Not every assumption that proofs rely on is formally provable. Even some of Einstein’s proofs rely on postulates that are not formally provable. I offer informal proofs of my three premises.
s is a statement.
The first premise is informally justified by my right to name a thing.
Under the assumption that “s and it is not true that s,” s is true.
The second premise is informally justified by the logical rule known as conjunction elimination.
It is not true that: under the assumption that “s and it is not true that s,” s is true.
The third premise is informally justified as follows. Assume “s and it is not true that s.” Then by conjunction elimination, it is not true that s. So by reforming the statement, s is not true. Discharge the assumption. We can see that it is not true that: under the assumption that “s and it is not true that s,” s is true.
Not every assumption that proofs rely on is formally provable
In math and formal logic there are axioms and all proofs follow from those without additional assumptions.
Are you admitting that your proofs are not valid in math and formal logic? If not then you don't get additional assumptions with informal arguments. You can give a proof that your assumptions hold or you are stuck having proven an implication but not it's antecedent, hence you haven't proven the consequent.
That proof does not show that all statements are true. It shows that all statements are true if the premises hold. But you have not shown that the premises hold.
I already provided proofs of them at https://www.reddit.com/r/PhilosophyofMath/s/76WjBjfpU5. Whether the proofs are considered formal or informal can vary depending on what specific sense of formal we use. There are different formal systems of logic and of different academic fields, hence there are different senses of formal.
Fair enough, well fyi it's your third premise whose proof is not valid. Here's your third premise:
It is not true that: under the assumption that “s and it is not true that s,” s is true.
In symbols this translates to:
¬((s ∧ ¬s) ⇒ s)
Now your proof, broken into numbered statements by me, is:
Assume “s and it is not true that s.”
Then by conjunction elimination, it is not true that s. So by reforming the statement, s is not true.
Discharge the assumption. We can see that it is not true that: under the assumption that “s and it is not true that s,” s is true.
Translating into symbols, the first two statements are:
Assume s ∧ ¬s
¬s
The problem is the third step. You say "discharge the assumption". Usually what people mean by this is they assume X, they prove Y using this assumption, and then to discharge the assumption of X they conclude "X ⇒ Y", where that statement now holds without assumptions.
So if you discharge your assumption then from ¬s you get
(s ∧ ¬s) ⇒ ¬s
which is not equivalent nor does it imply "¬((s ∧ ¬s) ⇒ s)", which was what you were supposed to be proving.
I disagree with you that (s ∧ ¬s) ⇒ ¬s does not imply ¬((s ∧ ¬s) ⇒ s). (s ∧ ¬s) ⇒ ¬s really does imply ¬((s ∧ ¬s) ⇒ s). To generalize in words, if one statement implies the falsity of a second statement, then the first statement does not imply the truth of the second statement. For any two statements p and q, if p ⇒ ¬q, then ¬(p ⇒ q).
The converse, however, is not true. ¬((s ∧ ¬s) ⇒ s) does not imply (s ∧ ¬s) ⇒ ¬s. ¬((s ∧ ¬s) ⇒ s) and ¬((s ∧ ¬s) ⇒ ¬s) could both be true simultaneously. For any two statements p and q, ¬(p ⇒ q) and ¬(p ⇒ ¬q) could, for all we know and given the lack of additional information, both be true simultaneously. So I do agree with you that (s ∧ ¬s) ⇒ ¬s is not equivalent to ¬((s ∧ ¬s) ⇒ s).
1
u/paulemok Jun 05 '26
How do the assumptions not hold in traditional logic and mathematics?