Handbook of the History of Logic: - Fordham University Faculty

Proof.
Treatments of the Paradoxes of Self-reference 593
(1)* ‘a’ signifies only that ‘a’ is not true and that B is C Assumed
(2)* ‘a’ is not true Assumed
(3)* It is not wholly as ‘a’ signifies (2)*, (Def. of truth)
(4)* It is not the case that ‘a’ is not true and B is C
(1)*, (3)*, (T1.1)
(5)* ‘a’ istrueorB is not C (4)*
(6)* ‘a’ signifies that ‘a’ istrueorB is not C (1)*, (2)*–(5)*, (P2)
(7)* ‘a’ signifies that B is C (1)*, (P2)
(8)* ‘a’ istrueorB is not C, andBis C, therefore ‘a’ istrue
(9)* ‘a’ signifies that ‘a’ is true. (P2)
This reasoning is a bit more confusing, and not only because Bradwardine jumps
between first order claims ((4)*, (5)* and (8)*) and claims about the examined
paradoxical sentence and its signification ((1)*, (2)*, (3)*, (6)*, (7)* and (9)*)
more than in the previous proof. It is, however, possible to see what really happens.
After laying the ground with the two assumptions (1)* and (2)*, Bradwardine
construes the disjunctive signification (6)* through the conditional proof (2)*–
(5)*. The he excludes the latter disjunct at (7)*, and proceeds with the support
of the inference (8)* to the main goal (9)*.
At (7)* Bradwardine makes a choice. He could at that step exclude the former
disjunct by taking the corresponding proposition “a’ signifies that ‘a’ isnot
true’ and using it in a similar way, to achieve instead of step (9)* the claim
“a’ signifies that B is not C.’ That is, he could reason as follows. Assume
‘a’ :(¬T ‘a’&p). Thus ¬T ‘a’ →¬(¬T ‘a’&p) → (T ‘a’ ∨¬p) →¬p. By (P2)*,
it follows that ‘a’ :(¬T ‘a’&p&¬p), and since any sentence follows from the contradiction
(¬T ‘a’&p&¬p), ‘a’ signifies everything, including its own truth.
For quite obvious reasons, this is not the way Bradwardine goes. He wants to
prove the thesis (T2) in a sensible way, and for this he needs “a’ signifies that ‘a’
is true’ but not much more. Thus, he reasons as follows. Assume ‘a’ :(¬T ‘a’&p).
Thus ¬T ‘a’ → ¬(¬T ‘a’&p) → (T ‘a’ ∨¬p) → T ‘a’. By (P2)*, it follows now
that ‘a’ :(¬T ‘a’&p&T ‘a’), which is a much more sensible kind of inconsistence.
However, this way of looking at the proof makes it apparent that Bradwardine is in
itself is not true, for example, that B is C, then supposing that A is not true, it follows that it
is not wholly as A signifies, that is, that A is not true and B is C, whence it is not the case that
A is not true and that B is C, and so, by postulate (P4), A is true or B is not C, andso,by
postulate (P2), A signifies that A is true or B is not C, and since it signifies the opposite of the
second part of this disjunction (we supposed), namely, that B is C, we have by postulate (P5),
from this disjunction with the opposite of its second part that the first part follows, namely, that
A is true. Hence by postulate (P2), A signifies A to be true.” [Bradwardine, internet, 45]. (P4)
and (P5) are ordinary principles of propositional logic; see [Bradwardine, internet, 39].