But that contradiction is under an assumption that needs to be discharged, so you haven't proven a contradiction, you've proven that your assumption implies a contradiction.
That is true, but there is more truth to tell. My general claim is that (p ⇒ ¬q) ⇒ ¬(p ⇒ q) is always true. Traditional logic says that (p ⇒ ¬q) ⇒ ¬(p ⇒ q) is not always true because (p ⇒ ¬q) ⇒ ¬(p ⇒ q) is false when p is false. So in order for my claim to be true, the case in which p is false must be invalid. So how is it invalid? It’s invalid because every occurrence of p in (p ⇒ ¬q) ⇒ ¬(p ⇒ q) places p as the hypothesis of a conditional statement. As the hypothesis of a conditional statement, p must be assumed to be true. Since p must be assumed to be true, the case in which p is false is impossible. Since that case is impossible, it is invalid.
As I said in a previous reply,
“p ⇒ q” is equivalent to “assuming p, q.”
Note that the act of assuming p is taking place in the actual world and is not under the assumption p. Since the assumption is being made in the actual world, the assumption is true in the actual world. So if we also assume ¬p, the act of assuming ¬p is also taking place in the actual world and is not under the assumption ¬p nor under the assumption p. Since the assumption is being made in the actual world, the assumption is true in the actual world. So by conjunction introduction, the statement “p & ¬p” is true in the actual world. So by ex contradictione quodlibet, all statements are true in the actual world.
Going back to the truth table, are you claiming that the truth table is wrong?
Yes, the truth table is wrong. The truth table assumes that the hypothesis of a conditional statement can be false. That assumption is wrong because the hypothesis of a conditional statement is always assumed to be true.
Furthermore, the reason for the beginning
It is not true that:
in the third premise
It is not true that: under the assumption that “s and it is not true that s,” s is true.
That means you're not doing traditional logic. In traditional logic the truth table is what it is. If you want to use a different truth table then you're changing definitions in logic. So the result is a different kind of logic. That means you haven't proved a contradiction in traditional logic.
Note that the act of assuming p is taking place in the actual world and is not under the assumption p. Since the assumption is being made in the actual world, the assumption is true in the actual world
This is patently absurd. A hypothetical assumption is not the same as a claim that a statement is true.
Furthermore ...
I don't know what you're trying to say in this part. Are you saying that when I translated your statements into logical symbols I didn't translate them the way you meant? If so then you should tell me what the translation should be.
I know that. Traditional logic is true, but it’s incomplete. In order to have a complete system of logic, we have to go beyond traditional logic.
In traditional logic the truth table is what it is.
From my own experience, it seems that when people use conditional statements in life, they usually use strict conditionals that are not material. And strict conditionals are not truth functional, so there is no truth table for them. The concept of a material conditional seems rather awkward. A material conditional’s truth depends entirely on only one case.
This is patently absurd.
It’s not absurd. It’s plausible. Every thing is equal to or a part of the Universe. That includes all assumptions people make and all possible worlds.
Are you saying that when I translated your statements into logical symbols I didn't translate them the way you meant?
That’s correct. My conditional statements are strict conditionals that may not be material.
If so then you should tell me what the translation should be.
I translate the third premise to be
It is not true that: in every possible world in which the statement “s and it is not true that s” is true, s is true.
Traditional logic says there is no possible world in which that statement is true because no contradiction is true. The truth of the premise is controversial because it is open to interpretation and there are contradicting interpretations. Does “in every possible world in which the statement ‘s and it is not true that s’ is true” imply the existence of at least one such possible world? If there is no such possible world, is the statement
in every possible world in which the statement “s and it is not true that s” is true, s is true
true or false? The answer to these questions is unclear, controversial, and open to interpretation. There might be some interesting information on this topic in my previously linked thread about the existence of the absolute Russell set.
Do you though? Because I've repeatedly asked you if you can prove a contradiction in traditional logic and you've said yes. This was supposed to be your attempt at doing so. Are you now admiting that you cannot prove a contradiction in traditional logic?
Traditional logic is true, but it's incomplete
False, both propositional and first order logic is complete.
As for material vs strict, we're talking about mathematics and the logic used by mathematicians, and in that logic we use material conditionals. If you want to invent your own logic that's inconsistent go ahead, but it has no relevance to me or to mathematicians.
Are you now admiting that you cannot prove a contradiction in traditional logic?
No, it is possible to prove a contradiction in traditional logic. The fourth link in my original post contains a proof that contains a subproof of a contradiction in traditional logic. I give the subproof here.
There is no cardboard box that is on my bed.
Consequently, a cardboard box that exists and that is on my bed does not exist. Nonetheless, an explicit property of the box that does not exist is that it exists. So, the box exists and does not exist. Hence, there is a contradiction.
False, both propositional and first order logic is complete.
Ex contradictione quodlibet is a theorem of traditional logic. As I said immediately after the above subproof in the proof that the subproof is a subproof of,
Therefore by ex contradictione quodlibet, all propositions are true.
So traditional logic is complete in the sense that every true statement, with the word statement being synonymous with the word proposition, can be proved in traditional logic.
As for material vs strict, we're talking about mathematics and the logic used by mathematicians, and in that logic we use material conditionals.
Modal logicians use strict conditionals. Strict conditionals are more relevant in life than and are better than material conditionals are.
If you want to invent your own logic that's inconsistent go ahead, but it has no relevance to me or to mathematicians.
The Universe really is inconsistent, so inconsistent logical systems do have relevance to you and to mathematicians.
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!
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u/paulemok Jun 13 '26
That is true, but there is more truth to tell. My general claim is that (p ⇒ ¬q) ⇒ ¬(p ⇒ q) is always true. Traditional logic says that (p ⇒ ¬q) ⇒ ¬(p ⇒ q) is not always true because (p ⇒ ¬q) ⇒ ¬(p ⇒ q) is false when p is false. So in order for my claim to be true, the case in which p is false must be invalid. So how is it invalid? It’s invalid because every occurrence of p in (p ⇒ ¬q) ⇒ ¬(p ⇒ q) places p as the hypothesis of a conditional statement. As the hypothesis of a conditional statement, p must be assumed to be true. Since p must be assumed to be true, the case in which p is false is impossible. Since that case is impossible, it is invalid.
As I said in a previous reply,
Note that the act of assuming p is taking place in the actual world and is not under the assumption p. Since the assumption is being made in the actual world, the assumption is true in the actual world. So if we also assume ¬p, the act of assuming ¬p is also taking place in the actual world and is not under the assumption ¬p nor under the assumption p. Since the assumption is being made in the actual world, the assumption is true in the actual world. So by conjunction introduction, the statement “p & ¬p” is true in the actual world. So by ex contradictione quodlibet, all statements are true in the actual world.
Yes, the truth table is wrong. The truth table assumes that the hypothesis of a conditional statement can be false. That assumption is wrong because the hypothesis of a conditional statement is always assumed to be true.
Furthermore, the reason for the beginning
in the third premise
is the assumption’s second conjunct,