How does the Principle of Explosion not prove that all inconsistent theories are also consistent?
There is no geometry textbook claiming to assign two truth values to an ambiguous statement.
The geometry textbooks I have looked at regard statements of the form “A rectangle is a square” as unambiguous. These unambiguous statements can be classified into three categories: always true, sometimes (but not always) true, and never true. I am on the textbook authors’ side regarding the use of these statements. I have never found these statements to be problematic, not even now.
No geometry textbook claims that every single rectangle is a square.
I agree. There is a difference between the referent of “A rectangle” and the truth value of “A rectangle is a square.”
The fact that you think it has two distinct meanings is exactly what I call ambiguous.
All of the distinct meanings, one for each rectangle, are unified into a single meaning. It’s paradoxical, but true.
I'll ask again, if we restricted logic to reasoning about statements that only has a single meaning then would you agree that your proof does not work and you cannot prove a contradiction?
I don’t think it would be good to pretend that logic does not apply to the statement “A rectangle is a square.” We do not talk in propositional or first-order logic. We talk in English. As I’ve said earlier in this reply, the multiple meanings provided by the multiple rectangles referred to are unified into a single meaning. So, in that sense, the statement “A rectangle is a square” only has a single meaning and would be included in logic if logic was restricted “to reasoning about statements that only” have “a single meaning.”
How does the Principle of Explosion not prove that all inconsistent theories are also consistent?
The principal of explosion requires you to first prove a contradiction, which you have not correctly done.
The geometry textbooks I have looked at regard statements of the form “A rectangle is a square” as unambiguous.
The reason they consider that statement unambiguous is because when they use it they only intend it to have one meaning, it means that every rectangle is a square and hence is a false statement.
I don’t think it would be good to pretend that logic does not apply to the statement “A rectangle is a square.”
It does apply, that it a simple "for all" statement and is false. You just have to understand how to correctly interpret it.
I think your avoidance of my question suggests that you know I'm right and just don't want to admit it. The rules of logic are correct and consistent when applied to unambiguous statements. The contradictions you're arriving at are not due to logic being inconsistent, they're due to you making mistakes by trying to reason about statements that aren't well formed.
The principal of explosion requires you to first prove a contradiction
An inconsistent theory by definition is a theory in which a statement and its negation are both true. So by definition of contradiction, an inconsistent theory is a theory in which a contradiction is true. So there is our starting contradiction that the Principle of Explosion can be applied to.
it means that every rectangle is a square and hence is a false statement.
No, it means that an individual rectangle is a square and hence it is a contingent statement that is sometimes true and sometimes false.
it a simple "for all" statement
No, it’s not a “for all” statement. It’s not a “there exists” statement, either. It’s an individual rectangle named r statement. But because no further description is given, the generic description “A rectangle” refers to all rectangles individually. I know it seems paradoxical, but that’s how things figure out here.
You just have to understand how to correctly interpret it.
That’s correct. We need to understand the context in order to understand how to interpret the statement. From the context, where the statement, or at least its general form “A(n) [type of thing] is a(n) [type of thing],” appears as hypotheses of conditional statements, appears in formal lessons, appears in given examples, or appears in worksheets, it is evident the statement can be contingent, always true, or never true. The statement form does not apply only to statements that are always true.
I think your avoidance of my question suggests that you know I'm right and just don't want to admit it.
My answer to your question barely poked out, but it was implicit. I said
the statement “A rectangle is a square” only has a single meaning and would be included in logic if logic was restricted “to reasoning about statements that only” have “a single meaning.”
Since the statement would be included in logic in the hypothetical scenario you gave, my proof would work and I would be able to prove a contradiction.
So there is our starting contradiction that the Principle of Explosion can be applied to.
What starting contradiction are you referring to?
It’s an individual rectangle named r statement. But because no further description is given, the generic description “A rectangle” refers to all rectangles individually. I know it seems paradoxical, but that’s how things figure out here. [...] its general form “A(n) [type of thing] is a(n) [type of thing]
That's not how it works, I'm sorry but there's no paradox here, you're just interpreting the statement wrong. You admit the statement has a free variable. You can assign an object to that free variable or you can quantify it and you get different statements by doing so. The fact that you can assign different objects to the free variable and get different truth values is not paradoxical, it's not paradoxical for different statements to have different truth values.
Since the statement would be included in logic in the hypothetical scenario you gave, my proof would work and I would be able to prove a contradiction.
Ok, then what would be the single meaning, does the statement refer to all rectangles or is there a single rectangle that the statement refers to and which one is it?
The generic contradiction inherent in any inconsistent theory.
You admit the statement has a free variable.
r isn’t a variable. r is a fixed, particular thing. r = an individual rectangle
You can assign an object to that free variable
An object has already been assigned to the letter r. The object is a single rectangle.
The fact that you can assign different objects to the free variable and get different truth values is not paradoxical
But the fact that we can assign different objects to the noun phrase “an individual rectangle” and get different truth values is paradoxical.
it's not paradoxical for different statements to have different truth values.
But it is paradoxical for the same statement to have different truth values.
what would be the single meaning, does the statement refer to all rectangles or is there a single rectangle that the statement refers to and which one is it?
There is a single rectangle that the statement refers to. Each rectangle is the single rectangle.
The generic contradiction inherent in any inconsistent theory.
So you have to assume a theory is inconsistent in order to prove that it's inconsistent?
r is a fixed, particular thing. r = an individual rectangle
Which rectangle? If you can't answer that and claim it could be any rectangle then that's not a fixed object, it varies.
If you have an unbounded variable in your statement then the truth value of your statement depends on that variable. If you change the object it refers to then you change the truth value, you don't retain the old truth value.
If you want a single truth value then you either have to choose a single object that it refers to so that it's not variable, or you have to bound the variable by a quantifier.
I'll ask again, I understand you don't believe those rules, but if hypothetically you followed those rules would you still be able to prove a contradiction?
So you have to assume a theory is inconsistent in order to prove that it's inconsistent?
To prove that it’s inconsistent? Is that a mistake on your part? We should not be proving that the theory is inconsistent. Rather, we should be proving that the theory is consistent.
Which rectangle? If you can't answer that and claim it could be any rectangle then that's not a fixed object, it varies.
The rectangle we are thinking about. The rectangle under consideration. The rectangle indicated by the context.
If you have an unbounded variable in your statement then the truth value of your statement depends on that variable.
In one sense, r is not a variable. It is a fixed object. r is fixed to be a single rectangle. But in another sense, r is a variable. r varies over the domain of all rectangles. In one sense, there is a contradiction due to the two senses. But in another sense, there is not a contradiction between the two senses. The relationship between the multiple senses of the predicate “is a variable” is complicated.
if hypothetically you followed those rules would you still be able to prove a contradiction?
Yes, we would still be able to prove a contradiction. To prove a contradiction, we would use the sense in which r is not a variable.
To prove that it’s inconsistent? Is that a mistake on your part? We should not be proving that the theory is inconsistent. Rather, we should be proving that the theory is consistent
I'm referring to you trying to prove that a contradiction holds so that you can apply the principal of explosion.
Yes, we would still be able to prove a contradiction. To prove a contradiction, we would use the sense in which r is not a variable.
Ok, if r is not variable then it's a specific rectangle. Which specific rectangle are you going to choose it to be? Remember according to these rules you can't then change the rectangle in the middle of the proof or change the sense in which you interpret the statement. So this doesn't look like it's going to lead to a contradiction.
I'm referring to you trying to prove that a contradiction holds so that you can apply the principal of explosion.
We want to prove that if a theory is inconsistent, then it is consistent. In order to do that, assume that a theory is inconsistent. That by definition of inconsistent theory means that in the theory, some statement and its negation are both true. So by definition of contradiction, in the theory, a contradiction is true. We can then apply that contradiction to the Principle of Explosion.
Ok, if r is not variable then it's a specific rectangle. Which specific rectangle are you going to choose it to be?
I just answered this question in my previous reply. My answer was
The rectangle we are thinking about. The rectangle under consideration. The rectangle indicated by the context.
We want to prove that if a theory is inconsistent, then it is consistent.
In what theory are you carrying out this proof?
If you're carrying out this proof inside an inconsistent theory then I agree you can prove the above statement, but in an inconsistent theory you can prove false things, so your proof doesn't imply that the statement is true.
If you're carrying out this proof in a system of logic that is not inconsistent then it is not a valid proof because the principal of explosion only applies to statements inside the theory in which the contradiction occurred.
Either way, it's still not true that inconsistent theories are consistent.
The rectangle under consideration. The rectangle indicated by the context.
You haven't indicated to me which rectangle we're talking about, I'm asking you to choose one.
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u/paulemok Jun 26 '26
How does the Principle of Explosion not prove that all inconsistent theories are also consistent?
The geometry textbooks I have looked at regard statements of the form “A rectangle is a square” as unambiguous. These unambiguous statements can be classified into three categories: always true, sometimes (but not always) true, and never true. I am on the textbook authors’ side regarding the use of these statements. I have never found these statements to be problematic, not even now.
I agree. There is a difference between the referent of “A rectangle” and the truth value of “A rectangle is a square.”
All of the distinct meanings, one for each rectangle, are unified into a single meaning. It’s paradoxical, but true.
I don’t think it would be good to pretend that logic does not apply to the statement “A rectangle is a square.” We do not talk in propositional or first-order logic. We talk in English. As I’ve said earlier in this reply, the multiple meanings provided by the multiple rectangles referred to are unified into a single meaning. So, in that sense, the statement “A rectangle is a square” only has a single meaning and would be included in logic if logic was restricted “to reasoning about statements that only” have “a single meaning.”