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.

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?

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?

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.

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 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.

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!

---------- 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.

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.

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.

It could not be that the mere operation of negation transforms a meaningless sentence into a meaningful sentence.

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?

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?

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.

And no, I don't see why you would think that the expression in question implies, "Possibly (P and not-P)".