Assumption...
Assumption...
Assumption.
Master of the obvious indeed.
The only irrefutable statement: "Assumption is necessary."
First, you are confusing
statements with
propositions: statements can differ syntactically, but be equivalent semantically. In other words, the same proposition can be expressed by many different statements. For example:
P or Q = ~(~P & ~Q)
Here, the disjuction "or" is nonexclusive, perhaps better read as "or/and." The statement "P or Q" is false, if and only if, both "P" and "Q" are false. Now, it follows, that is "P or Q" is true, then "~P & ~Q" is is false, therefore "~(~P & ~Q)." In other words, these two formulas are equivalent, in that they both express the same proposition. In natural language:
It is raining outside or cloudy outside
=
It is not the case that is not raining outside and not cloudy outside
These
different statements, express the
same proposition. In fact, the syntactical rules of natural language allow for a far greater freedom sentence formation, and that's ignoring the thousand of languages we have to choose from. The point is that "Assumption is necessary" cannot be the only statement which is irrefutable, since the same proposition can be expressed by the statement, "it is not case that it is not the case that an assumption is necessary." Indeed, I might even say "it is necessary to have assumptions," or whatever. In any case, we are still dealing with
different statements.
You might claim that I am nitpicking here, and that you really meant that "Assumption is necessary" to be the only
proposition that is irrefutable. However, if so then you have generated a paradox.
(A) The only irrefutable proposition: "Assumption is necessary"
(B) A is refutable
Iff "Assumption is necessary" is the only irrefutable proposition, then the proposition "The only irrefutable proposition: "Assumption is necessary"" is refutable. Therefore, B follows from A.
Now, if B is false then A is irrefutable, though if A is irrefutable then "Assumption is necessary" not the only irrefutable proposition and A is false. But if A is false then B is true. Therefore, B is always true and irrefutable.
That is bad enough, but now if B is always true, then A is always refutable, now it follows that if A is refutable then so is "Assumption is necessary." Thus "Assumption is necessary" is not irrefutable!!!
There are many semantic paradoxes, tautologies and theorems that are irrefutable. That is, true under all interpretations. In fact, all logically valid arguments employ one connective which is true under all interpretations. For example:
(P then Q) then Q
T...T....T....T....T
T...F....F....T.....F
F...T....T....T....T
F...T....F....T.....F
This truth table makes the point well. Whatever forumula we end up with on the conclusion side of an argument, we can simply add to the original set of premises as the consequent of a conditional. Pay attention to the second "then" in the sequence, which is True under all interpretations of the argument. It doesn't matter what combination of truth and falsity we apply to "P" and "Q," that second "then" is always true and irrefutable.
There are infinite irrefutable propositions. However, irrefutability does not necessarily mean true, it simply means that we cannot show it to be false.