Law of excluded middle

[carnap]

What is broader than classical logic?

The point of my comment was that not all logics are sub-logics of classical logic. But an example of a type of logic that would be "broader" than classical logic would be many-valued logic. Classical logic has just two truth values but you can formulate logics with 3 or more values, for example "true", "false", "undetermined".

[Cirion Spellbinder]

The point of my comment was that not all logics are sub-logics of classical logic. But an example of a type of logic that would be "broader" than classical logic would be many-valued logic. Classical logic has just two truth values but you can formulate logics with 3 or more values, for example "true", "false", "undetermined".

Wait, but doesn’t constructive logic have three truth values? I thought statements in constructive logic could either be true, false, or neither true nor false since LEM can’t be proved.

[carnap]

Wait, but doesn’t constructive logic have three truth values? I thought statements in constructive logic could either be true, false, or neither true nor false since LEM can’t be proved.

No, there are only two values in constructive logic. Truth and false. This was actually something proved long ago, there is no intermediate truth value in constructive logic. The law of the excluded middle can't be proved because you lack the reductio ad absurdum rule. The issue here is that the "or" operator (as well as the other operators) have different meaning in constructive logic. To assert the truth of falsity of a statement in constructive logic you need to have a proof (or disproof) of the statement, without a proof you cannot assert truth or falsehood. Its just unknown which is different than "neither true nor false".

The semantics for constructive logic are a lot more complex than classical logic. For example you don't have truth tables for propositional constructive logic.

[mkm]

But isn’t all science formulated based on classical logic, as in it used math which was formulated with classical logic?

That's probably why the quantum physics is so counterintuitive and many physicists started to flirt with the notion of multiversum, which really resembles semantics of possible worlds, i.e. the modal logic.

But we could also create an approximation of the probability of such an event given other observed criteria, no? And like I mentioned before, I believe the math of physics is all based on classical logic.

In the "reality"? Yeah, we do this all the time. The classical logic is just another tool to talk precisely (as much as possible at least...) about things around us, so one shouldn't get too attached to it, especially when it ceases to be that useful.

How does constructive logic solve this problem?

I'm not sure it does, I just referred to it to highlight that it's not a new problem, and it actually doesn't have clear-cut solution. We're roaming now in the land, where you can just assume some system and see if it suits you (or solves your problems; or it's aesthetically pleasing ). No one can prove that the classical logic, constructive logic, or any other logic, is "more valid".

[brimstoneSalad]

We reason about metaphysics which isn't the same as applying some formal logical system.

What do you think reason is?

"Logic gates" are based on classical operators.

Then there's your answer.

A logical system defines what is and isn't valid so defining a logical system that says "all invalid things is valid" makes no sense.

All things valid in another logical system are invalid, all things invalid in that system are valid; defined relatively.

A logic doesn't even need to be consistent,

It does in order to say anything meaningful.

Human reasoning doesn't represent a consistent system, that is, people often hold contradictory points of view.

I don't think you quite understand what a contradiction is, very rarely do people believe in direct contradictions, and when they do they're typically just unaware of the conflicting beliefs. Making them apparent usually results in feeling of cognitive dissonance.

There are a small minority of nutcases who think contradictions are fine. If you think they're right, that's your problem. I'm not having a debate over whether contradictions can be true or not.

The lack of the existence-property puts it at a disadvantage, part of "describing reality" would be the ability to construct objects that exist in the "reality".

What do you think existence is? What do you think reality is?

[brimstoneSalad]

But can’t we formulate a mathematical description of probability and gradients based on classical logic?

Maybe.

Could you elaborate? Doesn’t probability based on mathematics based on classical logic already exist?

I’ll check out what you’ve said otherwise.

I think it requires some axioms, that it's not a simple derivation from the laws of logic alone. Not sure.

[carnap]

What do you think reason is?

The human ability to think about abstract notions. Our ability for reason should not be mistake for logic, a logic is a formal system while our ability to reason is a fuzzy system riddled with bias and emotion.

Then there's your answer.

That doesn't answer the question, the classical "or" operator is used to connect two propositions not two logical systems. You'd have to define the operator for logical systems.

All things valid in another logical system are invalid, all things invalid in that system are valid; defined relatively.

The still doesn't make sense, there would be no away to evaluate the proposition "is valid in another system" within a given logical system.

It does in order to say anything meaningful.

Logics don't have to be "meaningful". What does it mean for a logic to be useful anyways? There are inconsistent logics that have applications, for example, models of belief revision.

quote=brimstoneSalad post_id=38700 time=1525213878 user_id=106]
I don't think you quite understand what a contradiction is, very rarely do people believe in direct contradictions, and when they do they're typically just unaware of the conflicting beliefs.
[/quote]
I didn't say anything about "direct contradictions", people hold beliefs that contradict in the sense that they are not logically compatible. Any model of human thinking has to account for the fact that people hold proposition as true that are contradictory. Whether or not the propositions obviously or directly conflict isn't the issue but rather that the beliefs, take as a whole, represent an inconsistent system.

[Cirion Spellbinder]

That's probably why the quantum physics is so counterintuitive and many physicists started to flirt with the notion of multiversum, which really resembles semantics of possible worlds, i.e. the modal logic.

Is that a problem?

In the "reality"? Yeah, we do this all the time. The classical logic is just another tool to talk precisely (as much as possible at least...) about things around us, so one shouldn't get too attached to it, especially when it ceases to be that useful.

Don't you think its more useful to have one broad system that can explain all things rather than two disconnected ones that can only explain things in their domain (e.g. quantum physics and relativity)?

I'm not sure it does, I just referred to it to highlight that it's not a new problem, and it actually doesn't have clear-cut solution.

Why is it a problem? And if it is, what is wrong with Aristotle's solution? Also I'm going to harass @brimstoneSalad because I know he is also familiar with physics and may have an interesting solution.

No one can prove that the classical logic, constructive logic, or any other logic, is "more valid".

I suspect that you can't prove that either. I'm inclined to favor the proposition that you can, as I can't see any sense in living a life without being able to deduce things. That seems like absolute insanity and how could I even reach such a conclusion without deduction?

[Cirion Spellbinder]

Given what you've said @carnap I'm somewhat inclined to believe that the broadest possible system is the best for describing reality. Is there any evidence of an upper limit on the number of possible truth values? If not, I'd imagine there would have to be a practical limit since its probably harder to develop proof rules for a consistent system as we introduce more truth values.

@brimstoneSalad, why do you stop at classical logic if you can go broader?

[brimstoneSalad]

The human ability to think about abstract notions.

Quite the anthropocentric definition.

Our ability for reason should not be mistake for logic, a logic is a formal system while our ability to reason is a fuzzy system riddled with bias and emotion.

See, that's not how I define reason; bias and emotion have no place in reason, that's something else.

That doesn't answer the question,

I think it pretty clearly does when we look at the output. Not sure what you're confused about.

The still doesn't make sense, there would be no away to evaluate the proposition "is valid in another system" within a given logical system.

I don't know how you come to that conclusion.

Just define the system you want to negate and throw a not operator on the end to reverse the results.

There are inconsistent logics that have applications, for example, models of belief revision.

For dealing with inconsistencies maybe; not an issue we have to contend with in the case of facts in actual reality vs. faulty models of it.

I didn't say anything about "direct contradictions", people hold beliefs that contradict in the sense that they are not logically compatible. Any model of human thinking has to account for the fact that people hold proposition as true that are contradictory. Whether or not the propositions obviously or directly conflict isn't the issue but rather that the beliefs, take as a whole, represent an inconsistent system.

What's your point then?

If it's not a direct contradiction, we should have no trouble dealing with it in classical logic given appropriate axioms about how the mind processes these models.