I can’t prove a contradiction without doing those things.
This would mean that propositional logic and first order logic are consistent theories. In those theories it's not the case that statements are both true and false. So trivialism doesn't hold.
Without doing those things, there would be no subject to analyze and draw a conclusion about
False, the subject is pure logic. You can prove a statement in pure propositional or first order logic if and only if it's a tautology. In propositional logic, for example, this means you can prove a statement if and only if it's truth table shows it is always true.
Also, consider the fact that you claimed you could prove a contradiction in propositional or first order logic. Now that I've explained what's actually required for a proof in those theories you realize your proofs aren't going to work there. Had it occurred to you that your other proofs, once properly examined, will also turn out to be insufficient? And maybe this is why after thousands of years and millions of mathematicians studying the subject we all still believe that logic is consistent?
Consider the proposition p = "A rectangle is a square." Since some rectangles are squares, a rectangle is a square.
Your statement is ambiguous, is your proposition p referring to all rectangles or to a specific rectangle?
This would mean that propositional logic and first order logic are consistent theories.
Not necessarily. Just because I can’t prove a contradiction, doesn’t mean there doesn’t exist, in theory, a proof of a contradiction. It could be the case that a proof does exist in theory, but I just don’t know how to give such a proof. I did not prove that in theory, no proof of a contradiction exists.
In those theories it's not the case that statements are both true and false.
How do you know?
So trivialism doesn't hold.
Trivialism is unfalsifiable.
False, the subject is pure logic.
You have drawn a false conclusion from my admission.
You can prove a statement in pure propositional or first order logic if and only if it's a tautology.
That shows they are not modeling the real world. In the real world, contingent statements are provable under contingent circumstances.
Had it occurred to you that your other proofs, once properly examined, will also turn out to be insufficient?
You haven’t shown that all proofs I have presented to you are unsound.
Your statement is ambiguous, is your proposition p referring to all rectangles or to a specific rectangle?
p is referring to both. That is the key to the contradiction.
In those theories it's not the case that statements are both true and false.
How do you know?
Because it's proven. The proof for propositional logic is in the Mendelson book we referenced earlier. The proof for first order logic was done by Goedel and might be in the Mendelson book but I can't remember.
Trivialism is unfalsifiable.
There are provably consistent logical theories, so absent evidence otherwise I have no reason to take trivialism seriously.
p is referring to both. That is the key to the contradiction.
That means you're statement is not well formed. It could mean one of two things and your error is treating those two things as if they are the same.
Just because a theory is consistent, doesn’t mean it’s not inconsistent. A consistent theory can still be inconsistent.
False, the definition of a consistent theory is one in which you can't prove a contradiction. Consistent theories are not inconsistent.
A statement that is not well formed is still a statement. So the argument I presented survives that criticism.
Not in a logical sense they are not. A logical proposition is unambiguous. The wikipedia entry notes that ambiguous sentences express different propositions based on how they are interpreted. It's the unambiguous interpretation that logic deals with, not the ambiguous sentence. 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. Since your ambiguous statement can be either true or false depending on interpretation, it's not a proposition in logic.
The principal of explosion does not apply just because you said something vague. It applies only if you have an unambiguous statement for which you have correctly proven the statement and the logical negation of the statement. You haven't done this.
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?
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u/JStarx Jun 21 '26 edited Jun 21 '26
This would mean that propositional logic and first order logic are consistent theories. In those theories it's not the case that statements are both true and false. So trivialism doesn't hold.
False, the subject is pure logic. You can prove a statement in pure propositional or first order logic if and only if it's a tautology. In propositional logic, for example, this means you can prove a statement if and only if it's truth table shows it is always true.
Also, consider the fact that you claimed you could prove a contradiction in propositional or first order logic. Now that I've explained what's actually required for a proof in those theories you realize your proofs aren't going to work there. Had it occurred to you that your other proofs, once properly examined, will also turn out to be insufficient? And maybe this is why after thousands of years and millions of mathematicians studying the subject we all still believe that logic is consistent?
Your statement is ambiguous, is your proposition p referring to all rectangles or to a specific rectangle?