CONDITIONALS: WEEK 8, INTRO TO SUBJUNCTIVES
FABRIZIO CARIANI
1.
Strict Conditionals and Variably Strict Conditionals
Consider the following logical principles for an arbitrary conditional operator
99K
:
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(AS)
p
99K
r
,
∴
(
p
q
)
99K
r
(T)
p
99K
q
,
q
99K
r
,
∴
p
99K
r
(AS) and (T) are short for ‘Antecedent Strengthening’, and ‘Transitivity’.
(1)
If
99K
satisﬁes (T), then it satisﬁes (AS)
Suppose (T) holds, and assume
p
99K
r
. Since
99K
is a conditional operator, we
can plausibly assume that for all
q
, (
p
q
)
99K
p
is a valid schema of the logic of
99K
. Therefore, we have (
p
q
)
99K
p
, and
p
99K
r
, which by (T), legitimates (
p
q
)
99K
r
.
Deﬁnition
99K
is a
strict
conditional iﬀ
for some class
W
of possible worlds, and for every
p
,
q
,
p
99K
q
is true iﬀ
p
⊃
q
is
true at
w
, for every
w
∈
W
(2)
If
99K
is strict, then it satisﬁes (T) (and therefore (AS))
Just because
⊃
satisﬁes both.
Let
p > q
abbreviate ‘If it were the case that
p
, then it would be the case that
q
’.
(3)
Intuitively, neither (AS) nor (T) holds for
>
Contrast:
(a) If it had been sunny on Sunday, I’d have played soccer (on Sunday).
(b) If it had been sunny on Sunday and I had had a car accident on Saturday, I’d
have played soccer (on Sunday).
Intuitively, the truth of (a) does not guarantee the truth of (b) because the antecedent
is strengthened by a conjunct that pre-empts the consequent.
Therefore
>
is not strict
Deﬁnition
99K
is a
variably strict
conditional iﬀ
for every
p
,
q
there is a class
W
of possible worlds s.t.
p
99K
q
is true iﬀ
p
⊃
q
is true
at
w
, for every
w
∈
W
.
(4)
In general, (i) not every variably strict conditional satisﬁes (AS), how-
ever (ii) if
99K
is variably strict each instance of
p
99K
q
is strict.
We need a conceptual analysis of the truth-conditions of subjunctive conditionals
that explains the failure of (AS). An analysis of
>
as a variably strict conditional is
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Notation: I use ‘
99K
’ as a variable ranging over conditional operators (occasionally the range of
this is just restricted to the subjunctive and the indicative conditional).
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