Figs - These were an unexpected treat. The fig tree is only a few years old, brought here in a pot after I bought it from Wilkinsons for £3.50 in January 2017, ...
2 hours ago
To Be is to be Unbecoming
no man might buy or sell, save he that had the mark, or the name of the beast, or the number of his nameAnd even this very Blogger have facebook at the top of these pages (under “More”). Even the BBC, who are not supposed to advertise (they famously say “sticky-backed plastic” instead of sellotape) show us the Facebook symbol, the ancient Greek number of his name.
Suppose that your friend tells you: “If what I’m saying using this very sentence is true, then time is infinite”. It turns out that there is a short and seemingly compelling argument for the following conclusion:That may look rather formal to you, but formal logic is not even logic (it is mathematics); the above is just very well laid out. Note the two uses of modus ponens, the two sets of three steps, with the first three steps, (1), (2) and (3), all beginning “Under the supposition that”. You should note that because we cannot always use modus ponens within the scope of a supposition, e.g.:
(P) The mere existence of your friend’s assertion entails (or has as a consequence) that time is infinite.
Many hold that (P) is beyond belief (and, in that sense, paradoxical), even if time is indeed infinite.
Here is the argument for (P). Let k be the self-referential sentence your friend uttered, simplified somewhat so that it reads “If k is true then time is infinite”. In view of what k says, we know this much:
(1) Under the supposition that k is true, it is the case that if k is true then time is infinite.
But, of course, we also have
(2) Under the supposition that k is true, it is the case that k is true.
Under the supposition that k is true, we have thus derived a conditional together with its antecedent. Using modus ponens within the scope of the supposition, we now derive the conditional’s consequent under that same supposition:
(3) Under the supposition that k is true, it is the case that time is infinite.
The rule of conditional proof now entitles us to affirm a conditional with our supposition as antecedent:
(4) If k is true then time is infinite.
But, since (4) just is k itself, we thus have
(5) k is true.
Finally, putting (4) and (5) together by modus ponens, we get
(6) Time is infinite.
We seem to have established that time is infinite using no assumptions beyond the existence of the self-referential sentence k, along with the seemingly obvious principles about truth that took us to (1) and also from (4) to (5).
Let your friend say instead: “If what I’m saying using this very sentence is true, then all numbers are prime”. Now, mutatis mutandis, the same short and seemingly compelling argument yields (Q):My suspicion is based on the fact that one could conceivably have a valid argument for
(Q) The mere existence of your friend’s assertion entails (or has as a consequence) that all numbers are prime.