issue was whether a conditional implies the possibility of both its antecedent and its consequent. It clearly does not. (Nor does it imply the truth or either). What do you think the issue of this thread is? Or the latest stage anyway? It is not, as you seem to think, about whether the particular example of a conditional statement is true, or even possibly true. Is is about the relation of the truth conditions of conditional statements, and the truth or falsity of the component statements of conditional statement, or of the possibility of the truth of the components. The central point is that a conditional is true (and so it is possible for it to be true) except under only one condition: when the antecedent is true, and the consequent is false. Under any other conditions it is possible for it to be true. And that implies that whenever the antecedent of a conditional is false, the entire conditional is true, and, so, it is possible for the whole conditional to be true. (Here we are talking about the material condition. Not the ordinary language conditional. The ordinary language conditional does have one condition it shares with the material conditional, namely that it cannot be true if its antecedent is true, and its consequent false. 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. For example, the conditional, if you do not buy it today, then the item will be more expensive later is still true even if you do buy the item today.(That is, whether or not you do buy the item today, it will still be more expensive later. So the truth value of the ordinary language conditional does not depend on the truth value of the antecedent. since, in this case, the conditional is true whatever the truth value of the antecedent happens to be).

ACB

Fri 2 Oct, 2009 01:19 pm
@Zetherin,
Zetherin;94776 wrote:

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

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 ----------

kennethamy;94780 wrote:
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 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.

kennethamy

Fri 2 Oct, 2009 02:47 pm
@ACB,
ACB;94798 wrote:
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.

Yes. Ordinary language has no rules for these contingencies you mention. But, since they are propositions, and therefore, could appear in arguments, logicians have to know what truth value to assign to them. It will not do to have truth "gaps". The argument,

1. If no dogs are dogs, then no cats are cats.
2. No dogs are dog.

Therefore, 3. no cats are cats.

Has to be assessed as either valid or invalid. (and sound or unsound)

The above argument is valid, but unsound, since premise 2. is false. And, of course it is unsound, since it has a false conclusion, and all arguments with false conclusions are unsound.

I don't understand what you means by "ordinary language conditionals imply possibilities". Possibilities of what?

Emil

Fri 2 Oct, 2009 04:34 pm
@ACB,
ACB;94747 wrote:
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
ACB;94747 wrote:
and I rule out the following possibility:

(c) The day is cloudless and the sun does not shine.

---------- Post added 10-03-2009 at 12:51 AM ----------

Zetherin;94776 wrote:
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.
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 ----------

ACB;94798 wrote:
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.

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.

Zetherin

Fri 2 Oct, 2009 05:27 pm
@JeffD2,
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

Emil

Fri 2 Oct, 2009 05:44 pm
@Zetherin,
Zetherin;94827 wrote:
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!

:OK:

Zetherin;94827 wrote:
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

While I do think that you ought to pick up and textbook and study it. I also learn by teaching, as I have written before. I plan on teaching at the university in the future, so this is good training.

Don't feel bad about not understanding it. Formal logic is tough to learn. You can do it alone though if you're good disciplined. (I did.)

What is good is that your attitude is not extremely hostile. Some people don't have a clue about logic but still get angry etc. when people that do have a clue correct them. Not that I am always error free, I am most definitely not.

Kennethamy probably knows more logic than I do. I'm only 20.

kennethamy

Fri 2 Oct, 2009 05:59 pm
@Emil,
Emil;94818 wrote:

---------- 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.

The salient point is that truth-functional logic is-truth-functional. Which is to say that it strips off any relations among propositions except their what their relations are relevant to their truth-values. A simple example is: I put on my shoes and I put on my socks, as contrasted with, I put on my socks and I put on my shoes. In terms of truth-functional logic, if both components of the conjunction are true, then the whole conjunction is true. But contrast this with ordinary language. It would make a difference whether the sentence was, "I put on my socks, and I put on my shoes", or "I put on my shoes, and put on my socks". Since, in ordinary language, the former would be true, and the latter, false. That is because in ordinary language. "and" sometimes means temporal succession. But in truth-functional logic, temporal succession does not matter.

ACB

Fri 2 Oct, 2009 06:04 pm
@Emil,
Thank you all. I don't think I can pursue this matter any further, so I am withdrawing from this thread.