Lecture 24: Motivating Modal
Logic, Translating into It
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Goal Today
✤
The goal today is to motivate modal logic, a logic that extends
propositional logic with two operators ⬦ (diamond) and □ (box).
✤
We do this by examining how we talk and reason about words like
“might”, “possible”, “can” (which we translate by ⬦) and words like
“must” and “necessary” (which we translate by □), and their formal
similarities with words like “knows” and “believes”.
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A Problem Case
✤
Consider
✤
✤
✤
(1) It might rain, but it might
not rain.
Can we compositionally
translate this into propositional
logic in a way that preserves
truth-conditions? Our usual
method seems to deliver
incorrect results.
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For, usual method suggests:
✤
Translate “it might rain” as p
✤
Translate “but” as ∧
✤
Translate 2nd conjunct as ¬p
✤
So our usual method suggests
translating (1) as “p∧¬p.”
✤
But this seems wrong, since
contradictions are never true.
And it seems like (1) is true a lot
of the time.
Some Implausible Responses
✤
(1) It might rain, but it might
not rain.
✤
So our usual method suggests
translating (1) as “p∧¬p.”
✤
One implausible response is:
move to a 3-valued logic, where
p∧¬p has “third” truth-value. If
idea is to always give (1) the
“third” truth-value, then it’s
implausible because sometimes
(1) seem definitely true.
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✤
Another implausible response
would involve moving to
predicate logic. This is
implausible because there don’t
seem to be any uses of “for all”
or “there is” in the statement.
✤
But maybe it contains the
kernel of a plausible response:
the idea should be to enrich our
language to handle sentences
like (1), just like we enriched
our logic with ∃,∀.
Generalizing the Example
✤
Such enrichment would be
uninteresting if this was an
isolated example. But consider
other examples much like this
one:
✤
(2) Anne can attend the
meeting, but she does not.
✤
(3) Claire must do her work,
but she does not.
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✤
Again, if we employed the
method used so far, it seems
like we may be tempted to
translate these by p∧¬p.
✤
But clearly when someone
says (2), they are not
contradicting themselves, i.e.
they are not saying that Anne
can and Anne can’t do
something.
A Guiding Idea
✤
Consider the way we talk about
our beliefs and our other
attitudes like knowledge.
✤
✤
✤
✤
An obvious way to translate
these sentences would be to
introduce operators
K for knowledge and B for belief.
✤
Then we would translate as:
(4) Anne believes that
skidiving is safe, but it isn’t.
(5) If the earnings report is
low, then Claire knows it.
(6) Clark flies, but Lois
doesn’t know that.
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✤
(4’) B(p) ∧ ¬p
✤
(5’) ℓ➝K(ℓ)
✤
(6’) c ∧ ¬K(c)
More on the Idea
✤
So we introduce two operators,
K for knowledge and B for belief.
✤
And we translate
✤
✤
(4) Anne believes that
skidiving is safe, but it isn’t
by (4’) B(p) ∧ ¬p, with the key:
p = skidiving is safe
Bq = Anne believes q 7
✤
A key feature of this translation,
as revealed by how we write out
the key, is that for each formula
q, there is another formula Bq
and another formula Kq.
✤
Obviously if we wanted to get
more elaborate and subscript, we
could translate rather as
(4’’) Ba(p) ∧ ¬p
with key p = skidiving is safe
Ba(q) = Anne believes q
An Obvious Distinction
✤
We introduce two operators, K
for knowledge and B for belief.
✤
Introducing these operators
gives us a way to mark a very
intuitive and obvious
distinction. Consider
✤
✤
7) Anne does not know that
it is raining.
8) Anne knows that it is not
raining.
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✤
These are describing very
different situations. (7) is true
when Anne is ignorant of
weather facts, while (8) is
true if Anne knows certain
weather facts.
✤
This distinction is mirrored
in our translations:
✤
7’) ¬K(p)
✤
8’) K(¬p)
Importing the Guiding Idea
✤
Our original problem case was:
✤
✤
✤
✤
(1) It might rain, but it might not
rain.
Let’s first note that (1) seems
equivalent to:
✤
Let’s introduce a symbol ⬦ for
“might.’’ Then we translate “it might be the case that p” by ⬦p.
✤
This operator is pronounced
“diamond” and ⬦p is pronounced as “diamond p.”
Hence we see that we can translate
as (1) and (1*) as
✤
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(1*) It might be the case that it
rains, but it might be the case that
it does not rain.
(1’) ⬦p∧⬦¬p
with key: p = it rains
Seeing “Might” Distinctions
Original English Sentence:
It might rain
Original EnglishSentence:
It might not rain
Equivalent English Sentence:
It might be the case that it rains
Equivalent English Sentence:
It might be the case that it does not rain
Translated Sentence:
⬦p
Translated Sentence:
⬦¬p
Original English Sentence:
It’s false that it might not rain
Original English Sentence:
It’s false that it might rain.
Equivalent English Sentence:
It’s not the case that it might be the case that it does not
rain.
Equivalent English Sentence:
It’s not the case that it might be the case that it rains
Translated Sentence:
¬⬦p
Translated Sentence:
¬⬦¬p
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Seeing “Might” Distinctions
⬦p
⬦¬p
¬⬦p
¬⬦¬p
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From Might to Know
✤
When we write in terms of the symbols ⬦ and ¬, we automatically see
the distinctions.
✤
Part of the difficulty in the case of “might” is usually we first have to
write the English sentences involving “might” as “it might be the case
that” before we can translate into the symbols like ⬦ and ¬.
✤
When we do the analogous set of distinctions in the case of “know”
there’s just not this intermediary step, as the following examples
show.
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Seeing “Knows” Distinctions
Original English Sentence:
Anne knows that it is raining
Original EnglishSentence:
Anne knows that it is not raining
Translated Sentence:
Kp
Translated Sentence:
K(¬p)
Original English Sentence:
Anne does not know that it is raining
Original English Sentence:
Anne does not know that it is not raining
Translated Sentence:
¬K(p)
Translated Sentence:
¬K(¬p)
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Seeing “Knows” Distinctions
K(p)
K(¬p)
¬K(p)
¬K(¬p)
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Recipe + examples: “Might”
✤
✤
✤
First, replace each instance of
“it might . . . . “ with the more
cumbersome “it might be the
case that . . . “
Second, translate each instance
of “Anne/Bill/Claire might
yadayada” with “It might be
the case that Anne/Bill/Claire
does yadayada”
✤
Example: Bill might attend the
meeting.
✤
So we write the equivalent
sentence: “It might be the case
that Bill attends the meeting.”
✤
Third, translate “it might be the
case that p” by ⬦p.
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Then we translate as ⬦p with key: p = Bill attends the
meeting.
Recipe + examples: “Might”
✤
✤
✤
First, replace each instance of
“it might . . . . “ with the more
cumbersome “it might be the
case that . . . “
Second, translate each instance
of “Anne/Bill/Claire might
yadayada” with “It might be
the case that Anne/Bill/Claire
does yadayada”
✤
Example: Anne might attend
and Bill might not attend.
✤
First we write equivalent “It
might be the case that Anne
attends, and it might be the case
that Bill does not attend.”
✤
Third, translate “it might be the
case that p” by ⬦p.
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Then translate by: ⬦a ∧⬦¬b,
with key:
a = Anne attends, and b = Bill
attends meeting.
Recipe + examples: “Possible”
✤
✤
✤
The locution “it is possible that”
seems very similar to “it might
be the case that.”
For, if it is possible that p, then
it might be the case that p.
Likewise, if it might be the case
that p, then it’s possible that p.
However, unlike “might”, it
doesn’t seem that “possible”
needs to be “expanded”, and
hence it’s easier to translate.
✤
So there is simply one step in
the recipe:
✤
Translate “it is possible that p”
by ⬦p.
✤
Example: if it is possible that
Anne gets the job, then it is
possible that Anne is wealthy.
✤
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Translation: ⬦p➝⬦q, with key:
p = Anne gets the job, q = Anne is wealthy.
Recipe + examples: “Possible”
✤
So there is simply one step in
the recipe:
✤
Translate “it is possible that p”
by ⬦p.
✤
Example: If Anne attends the
meeting, then it’s possible that
Bill attends the meeting.
✤
✤
✤
✤
Translation: a→⬦b, with key:
a= Anne attends the meeting,
b= Bill attends the meeting.
✤
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Example: It’s not possible that
Anne attends the meeting [she
is traveling!], and it’s not
possible that Bill does not
attend the meeting [he is in
town and definitely coming!]
Translation: ¬⬦a ∧ ¬⬦¬b
Example: it is possible that if
Anne attends, then Bill attends.
Translation: ⬦(a➝b).
General Discussion: Can vs. Might
✤
✤
(1) Consider “Bill can attend the
meeting.” If we say this after
looking at Bill’s calendar and
seeing that he is available, then
this seems very close to “Bill
might attend the meeting.”
But there’s a use of “can” that
doesn’t fit well with “might”. (2) “Bill can read Chinese.” It
seems I’m ascribing a capacity
or skill to Bill, and not just
saying that it might happen.
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✤
Turns out modal logic doesn’t
have much to say about
capacities.
✤
But since things like “Bill can
attend the meeting” do occur
naturally, we just agree to
translate this as ⬦b, where of
course b = Bill attends the
meeting. In essence, this is
borne of a tacit agreement to
focus on examples more like (1)
and less like (2).
General Discussion: Must
✤
✤
In the past slides, we’ve seen
that “might” and “it is possible
that” and “can” have closely
related ranges of application,
and so we just translate with a
single symbol ⬦.
✤
Seems that (1) and (2) could be
false while (1’) and (2’) true:
✤
(1’) Bill might attend
✤
(2’) It might be the case that it
rained today in Irvine
✤
Hence, “might” means
something different from
“must.” It’s so different that we
need a new symbol.
But what about “must”?
✤
(1) Bill must attend.
✤
(2) It must be the case that it
rained today in Irvine.
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Basic recipe + examples: “Must”
✤
In short, translate “it must be
the case that p” by □p. This is
pronounced “box p.”
✤
Example: if it rains then it must
be the case that the sidewalks
are wet.
✤
Translation: r→□w, with key: r= it rains, and w = sidewalks are wet.
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✤
“It must be the case that Bill
attended the meeting. If Bill did
not attend the meeting, then
people would have noticed.”
✤
Translation: □b. ¬b➝p
with key: b = Bill attends the
metting, p = people would have
noticed.
General recipe+examples: “Must”
First, translate each instance of
“Anne/Bill/Claire must
yadayada” with “It must be the
case that Anne/Bill/Claire does
yadayada”
✤
✤
First we replace by the
equivalent “It must be the case
that Anne attends, and if it
must be the case that Anne
attends, then it must be the case
that Bill attends.”
✤
Second, translate by
□a ∧ □a→□b
with key:
a = Anne attends
b = Bill attends
Second, translate “it must be
the case that p” by □p.
✤
Example: Anne must attend,
and if Anne must attend, then
Bill must attend.
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General recipe+examples: “Must”
First, translate each instance of
“Anne/Bill/Claire must
yadayada” with “It must be the
case that Anne/Bill/Claire does
yadayada”
✤
✤
Second, translate “it must be the
case that p” by □p.
✤
✤
Example: “Anne must attend or
Bill must attend, but it must be
the case that not both
attend” (They don’t like each
other at all!).
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First we replace by the
equivalent “it must be the case
that Anne attends or it must be
the case that Bill attends, and it
must be the case that it’s not the
case that Anne attends and Bill
attends.”
Second, translate as (□a ∨ □b) ∧ □(¬(a ∧ b))
General Discussion: Necessity
✤
So as we saw earlier, there’s a
close connection between might
and possible
✤
Similarly, there’s a close
connection between
must
and necesssary
✤
One difference was “might”
occurs both in “it might” and
“Anne might” locutions, while
“possible” only occurs in “it is
possible that” locutions.
✤
Hence, we translate “it is
necessary that p” as □p.
✤
Examples: It is necessary that
Bill attends. If Bill does not
attend, then Bill loses his job.
✤
Translation: □b. ¬b➝ℓ
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Paradigm Examples
Original Sentence
Equivalent Sentence
Translation
It might snow
It might be the case that it snows
⬦p, key: p = it snows
Anne might attend
It might be the case that Anne attends
⬦p, key: p = Anne attends
It’s possible that Anne attends
⬦p, key: p = Anne attends
Anne can attend
⬦p, key: p = Anne attends
It must have rained
It must be the case that it rained
□p, p = it rains
Bill must attend
It must be the case that Bill attends
□p, p = Bill attends
It is necessary that Bill attends
□p, p = Bill attends
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Goal Today
✤
The goal today is to motivate modal logic, a logic that extends
propositional logic with two operators ⬦ (diamond) and □(box).
✤
We do this examining how we talk and reason about words like
“might”, “possible”, “can” (which we translate by ⬦) and words like
“must” and “necessary” (which we translate by □), and their formal
similarities with words like “knows” and “believes”.
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Ω
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