Law of excluded middle

[Cirion Spellbinder]

I'm not sure what brimstone is trying to say because they don't appear to be using the common mathematical definitions of notions like "consistent", etc. Many-valued logics can be consistent, a logical system is concerned "consistent" so long is that there isn't any statement that can be both proved and disproved.

What does it mean to disprove something in many valued logic then? I have only ever heard it with respect to proving a contradiction.

In fact quantum mechanics seems to show that there are statements that are neither true or false.

What about modal logic in addition to classical logic instead?

So it would seem that classical logic is more a representation of how we think about the world rather than reality itself. I think this is an interesting issue that often gets ignored, namely, the limitation to human thought. Our ability of reason was crafted evolutionary to solve real world problems....not research the nature of the universe. It may very well be that our modes of thought will limit what we can discover about the world. Our entire system of science may just be set of useful anthropocentric notions that has no connection with "reality".

This is a worrisome notion which is what I was trying to allude to @brimstoneSalad before with many valued logic. Do you think it is provable and if so, why, and do you think genetic modification can change that?

[Cirion Spellbinder]

It's not classical anymore. It becomes a problem if one wants to formulate everything in classical logic and keep the correspondence with the reality.[...]Aristotle's solution is basically intruducing modal logic. Nonclassical.

I don't think this is a problem given that modal logic is an extension of classical logic. I haven't, for example, rejected Geometry because it introduces new axioms because it is nonetheless classically based.

Sure, but the reality trumps our wants every time. I would rather have two disconnected which are useful in their domains, rather than one, which is only partially ok.

Agreed, but do you think evidence of that exists?

Any attempt of proof would be circular at best, so it doesn't really make sense.

What about a recursive proof? I think we can conserve logic if we have our logic nested in a broader logic with additional axioms justifying ours, nested in another justifying its, and so on. That is probably not demonstrable, but it is one way I could conceive of a single logic applying to us without a circular proof.

Anyway, you misunderstood me. I don't say we should abandon logic (any) altogether. I say we should be aware that there is no one and only "true" formal system in which we should formulate our thoughts.

If we can formulate our thoughts in more than in any system of logic, then discourse is incoherent. For example, if I remove the proof rules for negation from propositional logic, I can create a logic where every proposition is true.

[brimstoneSalad]

Quite the anthropocentric definition.

When talking about human reasoning, yes, its going to be an anthropocentric definition. Again "reason" shouldn't be mistaken for a logical system, human thinking doesn't represent a logical system. Human reasoning isn't the application of a set of inference rules, its much more than that.

If it is said that humans have moral value because of "reason™", and "reason™" is defined as an arbitrary standard of specifically human thought process, all you're really doing is begging the question.

Emotions are a critical component to our ability to reason.

No.
You need emotion to motivate reasoning (as it's needed to motivate any behavior), but reason as a tool works without emotion influencing its process.

Reasoning about the real world is typically done with imperfect information where conclusions, in the deductive sense, cannot easily be made.

So?
You can use reason without perfect information. Statistics is based on that; we can understand the probability of X observation being by chance, we can also reason based on induction without other information as long as we understand its limits.

The other comments you made in your post make the mistake of assuming that logical operators have a semantic independently of some logical system. The meaning of the "or" operator hinges on a specific logic.

They have some meaning in English. Nitpicking this isn't useful unless you're going to say something substantive to contradict my explanations.

If you don't like the definitions, provide your own and argue for their superiority.

[brimstoneSalad]

@carnap, is what I said to @brimstoneSaladcorrect (at least in the tripartite case)? It contradicts what you said early about systems with more truth values being broader.

I'm not sure what brimstone is trying to say because they don't appear to be using the common mathematical definitions of notions like "consistent", etc. Many-valued logics can be consistent, a logical system is concerned "consistent" so long is that there isn't any statement that can be both proved and disproved.

With respect to notions like dialetheism, that's precisely the kind of inconsistency you find. A statement can be true and not false, as well as false and not true. Real contradictions in the strictest logical sense are acceptable to them.

In terms of metaphysics, there is no obvious reason why every statement should be either true or false. In fact quantum mechanics seems to show that there are statements that are neither true or false.

Borderline Deepak Chopra nonsense.

Wave mechanics aren't magical things that invalidate logic and give credence to contradiction. Asking what state a wave is in is simply a misleading question, it's more like asking what breed of dog a car is than asking anything profound or relevant to logic.

Superposition is philosophically interesting due to the interpretation which tell us a LOT about metaphysics (primarily invalidating theism), but the moment you *measure* it, it has truth value relative to your reference frame/universe.

So it would seem that classical logic is more a representation of how we think about the world rather than reality itself.

No, it would not seem that way. But the fact that you think that gives some insight into why you're so resistant to logical arguments.

I think this is an interesting issue that often gets ignored, namely, the limitation to human thought.

It's an uninteresting issue that's ignored for a reason: if logic isn't true, there's no point to thinking about anything because it's all inaccessible to us.

Either we can or can not comprehend reality. If we can, then our actions have meaning and we need to consider very seriously moral realism. If we can't then nothing matters. It's the ultimate Pascal's wager: one that actually works because it's not a false dichotomy.

Our entire system of science may just be set of useful anthropocentric notions that has no connection with "reality".

[mkm]

It was meant as an analogy for tripartite systems. If it is a false analogy, tell me (and why) so I can tell him.

I've done it already, I don't know how to be more clear. I pointed out that even the analogy to integers doesn't work.

Is this incompleteness theorem?

Yup.

I don't think this is a problem given that modal logic is an extension of classical logic. I haven't, for example, rejected Geometry because it introduces new axioms because it is nonetheless classically based.

In what sense it extends the classical logic? "Old" sentences have now new meaning, and there are new sentences that are uninterpretable in the classical logic. What do you mean?
For what purpose you should or shouldn't reject Euclidean geometry (id that's what you are reffering to by Geometry)? Classical geometry and, let's say, geometry in relativity, are incompatible. You have to reject one to have the other.

Agreed, but do you think evidence of that exists?

What kind of eveidence do you expect? The collaps of the classical logic in the quantum physics is the best eveidence you will propably get. There are no deductive proofs in experimental sciences.

What about a recursive proof? I think we can conserve logic if we have our logic nested in a broader logic with additional axioms justifying ours, nested in another justifying its, and so on. That is probably not demonstrable, but it is one way I could conceive of a single logic applying to us without a circular proof.

You need to assume some sort of language and logic to have meaningful proofs in the first place. That's why it becomes circular before you even proceed.

If we can formulate our thoughts in more than in any system of logic, then discourse is incoherent. For example, if I remove the proof rules for negation from propositional logic, I can create a logic where every proposition is true.

Not really, it just means that we may discuss things in many different frameworks and come to different incompatible conclusions. We just need to establish the framework to communicate properly. Usually classical logic is enough, sometimes it's not, depends on what do we try to discuss.

[Cirion Spellbinder]

I've done it already, I don't know how to be more clear. I pointed out that even the analogy to integers doesn't work.

So then do you think that you can't break down a tripartite problem into a bipartite one or just that it wouldn't work on non-arithmetic problems?

Yup.

Would you humor question about incompleteness theorem from somebody who knows nothing of it besides the phrase "not all statements in arithmetic can be proven in arithmetic?"

In what sense it extends the classical logic? "Old" sentences have now new meaning, and there are new sentences that are uninterpretable in the classical logic. What do you mean?

As you've picked up on, I'm not really familiar with modal logic. I understand that it adds new modal operators for necessity and possibility. How would adding new operators give classical sentences new meaning? I understand that you wouldn't be able to interpret sentences with modal parts in classical logic though. Is that what you mean?

For what purpose you should or shouldn't reject Euclidean geometry (id that's what you are reffering to by Geometry)? Classical geometry and, let's say, geometry in relativity, are incompatible. You have to reject one to have the other.

I agree with this.

What kind of eveidence do you expect? The collaps of the classical logic in the quantum physics is the best eveidence you will propably get. There are no deductive proofs in experimental sciences.

Is deduction limited to classical logic?

You need to assume some sort of language and logic to have meaningful proofs in the first place. That's why it becomes circular before you even proceed.

Would it help if that language was justifying by the language containing it and so on?

Not really, it just means that we may discuss things in many different frameworks and come to different incompatible conclusions. We just need to establish the framework to communicate properly. Usually classical logic is enough, sometimes it's not, depends on what do we try to discuss.

Won't the same problems of selecting one logic transfer to the selection of a proper framework of communication if it is to not be arbitrary?

[brimstoneSalad]

What kind of eveidence do you expect? The collaps of the classical logic in the quantum physics is the best eveidence you will propably get. There are no deductive proofs in experimental sciences.

Why do you believe that quantum physics is a problem for classical logic?

[mkm]

@Cirion Spellbinder

So then do you think that you can't break down a tripartite problem into a bipartite one or just that it wouldn't work on non-arithmetic problems?

I guess it's the first one, although I'm not sure what do you mean by "breaking down a tripartite problem into a bipartite one". Is for example the problem with the law of excluded middle "break-downable"?

Would you humor question about incompleteness theorem from somebody who knows nothing of it besides the phrase "not all statements in arithmetic can be proven in arithmetic?"

I may try, but I'm not an expert.

As you've picked up on, I'm not really familiar with modal logic. I understand that it adds new modal operators for necessity and possibility. How would adding new operators give classical sentences new meaning? I understand that you wouldn't be able to interpret sentences with modal parts in classical logic though. Is that what you mean?

The second part - yes. For the first part, let's take that Aristotle's statement s="tomorrow will be a sea battle". What I mean by that statements have now a new meaning is that if you approach with the classical logic, by granting a truth value to s you admit it's determined to happen (or not), that tomorrow will be a sea battle. In, for example, three valued logic it may have "maybe" value, and it's incompatible with the implications of the assignment in the classical logic.

Is deduction limited to classical logic?

Deduction is limited to formal systems. That these deductions are meaningful, i.e. they apply to the reality, it's the assumption.

Would it help if that language was justifying by the language containing it and so on?

That's propably the definition of the circular argument

Won't the same problems of selecting one logic transfer to the selection of a proper framework of communication if it is to not be arbitrary?

What's wrong with arbitrariness? The very rule itself "arbitrary rules are not allowed" is arbitrary, since the only way you could justify such a rule would be by a rule that implies "arbitrary rules are not allowed".

@brimstoneSalad

Why do you believe that quantum physics is a problem for classical logic?

What's the truth value of "upon the measurement the electron's spin along the certain angle will be X"?

[Cirion Spellbinder]

I guess it's the first one, although I'm not sure what do you mean by "breaking down a tripartite problem into a bipartite one". Is for example the problem with the law of excluded middle "break-downable"?

Is the problem with the law of excluded middle evaluable in a tripartite system?

I may try, but I'm not an expert.

Can you prove things in arithmetic from things not relying on or being arithmetic?

The second part - yes. For the first part, let's take that Aristotle's statement s="tomorrow will be a sea battle". What I mean by that statements have now a new meaning is that if you approach with the classical logic, by granting a truth value to s you admit it's determined to happen (or not), that tomorrow will be a sea battle. In, for example, three valued logic it may have "maybe" value, and it's incompatible with the implications of the assignment in the classical logic.

Ah, okay. I see the problem now. Is modal logic more than two valued?

Deduction is limited to formal systems. That these deductions are meaningful, i.e. they apply to the reality, it's the assumption.

There are no deductive proofs in experimental sciences.

I don't think this means that logic cannot describe these things afterwards, given that induction works.

That's probably the definition of the circular argument

It is my understanding that circular arguments go somewhat like A→B, B→C, C→A to substantiate A, whereas I am proposing to substantiate An because [...]An-3→An-2→An-1→An.

What's wrong with arbitrariness? The very rule itself "arbitrary rules are not allowed" is arbitrary, since the only way you could justify such a rule would be by a rule that implies "arbitrary rules are not allowed".

A better rule is no arbitrary rules but this one are permitted. If we want any coherence, we need to permit the least amount of assumptions which we only value because we believe these assumptions will permit the most order. That is of course an arbitrary interest by an arbitrary organism with many other arbitrary characteristics, but concessions must be made for the first premises.

And if nothing is wrong with arbitrariness, I'm sure you'd be willing to arbitrarily concede to use classical logic?

[brimstoneSalad]

@brimstoneSalad

Why do you believe that quantum physics is a problem for classical logic?

What's the truth value of "upon the measurement the electron's spin along the certain angle will be X"?

Your question/statement lacks the necessary precision.

X must be: spin up in one universe (we'll call it the "up" universe) and spin down in another universe (we'll call it the "down" universe).

If you can be more precise, E.g. What's the truth value of "upon the measurement IN THE UP UNIVERSE the electron's spin along the certain angle will be X"?
That's easy to answer. It's tautological, even.