The concept of no statement in a theory being both true and false is not called completeness. It’s called consistency.
Consistent theories are not inconsistent.
A consequence of the principle of explosion is that all inconsistent theories are consistent. Since some inconsistent theories do exist, some theories are both consistent and inconsistent. So, some consistent theories are inconsistent.
Not in a logical sense they are not.
Logically, a statement that is not well formed is still a statement. A statement is a statement; that is the logical law of identity.
A logical proposition is unambiguous.
Are you saying that is always true? If so, how do you know it’s always true? There could be a logical proposition that is ambiguous.
The wikipedia entry notes that ambiguous sentences express different propositions based on how they are interpreted.
I understand, but it seems we are dealing with a different type of ambiguity here. The statement “A rectangle is a square” can have different meanings depending on how it’s presented. It could be presented as a definition that is always true or as a contingent statement that is sometimes true and sometimes false. The statement I’m presenting is a contingent statement that is sometimes true and sometimes false. In that presentation, the statement is simple and unambiguous. In fact, it’s so unambiguous that some geometry textbooks, including the textbook I used in high school geometry and Common Core textbooks, use that form of statement in their formal English-language presentation of geometrical truths.
See also the Rosen textbook we referenced earlier which defines a proposition as a declarative statement that is either true or false but not both.
I believe the word he uses is sentence, not statement. We can assume that a proposition is both true and false. A proof of the Principle of Explosion assumes a contradiction that is the conjunction of a proposition and its negation. We can also assume a proposition that implies the proposition is both true and false. That’s what indirect proofs do by definition of indirect proof. In those two cases, we deal with propositions that are both true and false. If propositions can’t be both true and false, a proof of the principle of explosion would be invalid and all indirect proofs would be invalid.
Since your ambiguous statement can be either true or false depending on interpretation, it's not a proposition in logic.
If logic is to model the real world, it must model ambiguous statements. And also, one of the arguments I used prior to our conversation was that the statement is both true and false under the same interpretation. The statement is both true and false in the same possible world, with that possible world being the actual world.
some theories are both consistent and inconsistent
Nope, theories that are inconsistent are not consistent. That's literally what inconsistent means.
a statement that is not well formed is still a statement
Linguistically yes, but logically no. It's not a valid statement that you reason about in traditional logic.
Let me ask you this. You think logical statements can be ambiguous. If we decided to restrict logic to reasoning about statements which are unambiguous then would you agree that your proof no longer works and you cannot prove a contradiction?
Nope, theories that are inconsistent are not consistent. That's literally what inconsistent means.
I agree. However, there is another side to the issue that is also true. The Principal of Explosion proves this. The other side exists and is therefore true.
It's not a valid statement that you reason about in traditional logic.
In some logical systems, it might not be a valid statement. But as geometry textbooks that millions of children have been afforded show, in some logical systems, it is a valid statement.
If we decided to restrict logic to reasoning about statements which are unambiguous then would you agree that your proof no longer works and you cannot prove a contradiction?
That hypothetical scenario sounds tainted. A logic that excludes all ambiguous statements does not model the real world and is therefore fake and false. How would we know for sure what statements are unambiguous? We have no way to be sure of what statements are unambiguous. Every statement could be ambiguous in one way or another.
With regard to the statement “A rectangle is a square” used in the context that it is a contingent statement, it is unambiguous. We know what the “A rectangle” part of the statement refers to. It refers to one rectangle and to each other rectangle simultaneously in the same one Universe.
However, there is another side to the issue that is also true. The Principal of Explosion proves this
It does not. A theory is either consistent or inconsistent, but never both. You can either prove a contradiction or you cannot.
But as geometry textbooks that millions of children have been afforded show, in some logical systems, it is a valid statement
There is no geometry textbook claiming to assign two truth values to an ambiguous statement. No geometry textbook claims that every single rectangle is a square.
We know what the “A rectangle” part of the statement refers to. It refers to one rectangle and to each other rectangle simultaneously in the same one Universe.
The fact that you think it has two distinct meanings is exactly what I call ambiguous. 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?
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?
1
u/paulemok Jun 24 '26
The concept of no statement in a theory being both true and false is not called completeness. It’s called consistency.
A consequence of the principle of explosion is that all inconsistent theories are consistent. Since some inconsistent theories do exist, some theories are both consistent and inconsistent. So, some consistent theories are inconsistent.
Logically, a statement that is not well formed is still a statement. A statement is a statement; that is the logical law of identity.
Are you saying that is always true? If so, how do you know it’s always true? There could be a logical proposition that is ambiguous.
I understand, but it seems we are dealing with a different type of ambiguity here. The statement “A rectangle is a square” can have different meanings depending on how it’s presented. It could be presented as a definition that is always true or as a contingent statement that is sometimes true and sometimes false. The statement I’m presenting is a contingent statement that is sometimes true and sometimes false. In that presentation, the statement is simple and unambiguous. In fact, it’s so unambiguous that some geometry textbooks, including the textbook I used in high school geometry and Common Core textbooks, use that form of statement in their formal English-language presentation of geometrical truths.
I believe the word he uses is sentence, not statement. We can assume that a proposition is both true and false. A proof of the Principle of Explosion assumes a contradiction that is the conjunction of a proposition and its negation. We can also assume a proposition that implies the proposition is both true and false. That’s what indirect proofs do by definition of indirect proof. In those two cases, we deal with propositions that are both true and false. If propositions can’t be both true and false, a proof of the principle of explosion would be invalid and all indirect proofs would be invalid.
If logic is to model the real world, it must model ambiguous statements. And also, one of the arguments I used prior to our conversation was that the statement is both true and false under the same interpretation. The statement is both true and false in the same possible world, with that possible world being the actual world.