Keynes. Expanding this we have--

Patience is painful;
Patience is a virtue:
.'. Some virtue is painful.

And then we see the equivocation of the inference; for though patience
be painful _to learn_, it is not painful _as a virtue_ to the patient
man.

The hypothetical, '_If Plato was not mistaken poets are dangerous
citizens_,' may be considered as an argument against the laureateship,
and may be expanded (informally) thus: 'All Plato's opinions deserve
respect; one of them was that poets are bad citizens; therefore it
behoves us to be chary of encouraging poetry.' Or take this
disjunctive, '_Either Bacon wrote the works ascribed to Shakespeare, or
there were two men of the highest genius in the same age and country_.'
This means that it is not likely there should be two such men, that we
are sure of Bacon, and therefore ought to give him all the glory. Now,
if it is the part of Logic 'to make explicit in language all that is
implicit in thought,' or to put arguments into the form in which they
can best be examined, such propositions as the above ought to be
analysed in the way suggested, and confirmed or refuted according to
their real intention.

We may conclude that no single function can be assigned to all
hypothetical propositions: each must be treated according to its own
meaning in its own context.

Sec. 5. As to Modality, propositions are divided into Pure and Modal. A
Modal proposition is one in which the predicate is affirmed or denied,
not simply but _cum modo_, with a qualification. And some Logicians have
considered any adverb occurring in the predicate, or any sign of past or
future tense, enough to constitute a modal: as 'Petroleum is
_dangerously_ inflammable'; 'English _will be_ the universal language.'
But far the most important kind of modality, and the only one we need
consider, is that which is signified by some qualification of the
predicate as to the degree of certainty with which it is affirmed or
denied. Thus, 'The bite of the cobra is _probably_ mortal,' is called a
Contingent or Problematic Modal: 'Water is _certainly_ composed of
oxygen and hydrogen' is an Assertory or Certain Modal: 'Two straight
lines _cannot_ enclose a space' is a Necessary or Apodeictic Modal (the
opposite being inconceivable). Propositions not thus qualified are
called Pure.

Modal propositions have had a long and eventful history, but they have
not been found tractable by the resources of ordi