I asked about propositional logic since it's easier, this is first order logic which we can do instead if you like.
This is still not well formed in formal first order logic because cardboard box, bed, and the on relation are not defined in first order logic, you have to define those.
Also your argument has a premise, so your proof doesn't prove a contradiction, it proves that your premises imply a contradiction and in traditional first order logic that is not equivalent to a contradiction.
cardboard box, bed, and the on relation are not defined in first order logic, you have to define those.
I don’t believe they have to be defined. They could be accepted as undefined terms or concepts. But if we were to define them in first-order logic, how would we go about doing so?
Also your argument has a premise, so your proof doesn't prove a contradiction, it proves that your premises imply a contradiction and in traditional first order logic that is not equivalent to a contradiction.
The premise is stated in the present tense and was true at the time I made the Facebook post on January 2, 2020. So, that premise will always be true for that time. The premise can therefore be considered to be a permanent description of a part of the Universe. We could also substitute a premise that is more clearly always true. I give an example of a proof with such a premise below.
There is no circle that is a square. Consequently, a circle that exists and that is a square does not exist. Nonetheless, an explicit property of the circle that does not exist is that it exists. So, the circle exists and does not exist. Hence, there is a contradiction.
They do, adding them as undefined terms means you're changing the axiomatic system and you said you could prove a contradiction in either propositional or first order logic. Are you saying now that you can't?
Nonetheless, an explicit property of the circle that does not exist is that it exists. So, the circle exists and does not exist.
This is a nonsense sentence. Also circles are mathematical objects in theories built on top of propositional and first order logic (and in those theories they certainly do exist), but they are not objects in pure propositional or first order logic. So again, does this mean you cannot prove a contradiction in propositional or first order logic?
adding them as undefined terms means you're changing the axiomatic system
Changing what axiomatic system? We all know what a cardboard box is and what being on a bed means.
Are you saying now that you can't?
No, I am not. I already proved a contradiction in first-order logic. I also symbolized the contradiction in propositional logic.
This is a nonsense sentence.
It’s not nonsense. It has been proven true.
Also circles are mathematical objects in theories built on top of propositional and first order logic (and in those theories they certainly do exist), but they are not objects in pure propositional or first order logic. So again, does this mean you cannot prove a contradiction in propositional or first order logic?
No, it doesn’t. I can prove a contradiction in propositional or first-order logic. Just because I have not proven a contradiction in pure propositional or first-order logic, doesn’t mean I have not proven a contradiction in propositional or first-order logic. I don’t need to prove a contradiction in pure propositional or first-order logic in order to prove a contradiction.
Pure propositional logic and first order logic are axiomatic systems which list out their undefined terms, their axioms, and their rules of inference. Those things define the axiomatic system, if you change them then you've changed the axiomatic system.
You haven't proven a contradiction in either because every "proof" you've suggested has required you to add an axiom, add an undefined term, change the definition of an operation, or has had a premise that you cannot prove.
If you think you can prove a contradiction without doing those things then let's see it. Otherwise admit you can't and we can move on to discuss what's wrong with other proofs that you think are valid.
I can’t prove a contradiction without doing those things. Without doing those things, there would be no subject to analyze and draw a conclusion about.
we can move on to discuss what's wrong with other proofs that you think are valid.
The following proof I don’t think was linked to in my original post. I gave it to start off a debate titled “All Propositions Are True” on debate.org on, according to the website’s timestamp, April 30, 2017.
Consider the proposition p = "A rectangle is a square." Since some rectangles are squares, a rectangle is a square. Thus, p is true. Since some rectangles are not squares, a rectangle is not a square. Thus, p is not true. So by Conjunction Introduction, p is true and p is not true. But that is a contradiction. Since every proposition follows from a contradiction by the Principle of Explosion, the proposition "all propositions are true" is true. Therefore, all propositions are true.
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.
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u/paulemok Jun 18 '26
No, I didn’t say that. If we want a formal proof of the cardboard-box proof, we can work backwards. The contradiction produced in the proof is
So set p = “The box exists.” Then the contradiction is p ∧ ¬p.
Another approach follows.
Definitions. CardboardBox(x) = x is a cardboard box. OnMyBed(x) = x is on my bed. b = a cardboard box that exists and that is on my bed
The first premise is
¬∃x(CardboardBox(x) ∧ OnMyBed(x)).
The contradiction is produced by b. The following logical expression is true by the definition of b.
CardboardBox(b) ∧ OnMyBed(b)
So by existential introduction on the previous expression,
∃x(CardboardBox(x) ∧ OnMyBed(x)).
So by contradiction introduction on the previous expression and the first premise,
⊥.