Law of excluded middle

[mkm]

A : Oswald hadn't killed Kennedy
B: Someone else would

A→B

We wan't to consider ¬(A→B) which is logically equivalent to A∧¬B
A B.........¬(A→B) || A∧¬B
T T.........F
T F.........T
F T.........F
F F.........F

I think that means if it is true that Oswald hadn't killed Kennedy and if it is not true that someone else would, then Oswald hadn't killed Kennedy and someone else wouldn't kill Kennedy

Ok, now what does it mean for B to be true or not? Is it even a sentence in classical logic? What I try to convey is that the sense of A→B is basicly that Kennedy had to die there and then, and the statements expressed in classical logic like "Kennedy was killed" and "Oswald killed Kennedy" (or maybe not ) don't give you a hint how to deal with such conditional statements. You mitigate it somehow tacitly assuming that B is a legit sentence, but is it?

[carnap]

Does it have a name so I can look into it?

Not really, its just a corollary of other key theorems. . If there was statement A such that |- A and ~A then then assuming the logic is sound that would mean that |= A and ~A which is contradictory since no model could satisfy it.

Couldn't it be said that logic describes the phenomena of valid argumentation?

This would be circular, a "valid argument" is defined by some respective logic. "Valid argument" isn't a natural phenomena that is being researched, its a definition with respect to a specific logic.

How do we know that all things can be constructed in intuitionistic logic? Also, why is it useful to construct things as opposed to simply have proof of their existence?

Because any proof of existence has to be established with a particular instance, the only alternative would be if your premises were inconsistent in which case you can derive anything. Take for example (E.x)P(x) (e.g., there exists an x such that P(x)), if you were going to prove that in construction logic you'd have to discover some "a" such that "P(a)". On the other hand in classical logic you could assume ~(E.x)P(x) and demonstrate a contradiction which would allow you to establish (E.x)P(x).

Its useful when you need to actually apply the theory. Just knowing that a function, property, etc exists doesn't give you the ability to apply the theory.

[brimstoneSalad]

This would be circular, a "valid argument" is defined by some respective logic. "Valid argument" isn't a natural phenomena that is being researched, its a definition with respect to a specific logic.

That's a pretty strong borderline-subjectivist metaphisical claim. Something doesn't have to be a "natural phenomena" to be objectively true.

We can regard the sum of all consistent systems to indicate what is valid in an objective sense. The broader systems would of course make the more narrow ones superfluous. Which means we shouldn't expect systems like constructive logic to be terribly useful in most cases since the scope is unnecessarily limited (at least in most applications).

The only place I see systems like that as being useful is in certain models of computation, and occasionally in convincing extremist skeptics of a proof (which if it can be made in such limited systems, may be less controversial to them... although it really shouldn't be controversial to any sensible person in classical logic).

[Cirion Spellbinder]

Ok, now what does it mean for B to be true or not? Is it even a sentence in classical logic? What I try to convey is that the sense of A→B is basicly that Kennedy had to die there and then, and the statements expressed in classical logic like "Kennedy was killed" and "Oswald killed Kennedy" (or maybe not ) don't give you a hint how to deal with such conditional statements. You mitigate it somehow tacitly assuming that B is a legit sentence, but is it?

Excuse my stupidity. Doesn't that just mean the natural language permits useless sentences?

[mkm]

Excuse my stupidity. Doesn't that just mean the natural language permits useless sentences?

Well, I don't know how about you, but I want conditional statements to make sense, and then seek logic that reflects that. And you have it backwards.

[Cirion Spellbinder]

Well, I don't know how about you, but I want conditional statements to make sense, and then seek logic that reflects that. And you have it backwards.

I think you’re right that I have it backwards.

Can you explain why it isn’t a sentence because I don’t really understand that?

[carnap]

That's a pretty strong borderline-subjectivist metaphisical claim. Something doesn't have to be a "natural phenomena" to be objectively true.

I'm not sure what you mean by "subjectivist metaphysical claim" but what does it mean for a logic to be objectively true? How do we determine which logic is objectively true?

The point of my comment is that "validity", in the context of logic, is just a definition. One can certainly make metaphysical claims about logical statements but you're no longer doing logic but instead metaphysics.

We can regard the sum of all consistent systems to indicate what is valid in an objective sense. The broader systems would of course make the more narrow ones superfluous. Which means we shouldn't expect systems like constructive logic to be terribly useful in most cases since the scope is unnecessarily limited (at least in most applications).

How do you sum two logical systems? Do you just combine the rules or do you just take the union of the set of all their provable statements? In either case the sum of two or more consistent logical systems isn't necessarily consistent. Also the set of consistent logical systems would be infinite, so we define what is objectively true by the sum of an infinite set of consistent logical systems?

Constructive logic is very useful because it has an existence-property. That is any constructive proof of the existence of an object or property can be used as a recipe to construct the object. Classical logic lacks this poverty.

The only place I see systems like that as being useful is in certain models of computation, and occasionally in convincing extremist skeptics of a proof (which if it can be made in such limited systems, may be less controversial to them... although it really shouldn't be controversial to any sensible person in classical logic).

The issue with classical logic isn't a matter of skepticism but instead that it lacks the existence-property which is problematic mathematically and metaphysically. Classical logic is utilized for largely historic and practical reasons, not because anybody has determined what its "objectively true" which is a claim that forces one to address a variety of metaphysical issues. Perhaps quantum logic is the objectively true logic? It, after all, is more consistent with our physical theories.

And as a whole, non-classical logics are used in a variety of contexts.

[brimstoneSalad]

I'm not sure what you mean by "subjectivist metaphysical claim" but what does it mean for a logic to be objectively true? How do we determine which logic is objectively true?

That's a question of metaphysics and epistemology.
A theistic presuppositionalist would say "ask God".

How do you sum two logical systems?

An or operator would probably be the correct way to do so.

Do you just combine the rules or do you just take the union of the set of all their provable statements?

That would be the latter, I think.

In either case the sum of two or more consistent logical systems isn't necessarily consistent.

Can you prove a case where it is not?

Also the set of consistent logical systems would be infinite, so we define what is objectively true by the sum of an infinite set of consistent logical systems?

Why not? Ever studied calculus?
Infinite sets are by no means impossible to assess.

Constructive logic is very useful because it has an existence-property. That is any constructive proof of the existence of an object or property can be used as a recipe to construct the object. Classical logic lacks this poverty.

Which means the latter is capable of discussing more than the former; doesn't mean those things aren't worth discussing or have no truth value.
Again, we'd be back into metaphysics of truth.

[carnap]

That's a question of metaphysics and epistemology.

That is precisely the point, its not a question of logic but rather metaphysics.

An or operator would probably be the correct way to do so.

In what sense? Also the meaning of the "or" operator itself differs from logic to logic.

Can you prove a case where it is not?

A proof would hinge entirely on how you defined the "sum" of two logical systems. But since logical systems can be arbitrary, it wouldn't be difficult to build two that were consistent with themselves but conflicting with each other.

Why not? Ever studied calculus?
Infinite sets are by no means impossible to assess.

Its not a question of whether you can evaluate infinite sets but rather how you'd use an infinite collection of logical systems to define "objective true". What would that even mean?

Which means the latter is capable of discussing more than the former; doesn't mean those things aren't worth discussing or have no truth value.
Again, we'd be back into metaphysics of truth.

It means more statements are provable in classical logic but that it comes at a cost, namely, you no longer have the existence property. I've never suggested that classical logic isn't worth discussing, I've brought up alternate logics because there is no obvious reason to favor one as the one true logic or better somehow at describing "reality".

[mkm]

Can you explain why it isn’t a sentence because I don’t really understand that?

The core of the problem is that in reality there may not be dichotomy "everything is eaither true or false". That's why I said that you dodged a question before with your answer. Applaying classical logic to the reality results in logical determinism, because there is no "maybe" between 1 and 0.