Falsity implies anything?!?!?

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Emil
 
Reply Mon 28 Sep, 2009 12:15 am
@ACB,
ACB;93949 wrote:
If the antecedent is false and the consequent is completely nonsensical, is the proposition still true? For example:

"If Paris is in Germany, green questions walk coldly."

And what if the antecedent is false and the consequent presupposes a contradiction? Such as:

"If Paris is in Germany, both of the square circles are green."


It's not a material conditional at all. A material conditional consists of two proposition tokens (may be the same proposition twice) and the relation of material conditionality. But your first one does not express two propositions, so it does not express a conditional. Sentences are not propositions.

As for the second. A material conditional with a necessarily false consequent is itself necessarily false.
 
kennethamy
 
Reply Mon 28 Sep, 2009 06:43 pm
@Emil,
Emil;94014 wrote:
It's not a material conditional at all. A material conditional consists of two proposition tokens (may be the same proposition twice) and the relation of material conditionality. But your first one does not express two propositions, so it does not express a conditional. Sentences are not propositions.

As for the second. A material conditional with a necessarily false consequent is itself necessarily false.



But [p > (q & ~q) ] is not a contradiction if p is false. So it is not necessarily false.
 
ACB
 
Reply Mon 28 Sep, 2009 07:33 pm
@kennethamy,
kennethamy;94148 wrote:
But [p > (q & ~q) ] is not a contradiction if p is false. So it is not necessarily false.


Where does one draw the line between a consequent that is a necessarily false proposition and one that is a non-proposition?

First of all, we have now established that a nonsensical 'consequent' is not a proposition at all, and hence not a real consequent.

Now consider the following two conditionals:

1. If Paris is in Germany, two circles are square.
2. If Paris is in Germany, both of the square circles are green.

In (1), the consequent asserts something necessarily false, but it does not presuppose anything necessarily false. The subject (two circles) and the predicate (are square) both make sense individually. In (2), however, the subject of the consequent (both of the square circles) does not make sense, since it contains a contradiction within itself; it presupposes that there are square circles. Is the consequent of (2) therefore to be classed as gibberish, like my "green questions walk coldly" example, and therefore not a proposition at all?
 
kennethamy
 
Reply Mon 28 Sep, 2009 11:54 pm
@ACB,
ACB;94166 wrote:
Where does one draw the line between a consequent that is a necessarily false proposition and one that is a non-proposition?

First of all, we have now established that a nonsensical 'consequent' is not a proposition at all, and hence not a real consequent.

Now consider the following two conditionals:

1. If Paris is in Germany, two circles are square.
2. If Paris is in Germany, both of the square circles are green.

In (1), the consequent asserts something necessarily false, but it does not presuppose anything necessarily false. The subject (two circles) and the predicate (are square) both make sense individually. In (2), however, the subject of the consequent (both of the square circles) does not make sense, since it contains a contradiction within itself; it presupposes that there are square circles. Is the consequent of (2) therefore to be classed as gibberish, like my "green questions walk coldly" example, and therefore not a proposition at all?


1. First of all, it is not true that in the case of a conditional statement, either the antecedent or the consequent is assertive. In the statement of the form, if p then q, neither p nor q assert, but the entire conditional does assert. (The same is true of the disjunction, p or q. Only the entire proposition (p or q) asserts, but neither p nor q asserts). When I say, if Jack goes up the hill, then Jill goes up the hill, I am neither asserting that Jack goes up the hill, nor that Jill goes up the hill. Although I am asserting that if Jack goes up the hill, then Jill goes up the hill. On the other hand, if I say that Jack goes up the hill and Jill goes up the hill, I am asserting both of the conjuncts.

2. Second of all (and more relevantly) some philosophers have held that contradictions are nonsensical. On the other hand, since contradictions are false, and no statement which is true or false (has a truth value) can be nonsense, contradictions are not nonsense. (And there are even philosophers now who argue that contradictions can be true "dialetheism"). So your question is leads to complications. Wittgenstein (in his Tractatus) distinguished between the German "unsinn", and "sinnloss" (nonsense, and senseless). And held that contradictions (and tautologies too) are senseless, but not nonsense.

As is always the case, even if no sharp line can be drawn between contradictions and nonsense, that does not mean that there are no clear cases of both. It is a fallacy, to give another example, to argue that because there are cases when we cannot say whether a man is bald or not because he is "sort of in between" that there are no clearly bald men, and that there are no clearly not-bald men.
 
Emil
 
Reply Tue 29 Sep, 2009 01:28 am
@kennethamy,
kennethamy;94148 wrote:
But [p > (q & ~q) ] is not a contradiction if p is false. So it is not necessarily false.


Right. I got it wrong. Here is the fixed version: A conditional with a necessarily true antecedent and a necessarily false consequent is itself necessarily false.

But a conditional with a necessary false antecedent is itself necessarily true, and a conditional with a necessary true consequent is itself necessarily true.

---------- Post added 09-29-2009 at 09:55 AM ----------

ACB;94166 wrote:
Where does one draw the line between a consequent that is a necessarily false proposition and one that is a non-proposition?

First of all, we have now established that a nonsensical 'consequent' is not a proposition at all, and hence not a real consequent.

Now consider the following two conditionals:

1. If Paris is in Germany, two circles are square.
2. If Paris is in Germany, both of the square circles are green.

In (1), the consequent asserts something necessarily false, but it does not presuppose anything necessarily false. The subject (two circles) and the predicate (are square) both make sense individually. In (2), however, the subject of the consequent (both of the square circles) does not make sense, since it contains a contradiction within itself; it presupposes that there are square circles. Is the consequent of (2) therefore to be classed as gibberish, like my "green questions walk coldly" example, and therefore not a proposition at all?


As Kenneth said, when someone asserts a conditional, it does not follow that one also asserts both the proposition tokens it is composed of.

I don't understand what you think the difference is between asserting something and presupposing something.

It does not follow that if two phrases or words make sense individually, then they make sense together. E.g. "Green" and "3" both make sense, but not together. It is nonsense to say "A green 3".

I would say that both (1) and (2) express conditionals. Sentences are not to be confused with propositions. It is a category mistake to say that sentences or statements are true or false. Propositions are true or false.

I also agree that contradictions are senseful, i.e. not-nonsense. Since if contradiction-sentences do not express propositions, that is, there are no contradiction-propositions, and only contradictions-propositions are impossible, then nothing is impossible. But something is impossible, and thus contradiction-sentences do express propositions.

Condensed version of the argument.
 
ACB
 
Reply Tue 29 Sep, 2009 05:20 am
@Emil,
Kennethamy - Thanks for your informative reply. Now to deal with Emil's points:

Emil;94214 wrote:
As Kenneth said, when someone asserts a conditional, it does not follow that one also asserts both the proposition tokens it is composed of.


OK. Instead of saying that the consequent "asserts something necessarily false", let's say it "makes a necessarily false predication". In other words, it says something that stands in a necessarily false relation to its grammatical subject.

Emil;94214 wrote:
I don't understand what you think the difference is between asserting something and presupposing something.

It does not follow that if two phrases or words make sense individually, then they make sense together. E.g. "Green" and "3" both make sense, but not together. It is nonsense to say "A green 3".

I would say that both (1) and (2) express conditionals. Sentences are not to be confused with propositions. It is a category mistake to say that sentences or statements are true or false. Propositions are true or false.

I also agree that contradictions are senseful, i.e. not-nonsense. Since if contradiction-sentences do not express propositions, that is, there are no contradiction-propositions, and only contradictions-propositions are impossible, then nothing is impossible. But something is impossible, and thus contradiction-sentences do express propositions.


I agree that contradictions can be senseful. However, let me try to clarify the distinction I was making.

"Some bachelors are married" is meaningful, but necessarily false because it is a contradiction. The phrases "some bachelors" and "are married" both make sense, and they are combined in a grammatically correct way (unlike your "green 3" example).

But what about "Some married bachelors have black hair"? This purports to say something about married bachelors. Does it say something true about them, or something false about them? Neither, since there are no married bachelors. As I see it, a contradiction is a statement, and you cannot have a statement without a verb. "Married bachelors" is therefore not a contradiction in the proper sense; it is just a meaningless phrase, equivalent to a made-up nonsense word. Therefore, it prevents the sentence from expressing a proposition.

But I appreciate that there can be differing views about this.
 
Emil
 
Reply Tue 29 Sep, 2009 06:17 am
@ACB,
Propositions
ACB;94220 wrote:
OK. Instead of saying that the consequent "asserts something necessarily false", let's say it "makes a necessarily false predication". In other words, it says something that stands in a necessarily false relation to its grammatical subject.


I don't know what this means.

ACB;94220 wrote:
I agree that contradictions can be senseful. However, let me try to clarify the distinction I was making.

"Some bachelors are married" is meaningful, but necessarily false because it is a contradiction. The phrases "some bachelors" and "are married" both make sense, and they are combined in a grammatically correct way (unlike your "green 3" example).


Technically I didn't state a sentence involving "green" and "3". I just stated a phrase. The idea was that the readers would understand the concept of being green and the concept of being the number 3.

It is easy to make an example sentence with these two concepts:

[INDENT]E1. All green 3's are equal to 5.
[/INDENT]
(E1) does not express a proposition because the sentence involves a category mistake/error.


ACB;94220 wrote:
But what about "Some married bachelors have black hair"? This purports to say something about married bachelors. Does it say something true about them, or something false about them? Neither, since there are no married bachelors.


I disagree. I think the sentence is just false. I interpret it like this:

[INDENT]E2. (∃x)(Mx∧Bx∧Hx) [With the obvious meanings of the predicates]
[/INDENT]
And since there is not at least one x that has all these three properties, then it follows that the proposition expressed by (E2) is false.

ACB;94220 wrote:
As I see it, a contradiction is a statement, and you cannot have a statement without a verb. "Married bachelors" is therefore not a contradiction in the proper sense; it is just a meaningless phrase, equivalent to a made-up nonsense word. Therefore, it prevents the sentence from expressing a proposition.


I agree that all verbs have a verb.

Questions

  1. What do you mean by "statement"?
  2. Do you think that sentences/statements are the bearers of truth and falseness?
  3. Do you think that "round earth" is a meaningless phrase too? That would be highly in contrast to common sense.


Note. I use "sentence" and "statement" interchangeably.

Concepts
"Married bachelor" is not a sentence at all. It is nearly concept expressing. If we make it into an open sentence it will be concept expressing:

[INDENT]E3. ... is a married bachelor.

[/INDENT](E3) is concept expressing.


A concept can be a contradiction too. Quoting a textbook on logic:

[INDENT]Concept C is necessarily nonapplicable (a self-contradictory concept, as we usually say) if and only if there is no possible world in which C has application to some item or other. (Possible Worlds p. 90)[/INDENT]
Meta-discussional remarks
ACB;94220 wrote:
But I appreciate that there can be differing views about this.


Sure. We have moved away from the area where there is nearly complete consensus. I love debating the philosophy of logic, propositions, sentences and related concepts. Wink
 
kennethamy
 
Reply Tue 29 Sep, 2009 06:55 am
@ACB,
ACB;94220 wrote:

But what about "Some married bachelors have black hair"? This purports to say something about married bachelors. Does it say something true about them, or something false about them? Neither, since there are no married bachelors. As I see it, a contradiction is a statement, and you cannot have a statement without a verb. "Married bachelors" is therefore not a contradiction in the proper sense; it is just a meaningless phrase, equivalent to a made-up nonsense word. Therefore, it prevents the sentence from expressing a proposition.



What you are asking is about sentences whose subjects do not exist. A famous one is Russell's, "The present king or France is bald" when there is no current king of France. (It is too bad that this famous example is spoiled by the misuse of "present" for "current"). Russell's view (and Emil's and mind) about such sentences is that they are false because they imply that there is something the subject term denotes, and there is no such thing. So. since there is no current king of France, it is false that the current king of France is bald (notice the difference between saying that it is false that the current king of France is bald, and saying that the current kind of France is not bald). And thus, some married bachelors have black hair is also false, because nothing is a married bachelor. "Married bachelor" could not be meaningless, since the proposition, "married bachelors have black hair" is false, and no false statement can be meaningless. Besides, there are no married bachelors is true, and what is true cannot be meaningless. (I use "statement" and "proposition" synonymously, with sentences expressing propositions/statements. Sentences are meaningful/meaningless. Propositions/statements, are true or false.

P.S. It you say that sentences whose subjects do not exist are meaningless, what would you say about the sentence, "Unicorns do not exist"? I hope you think that sentence expresses a true proposition!
 
Emil
 
Reply Tue 29 Sep, 2009 07:14 am
@kennethamy,
kennethamy;94242 wrote:
What you are asking is about sentences whose subjects do not exist. A famous one is Russell's, "The present king or France is bald" when there is no current king of France. (It is too bad that this famous example is spoiled by the misuse of "present" for "current"). Russell's view (and Emil's and mind) about such sentences is that they are false because they imply that there is something the subject term denotes, and there is no such thing. So. since there is no current king of France, it is false that the current king of France is bald (notice the difference between saying that it is false that the current king of France is bald, and saying that the current kind of France is not bald). And thus, some married bachelors have black hair is also false, because nothing is a married bachelor. "Married bachelor" could not be meaningless, since the proposition, "married bachelors have black hair" is false, and no false statement can be meaningless. Besides, there are no married bachelors is true, and what is true cannot be meaningless. (I use "statement" and "proposition" synonymously, with sentences expressing propositions/statements. Sentences are meaningful/meaningless. Propositions/statements, are true or false.

P.S. It you say that sentences whose subjects do not exist are meaningless, what would you say about the sentence, "Unicorns do not exist"? I hope you think that sentence expresses a true proposition!


---------- Post added 09-29-2009 at 05:10 PM ----------

I wrote an essay about this and posted it in the forum, so we can stop the discussion of missing subjects in here. Or, the mods can move the posts to my new thread.
 
kennethamy
 
Reply Tue 29 Sep, 2009 09:15 am
@Emil,
Emil;94245 wrote:


---------- Post added 09-29-2009 at 05:10 PM ----------

I wrote an essay about this and posted it in the forum, so we can stop the discussion of missing subjects in here. Or, the mods can move the posts to my new thread.


Yes. It is a matter of the scope or the negation. I mentioned this point in my reply to ACB.

In fact, "Unicorn do not exist" asserts that its subject is missing, so it is true because it is true. Negative existentials are very interesting specimens.
 
ACB
 
Reply Tue 29 Sep, 2009 03:50 pm
@kennethamy,
So you are both basically arguing (1) that an expression (e.g. "married bachelor" or "the current king of France") need not refer to anything in order to have meaning and thus form part of a proposition; (2) that an internal contradiction does not prevent an expression from having meaning; but (3) that the individual words of the expression need to make sense, so we cannot substitute nonsense words. And there are borderline cases as regards meaningfulness.

Is that right? If so, I'll settle for it.
 
Emil
 
Reply Tue 29 Sep, 2009 04:16 pm
@ACB,
ACB;94316 wrote:
So you are both basically arguing (1) that an expression (e.g. "married bachelor" or "the current king of France") need not refer to anything in order to have meaning and thus form part of a proposition; (2) that an internal contradiction does not prevent an expression from having meaning; but (3) that the individual words of the expression need to make sense, so we cannot substitute nonsense words. And there are borderline cases as regards meaningfulness.

Is that right? If so, I'll settle for it.


Yes, with a possible reservation about (3). The following sentence is meaningful:

[INDENT]E1. "Balsadlkjnsad" is not a sentence.

[/INDENT]But it involves a meaningless 'word'*.

Proper name theory
At least in some sense. It may be said that "Balsadlkjnsad" is a proper name, and thus there is no problem with ""Balsadlkjnsad"" as it can be a proper name even though "Balsadlkjnsad" isn't meaningful. Pay close attention to the number of quotation marks.

It may be said that most people would not know what ""Balsadlkjnsad"" referred to if they heard it, but so it is too with many proper names.
It may also be said that many people would not recognize it as a proper name, but then so it is with many proper names. Especially in the information/internet age where there are all kinds of nicknames.

*. "word" is sometimes defined as being meaningful. I don't mean that definition.

Small fix
An alternative fix is just to add "unless the meaningless word in in quotation marks." to the thesis.
 
kennethamy
 
Reply Wed 30 Sep, 2009 06:51 am
@ACB,
ACB;94316 wrote:
So you are both basically arguing (1) that an expression (e.g. "married bachelor" or "the current king of France") need not refer to anything in order to have meaning and thus form part of a proposition; (2) that an internal contradiction does not prevent an expression from having meaning; but (3) that the individual words of the expression need to make sense, so we cannot substitute nonsense words. And there are borderline cases as regards meaningfulness.

Is that right? If so, I'll settle for it.



Yes, roughly. In general, reference and meaning ought to be distinguished. Contradictory sentences are meaningful since they are false, and they could not be false unless they were meaningful. Furthermore, the negations of contradictions are tautologies, and tautologies are true. It could not be that the mere operation of negation transforms a meaningless sentence into a meaningful sentence. Save for the kind of exception discussed by Emil, if a sentence is meaningful, then its individual terms are meaningful. But, of course, not conversely. "Proclivity drinks procrastination" is not a meaningful English sentence.
 
Emil
 
Reply Wed 30 Sep, 2009 09:42 am
@JeffD2,
kennethamy wrote:
It could not be that the mere operation of negation transforms a meaningless sentence into a meaningful sentence.


I will discuss this is a future essay I think.
 
ACB
 
Reply Thu 1 Oct, 2009 06:39 am
@Emil,
Returning to the main topic of this thread, i.e. "falsity implies anything", I wonder about the following. Consider this statement:

If Paris is in Germany, it is not the case that Paris is in Germany.

This conditional is apparently true, as it has a false antecedent. Does this mean that it is logically possible that Paris both is and is not in Germany, and hence logically possible for a contradiction to be true? I am referring purely to the logical form of the statement; this should be a separate matter from the facts of the case.

It seems to me that the answer is 'no', since there can be no possible world in which Paris both is and is not in Germany. But on the other hand, the sentence in bold has the form "If P, not-P", which logically implies "Possibly (P and not-P)", does it not?

How can this paradox be resolved?
 
kennethamy
 
Reply Thu 1 Oct, 2009 06:55 am
@ACB,
ACB;94563 wrote:
Returning to the main topic of this thread, i.e. "falsity implies anything", I wonder about the following. Consider this statement:

If Paris is in Germany, it is not the case that Paris is in Germany.

This conditional is apparently true, as it has a false antecedent. Does this mean that it is logically possible that Paris both is and is not in Germany, and hence logically possible for a contradiction to be true? I am referring purely to the logical form of the statement; this should be a separate matter from the facts of the case.

It seems to me that the answer is 'no', since there can be no possible world in which Paris both is and is not in Germany. But on the other hand, the sentence in bold has the form "If P, not-P", which logically implies "Possibly (P and not-P)", does it not?

How can this paradox be resolved?


I don't understand why you think there is a contradiction? The expression is not, Paris is in Germany, and Paris is not is Germany. That would be a contradiction. But why would you think that the expression, If Paris is in Germany, it is not the case that Paris is in Germany is a contradiction? And no, I don't see why you would think that the expression in question implies, "Possibly (P and not-P)".
 
Emil
 
Reply Thu 1 Oct, 2009 07:15 am
@ACB,
ACB;94563 wrote:
Returning to the main topic of this thread, i.e. "falsity implies anything", I wonder about the following. Consider this statement:

If Paris is in Germany, it is not the case that Paris is in Germany.

This conditional is apparently true, as it has a false antecedent. Does this mean that it is logically possible that Paris both is and is not in Germany, and hence logically possible for a contradiction to be true? I am referring purely to the logical form of the statement; this should be a separate matter from the facts of the case.

It seems to me that the answer is 'no', since there can be no possible world in which Paris both is and is not in Germany. But on the other hand, the sentence in bold has the form "If P, not-P", which logically implies "Possibly (P and not-P)", does it not?

How can this paradox be resolved?
http://img143.imageshack.us/img143/6517/shot00101oct091510.jpg
 
kennethamy
 
Reply Thu 1 Oct, 2009 07:32 am
@Emil,
Emil;94570 wrote:
http://img143.imageshack.us/img143/6517/shot00101oct091510.jpg



In any case, (P>~P) is true, if P is false, and so, could not be a contradiction. It is also L-equivalent to ~(P & ~~P) and that is certainly not a contradiction. Therefore, (P > ~P) is not a contradiction.
 
Emil
 
Reply Thu 1 Oct, 2009 07:49 am
@Emil,
Giving it more thought. I can prove that you are wrong. Here is the possible worlds diagram for your claim:
http://img230.imageshack.us/img230/1011/shot00201oct091541.jpg

As you can see, the implication is false. Recall the truth condition for a logical/strict implication:

[INDENT]A proposition P implies a proposition Q if and only if there is no possible world where P is true and Q false. (Possible Worlds, p. 31)

[/INDENT]As it can be seen on the worlds diagram. There is at least one world where P is true and Q is false. Q is always false, of course, it is a contradiction.

---------- Post added 10-01-2009 at 03:50 PM ----------

kennethamy;94574 wrote:
In any case, (P>~P) is true, if P is false, and so, could not be a contradiction. It is also L-equivalent to ~(P & ~~P) and that is certainly not a contradiction. Therefore, (P > ~P) is not a contradiction.
 
ACB
 
Reply Thu 1 Oct, 2009 08:20 am
@kennethamy,
kennethamy;94567 wrote:
And no, I don't see why you would think that the expression in question implies, "Possibly (P and not-P)".


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.
 
 

 
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