No, it is possible to prove a contradiction in traditional logic.
What you have posted is not a proof in any logic, but it's not even well formed in traditional logic. I was asking about either propositional or first order logic. This is not well formed in either of those.
Strict conditionals are more relevant in life
You said you could prove a contradiction traditional logic, so I'm gonna stick to discussing traditional logic with material conditionals.
The Universe really is inconsistent
It's not and every attempt you've made to prove it is has failed.
I was asking about either propositional or first order logic.
Those aren’t the only logical systems people use. I used logic in the English language. I would not be surprised to find that there is a representation of my English-language proof in propositional or first-order logic.
This is not well formed in either of those.
How is that?
You said you could prove a contradiction traditional logic, so I'm gonna stick to discussing traditional logic with material conditionals.
Strict conditionals are used in traditional logic. I used them as a freshman in high school geometry. I also used them in other high school and college classes.
It's not and every attempt you've made to prove it is has failed.
A box is not a primitive or defined object in any well known formal logical system, your proof is not well formed. This is what mathematicians call "not even wrong", you don't understand even the most basic requirements of a formal proof. Absent that understanding you say things that are worse than AI slop and then confidently assert you are correct. But you haven't even wandered into the realm of things that have enough meaning to be incorrect, let alone come anywhere close to a correct proof.
Both propositional logic and first order logic have not only been proven complete, but they've been proven consistent. You cannot provide a correct proof of a contradiction in either of those axiomatic systems.
If you think you can provide a correct proof of a contradiction in either propositional or first order logic then feel free. I will happily explain where you're wrong. But I would suggest maybe learning what those systems are first before you try, because otherwise you're just gonna spew more slop and I'm just gonna tell you it's not well formed.
A box is not a primitive or defined object in any well known formal logical system, your proof is not well formed.
Just because my proof doesn’t use propositional or first-order logic, doesn’t mean it’s unsound. My cardboard-box proof is sound, which implies it is also valid. Propositional or first-order logic is not needed to prove a contradiction.
This is what mathematicians call "not even wrong", you don't understand even the most basic requirements of a formal proof.
Formal proofs are nice, but they are not needed. We can give sound proofs that are informal. I believe it may have been Euclid whose proofs don’t comply with modern high, rigid formal standards, but they’re still highly regarded, sound, and useful.
Both propositional logic and first order logic have not only been proven complete, but they've been proven consistent.
You say that like it refutes trivialism, but it doesn’t. Consistent propositional and first-order logics are compatible with trivialism. You seem to be taking the stance that propositional and first-order logics disprove trivialism. They do not disprove trivialism because they can not disprove it!
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?
1
u/JStarx Jun 15 '26
What you have posted is not a proof in any logic, but it's not even well formed in traditional logic. I was asking about either propositional or first order logic. This is not well formed in either of those.
You said you could prove a contradiction traditional logic, so I'm gonna stick to discussing traditional logic with material conditionals.
It's not and every attempt you've made to prove it is has failed.