Is classical logic inconsistent?

[Owen Holden]

If classical logic is inconsistent then so is classical mathematics.

A. x=y -> (Fx <-> Fy). This is an axiom of first order logic and it is a theorem of second order logic. (Leibnitz's Law)

A is a theorem of Principia Mathematica, *13.15.

1. x=y -> ([](x=x) <-> [](x=y)).
therefore,
2. x=y -> [](x=y)?? (Because [](x=x) is an axiom or a theorem)

3. Herkyl=Herkyl -> [](Herkyl=Herkyl) ??

Surely it is true that Herkyl is self identical, but, it is false to say that Herkyl=Herkyl is logically true.


B. y=(the x:Gx) -> (Fy <-> F(the x:Gx)).

B is a theorem of Principia Mathematica *14.15.

1a. (the x:Gx)=y -> ([](y=y) <-> []((the x:Gx)=y).

2a. (the x:Gx)=y -> []((the x:Gx)=y)??

3a. (the number of planets)=9 -> []((the number of planets)=9)??

It is clearly false to say that (Herkyl=Herkyl) is necessarily true, and,
It is clearly false to say that (the number of planets)=9, is necessarily true.

That is to say the logical deductions, 3 and 3a, are false.

Therefore classical logic is inconsistent!

 

[Hurkyl]

I'm not familiar with the [] symbol; TMK, it's not a standard notation of formal logic.

x=x is certainly a tautology, is true relative to any truth valuation, and valid in any interpretation. I'm not sure what you could possibly mean by "it's false to say x=x is logically true". And there is ambiguity arising from your use of natural language -- what precisely did you mean to say by that? Did you mean to assert something like
$$\neg \forall x:\left( (x = x) \Leftrightarrow T \right)?$$
(This, of course, is a contradiction, because its negation is a tautology)

It is clearly false to say that (the number of planets)=9, is necessarily true.

But it necessarily follows from the hypothesis "(the number of planets)=9".

 

[Owen Holden]

I'm not familiar with the [] symbol; TMK, it's not a standard notation of formal logic.

[] means it is logically necessary that. <> means it is logically possible that.

These symbols have been a part of standard modal logic since 1918, C. I. Lewis, A Survey of Symbolic Logic. ...where have you been?

x=x is certainly a tautology, is true relative to any truth valuation, and valid in any interpretation. I'm not sure what you could possibly mean by "it's false to say x=x is logically true". And there is ambiguity arising from your use of natural language -- what precisely did you mean to say by that? Did you mean to assert something like
$$\neg \forall x:\left( (x = x) \Leftrightarrow T \right)?$$
(This, of course, is a contradiction, because its negation is a tautology)

But it necessarily follows from the hypothesis "(the number of planets)=9".

It is empirically true that '(the number of planets)=9', not logically true.
It is logically true that: (the number of planets)=9, is false.
Surely it was a different number at some point in time.
Surely there was no sun and no planets at some time, in which case 'the number of planets is nine' is gibberish.

If there are no people then there are no statements at all, are there?

Does your understanding of 'necessary' entail for all times??

Even the logical necessity that 2+2=4, is true ..iff there are minds to understand it.

All of truth is time dependent, ie., there cannot be absolute truths, including this one.

Hurkyl=Hurkyl, is true iff Hurkyl is alive.
That Hurkyl posts on this board imples that Hurkyl exists.

Hurkyl=Hurkyl, is false when there is no Hurkyl.

x=x <-> Ey(x=y) <-> EF(Fx) <-> x exists.

If Hurkyl exists, does it necessarily exist?? I don't think so, do you?

Do you also believe that (the present king of France)=(the present king of France) is true, because: F(the present king of France) <-> F(the present king of France), is tautologous for all F??

We can easily prove that (the present king of France) does not exist, because of the fact that there are no present Kings of France.

That is to say (the present king of France) cannot have the property of being self identical...It has no properties at all.

 

[CRGreathouse]

[] means it is logically necessary that. <> means it is logically possible that.

These symbols have been a part of standard modal logic since 1918, C. I. Lewis, A Survey of Symbolic Logic. ...where have you been?

I don't consider that notation standard.

If there are no people then there are no statements at all, are there?

I'm not sure what philosophy holds this, but not one that I subscribe to. Of course that's entirely nonmathematical -- if you want to discuss this I think we have a forum for epistemology.

Hurkyl=Hurkyl, is true iff Hurkyl is alive.
That Hurkyl posts on this board imples that Hurkyl exists.

Hurkyl=Hurkyl, is false when there is no Hurkyl.

How odd. I think the set of odd perfect numbers = the set of odd perfect numbers, even if there are no odd perfect numbers.

 

[DeadWolfe]

This post is not mathematics, but a nonstandard interpretation of logic. It belongs on the Philosophy boards.

 

[Owen Holden]

Originally Posted by Owen Holden
[] means it is logically necessary that. <> means it is logically possible that.

These symbols have been a part of standard modal logic since 1918, C. I. Lewis, A Survey of Symbolic Logic. ...where have you been?

I don't consider that notation standard.

Originally Posted by Owen Holden
If there are no people then there are no statements at all, are there?

I'm not sure what philosophy holds this, but not one that I subscribe to. Of course that's entirely nonmathematical -- if you want to discuss this I think we have a forum for epistemology.

How is it that there is: language, logic or mathematics, if there are no minds.

2+2=4, is only a scrible, if there cannot be an interpretation of the symbols.

There cannot be knowledge of: truth, fact, or existence, if there is nobody to understand, can there?

We 'believe' there were things before mind, but surely we cannot know such things.

Originally Posted by Owen Holden
Hurkyl=Hurkyl, is true iff Hurkyl is alive.
That Hurkyl posts on this board imples that Hurkyl exists.

Hurkyl=Hurkyl, is false when there is no Hurkyl.

How odd. I think the set of odd perfect numbers = the set of odd perfect numbers, even if there are no odd perfect numbers.

I think that 'your' view is indeed odd.
Only existent sets are equal.
If the set of odd perfect numbers does not exist, then it cannot have properties such as self-identity.
To exist is to have some property.

The (empty) null set exist by axiom.

If there are no odd perfect numbers then the set of odd perfect numbers is not the empty set, as you seem to infer, rather it is a non-existent set.

Those things which are and are not are non-existent, and they are not members of the null set.
The empty sets has no members at all, real or fictitious.

For example: the present king of France does not exist, but, we cannot infer that the present king of France is a member of the empty set, can we??

 

[CRGreathouse]

How is it that there is: language, logic or mathematics, if there are no minds.

2+2=4, is only a scrible, if there cannot be an interpretation of the symbols.

There cannot be knowledge of: truth, fact, or existence, if there is nobody to understand, can there?

How not? I'm not claiming that someone can interpret "2+2=4", just that it holds.

I claim:

* There is logic, independent of minds.
* There is mathematics, independent of minds.

These two statements are philosophical statements that follow from mathematical realism. I may be the only realist on these boards -- I think formalism is most common. I don't think they believe in quasi-Platonic existence at all.

For example: the present king of France does not exist, but, we cannot infer that the present king of France is a member of the empty set, can we??

For K = the set of current kings of France, I would say that $K\subseteq\emptyset$.

 

[Owen Holden]

Originally Posted by Owen Holden
How is it that there is: language, logic or mathematics, if there are no minds.

2+2=4, is only a scrible, if there cannot be an interpretation of the symbols.

There cannot be knowledge of: truth, fact, or existence, if there is nobody to understand, can there?

How not? I'm not claiming that someone can interpret "2+2=4", just that it holds.

There cannot be a meaning to the phrase 'just that it holds' if there are no minds.

I claim:

* There is logic, independent of minds.
* There is mathematics, independent of minds.

How is it possible to know these things if there are no people.

These two statements are philosophical statements that follow from mathematical realism. I may be the only realist on these boards -- I think formalism is most common. I don't think they believe in quasi-Platonic existence at all.

Platonism (realism) is an illusion of many mathematical minds.
That there is an abstract universe of 'concepts' is pure silly talk.
There is only empirical things and mental things.
Surely mental things only exist when there are minds.
There cannot be 'platonic' existences beyond mind.

For K = the set of current kings of France, I would say that $K\subseteq\emptyset$.

The set of current kings of France is included in the empty set, is true.
Because the empty set is included in itself.
All empty sets are included in the empty set.

The set of current kings of France is included in the null set, does not entail a property of any present king of France.

 

[Hurkyl]

Own Holden: your original indicated you wanted to talk about the logic used by mathematics. However, you have written absolutely nothing on that topic.

I strongly encourage you to either revise your intent for this discussion, or quickly switch gears into one where you are trying to learn mathematical logic.


Incidentally

These symbols have been a part of standard modal logic since 1918, C. I. Lewis, A Survey of Symbolic Logic. ...where have you been?

I've been in the 21st century and the latter part of the 20th. :tongue:

 

[LightbulbSun]

Originally Posted by Owen Holden
How is it possible to know these things if there are no people.

Do planets suddenly disappear because we're not around? Well there's your answer. It is possible.

 

[Phred101.2]

Surely it is true that Herkyl is self identical, but, it is false to say that Herkyl=Herkyl is logically true.

Your first post isn't a logically consistent argument, for starters.
If something is 'self identical' then it is 'itself', by logic. You could say that 'entropy=time', is empirically true, which surely can't mean or imply that it is logically false, as well? However, some idea or postulate, which may or may not be logically possible, and so logically perhaps also empirically true (if it happens to be a conjecture about the physical existence of some other 'thing'), is a contention. If you're otherwise saying that a symbol (like 'Herkyl') can represent more than one thing (and so logically be also a different thing, in the sense of being named the same by someone), that isn't supporting your apparent conclusion in the slightest...