Lecture 27: Pragmatics and the
Paradoxes of Material Implication
1
Goals Today
✤
Recall and further motivate the paradoxes of the material conditional,
also known as the paradoxes of material implication.
✤
Discuss some responses to these paradoxes, primarily from
pragmatics but also from modal logic.
✤
This topic is the ideal topic with which to end this course, since it
involves reviewing several different ideas from previous lectures.
2
Meaning of Logical Particles
✤
By “logical particles” we just
mean words like “and”, “or”,
“not”, “if . . . then. . . “ and
“some” and “all.”
✤
In some sense, the entire course
thus far has been concerned
with articulating one specific
answer to the question:
✤
what is the meaning of the
logical particles?
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✤
Our answer: the meaning of the
logical particles is given by
their translations into
propositional and predicate
logic via ∧, ∨, ¬, →, ∃, ∀, and
this answer is adequate because
the translation preserves truthconditions.
✤
Today we’re going to talk about
challenges to the claim that the
truth-table for → is a good
translation of “if . . . then . . . “
Recall: Validity
✤
Suppose ϕ1, …, ϕn , ψ, are formulas of propositional logic. Then we
say that
ϕ1, …, ϕn ⊨ ψ if a truth-table which contains columns for ϕ1, …, ϕn, ψ has this
feature: whenever a row contains a T in each of the ϕ1, …, ϕn columns,
this row also has a T in the ψ column.
✤
We call ϕ1, …, ϕn the premises and we call ψ the conclusion. And the
argument with premises ϕ1, …, ϕn and conclusion ψ is said to be valid
if ϕ1, …, ϕn ⊨ ψ.
✤
Intuitively, valid arguments have the feature that the truth of the
premises guarantees the truth of the conclusion.
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Bibliographic Remark
✤
In addition to our Gamut
textbook, the following
discussion owes much to the
books from which I learned:
✤
Chapter 1 of: Graham Priest.
An Introduction to NonClassical Logic.
✤
Chapter 1 of: Routley et. al.
of Relevant Logics and Their
Rivals
✤
✤
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Edgington Pragmatics of the
Logical Constants
For electronic copies of these,
simply search for “Priest” or
“Routley” on the course
website.
Four Paradoxical Features of the
Material Conditional
✤
One can check with truth-tables
that we always have these
validities:
✤
✤
✤
When we translate → by
“if . . . then . . .“ we get:
✤
1’. b. Therefore, if a then b.
✤
2’. Not a. Therefore, if a then b.
✤
3’. It’s not the case that if a
then b. Therefore, a.
✤
4’. If a and b then c. Therefore,
if a then c, or if b then c.
1. b ⊨ a→b
2. ¬a ⊨ a→b
✤
3. ¬(a→b) ⊨ a
✤
4. (a∧b)→c ⊨ (a→c)∨(b→c)
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Why they are paradoxical:
the first two in more detail
Supposing that → is a good
translation of “if . . . then . . .”,
we have that the following are
valid arguments:
✤
1’’. Seattle is in Washington.
Therefore, if Spokane is in
Washington then Seattle is in
Washington.
1’. b. Therefore, if a then b.
✤
2’’. Seattle is not in California.
Therefore, if Seattle is in
California then LA is in
California.
✤
What comes before the
“therefore” is true, but what
comes after it seems false.
✤
2’. Not a. Therefore, if a then b.
✤
This is paradoxical because
valid arguments are arguments
where truth of premises
guarantees truth of conclusion.
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Why they are paradoxical:
the second two in more detail
✤
Supposing that → is a good
translation of “if . . . then . . .”,
we have that the following are
valid arguments:
✤
3’. It’s not the case that if a
then b. Therefore, a.
✤
4’. If a and b then c.
Therefore, if a then c, or if b
then c.
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✤
3’’. It’s not the case that if there
is an earthquake then the stock
market will rise. Therefore there
is an earthquake.
✤
4’’. If you work hard and are
lucky then you will be
successful. Therefore, if you
work hard then you will be
successful, or if you are lucky
you will be successful.
The Response from Modal Logic
✤
One of the original aims of
modal logic was to find
responses to some of these
paradoxes.
✤
The suggestion from modal
logic was that “if a then b” should be translated not as
“a→b” but rather as “□(a→b).”
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✤
The motivation for this was that
many things can be true by
accident. It is sunny but it may
have rained. But if you say
something is necessary, you’re
saying it’s not true by accident.
✤
Hence, maybe when we’re
saying “if a then b” what we’re
saying is not merely that a→b,
but that this is no accident: it
must have been so.
Advantages of Modal Response
✤
✤
The suggestion from modal
logic was that “if a then b” should be translated not as
“a→b” but rather as “□(a→b).”
Again, the whole idea here is
that something can be possibly
true without being true.
✤
So there will be many natural
situations in which we have
⬦(a∧¬b) true while a is false.
Hence:⬦(a∧¬b) ⊭ a
Or what is same:¬□(a→b) ⊭ a
✤
So translating “if a then b” by
“□(a→b)” doesn’t commit us to
✤
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3’. It’s not the case that if a
then b. Therefore, a.
Disadvantages of Modal Response
✤
✤
The suggestion from modal
logic was that “if a then b” should be translated not as
“a→b” but rather as “□(a→b).”
✤
✤
The modal response does not
do too well with the inference:
✤
For, something closely
related comes out valid on
many modal semantics using
new translation of if-then:
✤
1’. b. Therefore, if a then b.
Hence,
✤
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1’’. Necessarily b.
Therefore, if a then b.
1’’’. Necessarily 2+2=4.
Therefore, if Spokane is in
Washington then 2+2=4.
The Response from Pragmatics
✤
✤
The response from pragmatics
is that the paradoxical features
of the conditional don’t indicate
that we’ve mistranslated “if . . .
then . . . “ by the arrow →.
Rather, these features of the
conditional merely seem
paradoxical to us because it
would be highly unusual to
actually say these things,
because very often they would
violate Grice’s maxims.
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✤
Consider again:
1’. b. Therefore, if a then b.
✤
If you actually knew b, then
why would you tell me “if a
then b”? One of the few reasons
we have for asserting “if a then
b” is that we want to figure out
whether b holds and rules like
this help us. Given this fact
about us and how we talk, it
makes sense you wouldn’t say
“if a then b” when you know b.
Advantages of Pragmatic Response
✤
Likewise, pragmatics handles
nicely the case
✤
2’. Not a. Therefore, if a then b.
✤
You know “not a.” I then come
up to you and inquire whether
a holds. Given what you know,
it would be misleading to
answer “if a then b”. For, this
rule is often used to establish
“not a.” So you implicate that
you don’t know “not a”.
✤
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There is a general principle at
work here: usually when we
assert an “if . . . then . . . “
statement in response to a
question, either the speaker or
listener is uncertain of how
exactly things stand with the
antecedent and consequent. For,
we communicate conditionals
to one another to help one
another figure out how things
stand with the antecedent and
consequent.
Disadvantages of the Pragmatic
Response
✤
One disadvantage of the
pragmatic response is that it
does not handle well the
inference:
✤
✤
✤
Everyone agrees that these
examples sound strange. The
challenge is to explain why
this is that doesn’t involve
radically changing the
meaning of “if . . . then . . . “.
✤
Saying “a” doesn’t violate
any of Grice’s norms. It’s
relevant to the conditional
and it’s much more specific
and informative than “not if
a then b.”
3’. It’s not the case that if a
then b. Therefore, a.
3’’. It’s not the case that if
there is an earthquake then
the stock market will rise.
Therefore there is an
earthquake.
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Disadvantage of the Pragmatic
Response
✤
Consider this conversation:
✤
What’s going on here?
✤
Person 1. “I think the economy
will not recover this year.”
✤
✤
Person 2. “Regrettably, I concur.
But if the economy did recover
this year, taxes would increase.”
Well, 1 and 2 agree that ¬a,
where a = economy gets better
this year. But 1 thinks that a→b
while 2 thinks that a→¬b, where
b = taxes increase.
✤
Truth-tables say both are
equally right. Experience
suggest that they aren’t equally
right. Pragmatics seems
irrelevant since they cooperate.
✤
Person 1. “I disagree. There are
certain mechanisms that would
prevent tax increases even in
the event of a recovery.”
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Recap
✤
We’ve looked today at three
traditional ways of interpreting
the word “if . . . then . . .”
✤
Let’s call the first strategy the
material strategy: this strategy
just says that “if a then b “
should simply mean “a→b”.
✤
The second strategy was the
modal strategy: this strategy
said just says that “if a then b”
should mean “□(a→b).”
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✤
Finally, we’ve looked at the
pragmatic strategy, which
agrees with the material
strategy that if a then b “ should
simply mean “a→b”, but
additionally counsels that we
should pay attention to the
effects of the ambient
conversation in understanding
our reactions to various
inferences.
✤
Let’s look at the scorecard. . . .
Scorecard. Key: ✕ means handles
poorly, ✓ means handles well.
Inference
Material
Modal
Pragmatic
b. Therefore, if a then b
✕
✕
✓
Not a. Therefore, if a then b
✕
✕
✓
It’s not the case that if a then b.
Therefore, a.
✕
✓
✕
If a and b then c. Therefore, if a
then c, or if b then c.
✕
✓
?
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Goals Today
✤
Recall and further motivate the paradoxes of the material conditional,
also known as the paradoxes of material implication.
✤
Discuss some responses to these paradoxes, primarily from
pragmatics but also from modal logic.
✤
This topic is the ideal topic with which to end this course, since it
involves reviewing several different ideas from previous lectures.
18
Ω
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