Are the laws of logic important to theology?

essentialsaltes said:

Conclusion:
The LEM is false. It can be used where it's sensible to use it. But there are places where it is unsensible to use it.

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I don't think that argument holds water, but it is a fact that certain approaches to mathematics toss out the LEM. See http://en.wikipedia.org/wiki/Constructivism_(mathematics)

zippy2006 said:

I don't think those systems do anything of the sort. They are just symptoms of a confused analytical philosophy that misunderstands what is meant by the law of identity, etc. This misunderstanding is apparently also present in essentialsaltes. ...But feel free to actually set out an argument in their favor.

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The fact they exist and work to solve problems in specific fields is all the argument I need. That shows what the axioms in the OP are not required by all logical systems.

Heck, even if they weren't useful and we just internally consistent that would be enough to show that you're wrong.

essentialsaltes said:

Either the present king of France is bald, or he is not bald (LEM)

Lemma1
If someone is bald, they will appear in the set of bald people.
The present King of France is not in the set of bald people.
Therefore the present King of France is not bald.

Lemma2
If someone is not bald, they will appear in the set of not bald people.
The present King of France is not in the set of not bald people.
Therefore the present King of france is not not bald.

Conclusion:
The LEM is false. It can be used where it's sensible to use it. But there are places where it is unsensible to use it.

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I do not see how those two "lemmas" prove the LEM is false. The first one, if all of the premises are true, would prove the King of France is bald. The second, if all of the premises are true, would prove the Kind of France is not bald. There is not inconsistency in that.

Obviously, not all of those premises can be true. In particular, the following cannot both be true:

The King of France is not in the set of bald people.
The King of France is nto in the set of not bald people.

essentialsaltes said:

Either the present king of France is bald, or he is not bald (LEM)

Lemma1
If someone is bald, they will appear in the set of bald people.
The present King of France is not in the set of bald people.
Therefore the present King of France is not bald.

Lemma2
If someone is not bald, they will appear in the set of not bald people.
The present King of France is not in the set of not bald people.
Therefore the present King of france is not not bald.

Conclusion:
The LEM is false. It can be used where it's sensible to use it. But there are places where it is unsensible to use it.

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...thanks for providing an actual argument.

Propositions require determinate subjects and predicates by definition. "The present king of France" is not a determinate entity, indeed it is a non-existent entity. In other words, nonsensical sentences are not truth-evaluative and thus not propositions at all.

Your set-theory also points to this fact. You say, "If someone is bald..." This "someone" limits the universe of discourse to existing persons. In other words, the implicit quantifier is "For all existing persons..." ∀x(P(x) -> (B(x) v ~B(x))) Either the present king of France counts as a "someone" or he does not. If he does, then the first line of your lemma is false; if he doesn't, then your syllogism is invalid by equivocation.

Predication of non-existent things is vacuous. This is because "The king of France is not" ...period. We can go on to include any mode of being in that non-existence. ...not blue, not red, not non-blue, not bald, not non-bald, not the king of France, etc.

This is also borne out by the more direct application of the LEM, which would require us to call the proposition, "The king of France is bald," false. But is it even proper to call such a proposition false? Certainly not without ambiguity, for it is not false in virtue of the king's baldness, which is where the predication derives its meaning, but rather in virtue of his non-existence. Since the truthmaker is his lack of existence, the corresponding negation must include his existence. But the negation you give pretends that the truthmaker is his lack of baldness, and negates that attribute. The relevance of the LEM here deals with his existence, not his baldness, for that is whence the truth or falsity of the statement derives. This commonly-used example is sophistical in the traditional sense.

Finally, it is worth noting that your example makes use of the LEM explicitly, thus assuming it:

The present King of France is not in the set of bald people.
Therefore the present King of France is not bald.

The present King of France is not in the set of not bald people.
Therefore the present King of france is not not bald.

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...why suppose that he must reside in the complementary set? The reason is the LEM itself. At the very least you would have to explain why the LEM applies here and not there.

KCfromNC said:

The fact they exist and work to solve problems in specific fields is all the argument I need. That shows what the axioms in the OP are not required by all logical systems.

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I already explained this. Non-classical logical systems exist, but being non-classical does not mean they avoid use of the law of identity. Again: feel free to demonstrate the contrary.

Conscious Z said:

I said his argument was valid but not sound. The unsound premise is so obvious it doesn't need to be said: My opinions are facts.

There was nothing that needed clarification in my post. By "true," I obviously meant the conclusion, and conclusions are true or false.

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I didn't mean to nitpick, but it becomes important in some instances. For example, you here talk about an unsound premise. But premises are true or false. Validity is a structural concept and soundness is concerned both with structure and truth (of premises). Truth has to do with reality, not structure. It is not a relation between premises and inferences but a relation between reality and a knowing subject.

Similarly, saying that a conclusion is true is very different from saying that a syllogism is sound. In fact a syllogism can be unsound while its conclusion remains true. An argument is a thing that is meant to present the conclusion as a secondary consideration that follows of necessity from other, more primary (and better-known) considerations.

...just wanted to make sure everyone was on the same page since you went ahead and started explaining these concepts.

zippy2006 said:

I didn't mean to nitpick, but it becomes important in some instances. For example, you here talk about an unsound premise. But premises are true or false. Validity is a structural concept and soundness is concerned both with structure and truth (of premises). Truth has to do with reality, not structure. It is not a relation between premises and inferences but a relation between reality and a knowing subject.

Similarly, saying that a conclusion is true is very different from saying that a syllogism is sound. In fact a syllogism can be unsound while its conclusion remains true. An argument is a thing that is meant to present the conclusion as a secondary consideration that follows of necessity from other, more primary (and better-known) considerations.

...just wanted to make sure everyone was on the same page since you went ahead and started explaining these concepts.

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I'm fully aware of what valid and sound both mean. I don't need a lesson.

In my initial post, I didn't mis-use either term. You literally chimed in for no good reason to explain a concepts that I used correctly.

Since you do seem to want to nitpick and offer unsolicited advice, however, perhaps I should correct you on this little diddy:

Truth has to do with reality, not structure. It is not a relation between premises and inferences but a relation between reality and a knowing subject.

A true proposition does not require a knowing subject in order for it to be true. For example, when it was first discovered that Mt. Everest was the tallest mountain in the world, it didn't suddenly become true that "Mt. Everest is the tallest mountain in the world." It was already true, we just didn't know it.

Truth is a relation between reality and proposition, not reality and subject.

Conscious Z said:

In particular, the following cannot both be true:

The King of France is not in the set of bald people.
The King of France is nto in the set of not bald people.

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Why can't they both be true? I submit that they are both true. Hint: there is no King of France. So he will not appear on any list of people.

So either the LEM is not always true, or you have to (like zippy) make some rules about some statements not really being statements. The LEM always works, except when it doesn't.

zippy2006 said:

Finally, it is worth noting that your example makes use of the LEM explicitly, thus assuming it:

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Is that not what a reductio ad absurdum is? You assume A, show that it leads to a contradiction, therefore ~A.

essentialsaltes said:

Why can't they both be true? I submit that they are both true. Hint: there is no King of France. So he will not appear on any list of people.

So either the LEM is not always true, or you have to (like zippy) make some rules about some statements not really being statements. The LEM always works, except when it doesn't.

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It is logically impossible for them to both be true at the same time. There need not actually be a referent for "King of France" for that to be the case. Similarly, X cannot both equal Y and equal "not-y." The question of whether X has a referent is irrelevant to the logic.

Conscious Z said:

It is logically impossible for them to both be true at the same time. There need not actually be a referent for "King of France" for that to be the case. Similarly, X cannot both equal Y and equal "not-y." The question of whether X has a referent is irrelevant to the logic.

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Well, reasoning with empty classes notoriously causes problems.

But in THIS case, your reductio has merely proved that there is no present king of France.

"The present king of France is bald" is not a proposition (since it carries within itself a false assumption).

Architeuthus said:

Well, reasoning with empty classes notoriously causes problems.

But in THIS case, your reductio has merely proved that there is no present king of France.

"The present king of France is bald" is not a proposition (since it carries within itself a false assumption).

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This is bringing back all sorts of hazy memories of my analytic tradition class. I remembered that Russell wrote a good bit about this, and that he argued the King of France example didn't disprove LEM (and that it was a proposition). After doing some research, I saw that Wiki has a pretty good rundown of his analysis: Definite description - Wikipedia, the free encyclopedia

Conscious Z said:

This is bringing back all sorts of hazy memories of my analytic tradition class. I remembered that Russell wrote a good bit about this, and that he argued the King of France example didn't disprove LEM (and that it was a proposition). After doing some research, I saw that Wiki has a pretty good rundown of his analysis: Definite description - Wikipedia, the free encyclopedia

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I'm not arguing with what the Wikipedia article says.

But back to my previous point; there are approaches to mathematics that toss out the LEM.

Conscious Z said:

It is logically impossible for them to both be true at the same time.

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Well, then in which set does the present King of France appear?

Conscious Z said:

It is logically impossible for them to both be true at the same time. There need not actually be a referent for "King of France" for that to be the case. Similarly, X cannot both equal Y and equal "not-y." The question of whether X has a referent is irrelevant to the logic.

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Actually, it depends upon your logical system and who you are. Aristotle, for example, believed the statement "some cats are brown" implies the existence of some brown cats. It also implies "some cats are not brown". However, modern logic tries to find ways around the existential import.

zippy2006 said:

I already explained this. Non-classical logical systems exist, but being non-classical does not mean they avoid use of the law of identity. Again: feel free to demonstrate the contrary.

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Are you backtracking to just the law of identity here? Because all along you've been talking about other axioms as well - noncontradiction, etc.

Chany said:

Actually, it depends upon your logical system and who you are. Aristotle, for example, believed the statement "some cats are brown" implies the existence of some brown cats. It also implies "some cats are not brown". However, modern logic tries to find ways around the existential import.

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I would have said that it was the other way around; modern logic puts emphasis on the existential import, and then tries to address the problems with counterexamples to ∀x∈S. P(x) => ∃x∈S. P(x), such as that "all unicorns have horns" does not imply "some unicorns have horns."

Architeuthus said:

I would have said that it was the other way around; modern logic puts emphasis on the existential import, and then tries to address the problems with counterexamples to ∀x∈S. P(x) => ∃x∈S. P(x), such as that "all unicorns have horns" does not imply "some unicorns have horns."

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I'm referring to the later situation when I use the term "get around". They try to find ways of eliminating the problem or creating a system of logic that does not have to deal with the problem. Modern logic does look into the existential import, yes.

essentialsaltes said:

Is that not what a reductio ad absurdum is? You assume A, show that it leads to a contradiction, therefore ~A.

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Yes, but this was your assumption that you reduced to an absurdity:

essentialsaltes said:

Either the present king of France is bald, or he is not bald (LEM)

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The instances that I pointed out, on the other hand, were actually presented as valid inferences within your reasoning. That is to say, they are necessary parts of your reductio argument. Yet these necessary parts rely on the LEM themselves, so if you undermine the LEM you undermine your own argument against the LEM (because you argument relies upon it).

I gave a number of objections, but at one point I said, "If [the king of France does count as a 'someone'], then the first line of your lemma is false; if he doesn't, then your syllogism is invalid by equivocation." Since it seems to be the most obvious objection, I want to reiterate that point. Consider your first premise:

essentialsaltes said:

If someone is bald, they will appear in the set of bald people.

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Your argument assumes that he is a "someone," so we can work from that. First, the king of France is non-existent but still counts as a "someone." Second, the "set of bald people" is restricted to existing things (and more specifically, persons). So the effective meaning of your premise includes this as an implication:


...this is obviously false, as I alluded to in my last post. This counter-argument seems quite solid, but perhaps I am making some quantificational presupposition.

Architeuthus said:

But back to my previous point; there are approaches to mathematics that toss out the LEM.

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Ignoring all the points I have already made about formal systems, your Wikipedia source doesn't "toss out the LEM." It merely remains agnostic with respect to the LEM when dealing with infinite collections. And the LEM is tertiary in the whole scheme, for Brouwer doesn't believe all mathematical problems have solutions and "to Brouwer, the law of the excluded middle was tantamount to assuming that every mathematical problem has a solution." On top of all this, his whole move is clearly controversial and its legitimacy is unproven.

Conscious Z said:

Truth is a relation between reality and proposition, not reality and subject.

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Propositions exist in minds, hence my point about subjects. Indeed truth is more properly a relation between reality and a (knowing) subject than between reality and a mind-dependent proposition.

Conscious Z said:

It is logically impossible for them to both be true at the same time. There need not actually be a referent for "King of France" for that to be the case. Similarly, X cannot both equal Y and equal "not-y." The question of whether X has a referent is irrelevant to the logic.

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You're begging the question. The only reason such a thing is said to be "logically impossible" is the LEM itself (or its corollary, the LNC). Or, if you don't think you're begging the question, what principle other than the LEM are you appealing to?

KCfromNC said:

Are you backtracking to just the law of identity here? Because all along you've been talking about other axioms as well - noncontradiction, etc.

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No, that wasn't my intention. The law of identity is just an easier reference point since the other two more or less follow from it.

Chany said:

They try to find ways of eliminating the problem or creating a system of logic that does not have to deal with the problem.

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And many such formal systems that attempt to prescind from the question of actual existence simultaneously detach from reality itself (as is particularly possible in stipulative systems like theoretical mathematics and computer science). Thus when fundamental laws are understood in a metaphysical way these aforementioned formal systems cannot even undermine the fundamental laws in principle. They are negating formal representations of a fundamental law and folks like KCfromNC and Architeuthus take that to mean that they have negated the metaphysical law itself.

It's something like saying that the existence of a trinary computer undermines the LEM since the binary computer has traditionally modeled the LEM. "The fact that trinary computers exist and work to solve problems in specific fields is all the argument I need. That shows that the axioms in the OP are not required by all logical systems."

zippy2006 said:

your Wikipedia source doesn't "toss out the LEM." It merely remains agnostic with respect to the LEM when dealing with infinite collections. And the LEM is tertiary in the whole scheme, for Brouwer doesn't believe all mathematical problems have solutions and "to Brouwer, the law of the excluded middle was tantamount to assuming that every mathematical problem has a solution." On top of all this, his whole move is clearly controversial and its legitimacy is unproven.

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I think you misread that Wikipedia article: "Much constructive mathematics uses intuitionistic logic, which is essentially classical logic without the law of the excluded middle. This law states that, for any proposition, either that proposition is true or its negation is. This is not to say that the law of the excluded middle is denied entirely; special cases of the law will be provable. It is just that the general law is not assumed as an axiom. The law of non-contradiction (which states that contradictory statements cannot both at the same time be true) is still valid."

It is a fact that constructive mathematics tosses out the LEM as an axiom, and it is also a fact that this is a legitimate branch of mathematics (see, for example, this book). Today, it's more the connections to computer science than Brouwer's ideas that drive the field.