There is no need to continue reading this post, if you do not agree entirely with the following and have no doubt as to the common sense and meaning of each of the words that are used in these AXIOMS:
(AX1) A STATEMENT is any grammatically complete English sentence.
(AX2) Every STATEMENT falls into one and exactly one of three SETS:
(T) The set of statements that we have decided and agree are TRUE.
(F) The set of statements that we have decided and agree are FALSE.
(U) The set of statements that we have NOT decided and agreed belong to either Set (T) or Set (F).
(AX3) The operation of this exercise is first, to propose statements for consideration; and second, to decide and agree which of the three sets it belongs to.
(D1) At the outset the set (T) contains exactly three statements that we agreed and decided (if you got this far) are in fact TRUE, namely Axioms (AX1), (AX2) and (AX3) which are all valid Statements, by inspection, and are decidedly and agreeably TRUE by our common sense and agreement of the meaning of "axiom".
(D2) For any new statement that is proposed, the first logical step to make is to decide and to agree whether or not it belongs to the set (U), that is, whether there is any logical hope or possibility that we can decide and agree about whether it definitely belongs to EITHER set (T) or Set (F).
(D3) I shall now demonstrate that the Set (U) is definitely not empty, that is the Set (U) is not the NULL set. Consider any paradoxical or logically inconsistent statement like the Liars Paradox:
(LP) "This very statement is false."
Assume first that (LP) belongs to the set (T), that is, that LP is TRUE. If (LP) is true then it must be FALSE, that is, it belongs to Set (F). But if (LP) is false, then (LP) must be true and so also belongs to set (T). And so on. Thus in order not to violate AX2, we are forced to decide and agree that (LP) belongs to one and only one set, the set (U) because we cannot decide that it belongs to either Set (T) or Set (F). Quod erat demonstrandum (Q.E.D.) -- The set (U) is not empty.
I believe, though we have not decided or agreed to this, that every paradoxical or illogical statement of this sort (LP) belongs to set (U), that is, we cannot logically decide which of the two disjoint sets, (T) or (F) it belongs to.
(D4) Consider another statement which is not a paradox but is nonetheless very strange:
(NAP): "This very statement is true."
Does (NAP) belong to Set (T), (F), or (U)?
I can't say at the moment, so let's hear your opinion in the Comment Thread.
(D5) I wish to propose for discussion the following Definition:
(D5D1) An INFALLIBLE statement is one that we decide and agree CANNOT belong to either Set (F) or Set (U), even if we cannot decide and agree that it belongs to set (T).
(D6) Please propose your own definition of 'infallible statement".
(D7) Is the set of all statements that don't belong to (T) (F) or (U) empty?
(D8) Is NAP an infallible statement? Am I the Pope?
(D9) Are there any identifiable DISJOINT sets of statements within the Set (U)?
(D10) I know, I know...this is the weirdest Philippine Commentary posting ever! But I want to test the hypothese that seemed so clear in my head before but looks really strange to me now that in a quintessentially Goedelesque way:
INFALLIBIITY IS DECIDEDLY AND AGREEABLY TRUE BUT ALSO HERETICAL!
The basic idea is that for Papal Infallibility to be "logical" it has to stand outside the Deposit of Faith that is Divine Revelation (Sacred Scripture and Sacred Tradition). It is no longer Religion but Mathematics.