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2250
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James Hildebrand
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Philosophy

2250

James Hildebrand

Spring

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MODAL SENTENTIAL LOGIC
(Many of the exercises and comments in these notes have been drawn from
John Nolt, Logics, Ch. 11 [Belmont, CA: Wadsworth, 1997]
which is recommended for further study of this topic.
I have also consulted John L. Bell, David DeVidi and Graham Solomon, Logical Options
[Peterborough: Broadview, 2001]
and J.C. Beall and Bas C. van Fraassen Possibilities and Paradoxes [Oxford: Oxford
University Press, 2003])
Consider the sentence, “Fish swim.”
As far as SL is concerned, we can affirm this sentence, or we can deny (or negate) it.
But affirmation and denial are not the only forms that the assertion of a sentence, like “Fish
swim,” can take. Here are some examples of what we might call “modifications” of the simple
assertion that fish swim:
Alethic modes:
Fish swim.
It is not the case that fish swim.
It is necessarily the case that fish swim.
It is possibly the case that fish swim.
It is contingently the case that fish swim.
Deontic modes:
It ought to be the case that fish swim.
It is permitted to be the case that fish swim.
Temporal modes:
It has always been the case that fish swim.
It has been the case that fish swim.
It will be the case that fish swim.
It will always be the case that fish swim.
Epistemic modes:
s knows that fish swim
s believes that fish swim
s doubts that fish swim
s fears that fish swim
s hopes that fish swim
Modal logic goes beyond sentential logic to recognize these various modes of assertion,
and others like them, to develop a symbolic language to translate them, and to work out the
formal syntax and semantics for reasoning that involves modal claims.
I propose to give you a brief introduction to modal logic. For simplicity’s sake we will look
just at some modes, the “alethic” or “truth” modifications.
In SL we recognized two alethic modes: assertion and negation. In the new formal language we are now going to develop, we will recognize four modes:
above assertion we will add assertion of necessity, and between assertion and denial we will
add assertion of possibility. We will use the box, “□,” to stand for necessity, and the diamond,
“◊” to stand for possibility.
So we will represent our four modes as follows, in order of decreasing strength of assertion:
□P – necessarily P
P – P
◊P – possibly P
~P –not P
Combining these connectives can enable us to translate expressions like:
Not necessarily P (~□P),
Possibly not P (◊~P),
Not possibly P (~◊P), and
Necessarily not P (□~P)
“Not necessarily” is logically equivalent to “possibly not” and “not possibly” is logically
equivalent to “necessarily not” so combining these intermediate modes with our original four
gives us the following more complete hierarchy of modal assertions:
□P –necessarily P
P – P
◊~P – possibly not P
◊P –possibly P
~P –not P
□~P – necessarily not P
We can also express the fact that while P is true, it is merely brute factually or contingently
(as opposed to necessarily) true. To say that something is contingently true is to say two things:
first, that it is as a matter of fact true, but second, that it could have been false or, equivalently,
that it is not necessarily true. That gives us:
P & ◊~P (or, equivalently: P & ~□P).
Inversely, to say that P is contingently false is to say first, that it is false, but second, that it
could have been true:
~P & ◊P.
To say that two different sentences are compatible with one another is to say that it is
possible for them to both be true:
◊(P & Q)
There are a number of other claims we might translate using our new notation:
“If grass is green, then it is necessarily possible that grass is green.” — “G ⊃ □◊G.” Necessarily, if grass is green, then it is necessarily possible that grass is green.” —
“□(G ⊃ □◊G).”
Green grass is compatible with tall grass, but incompatible with blue grass.” —
“◊(G & T) & ~◊(G & B)”
These translations are fairly straightforward, but there is one nicety of translation that merits
comment.
In English, when we want to say that a conditional is necessary, we will often “infix” rather
than “prefix” the necessity connective (that is, we will put it in the middle of the conditional rather
than out front). This can make it look as if the necessity connective attaches merely to the
consequent. But that is rarely what is actually intended. For example, understanding “must be”
to be another word for “is necessarily,” the statement:
If grass is green then it must be healthy.
Should properly be translated as:
□(G ⊃ H)
rather than as:
G ⊃ □H.
The first translation says that there is no case where the conditional sentence “if grass is
green then it is healthy” turns out to be false, and hence no case where grass is green but not
healthy. In contrast the second translation says that if there is any case where grass is green
then there is no case where it is not healthy, so that even in other cases where it is not green, it
still must be healthy (its being green in even one case guarantees that). The second is a quite
different (and much stronger) claim than the first, so we have to choose between them when
translating from English. Following our usual practice of interpreting ambiguous claims in the
way that is most likely for them to be true dictates that “□(G ⊃ H)” should be preferred as the
translation, even though the English places the necessity connective (“must”) in the consequent.
The time has come to be a bit more specific about the vocabulary and the syntax of the
formal language we have been using, MSL.
MSL is written on top of SL as follows (new elements are in blue for ease of comparison): The Formal Language MSL
(A Formal Language for Modal Sentential Logic)
I. The Vocabulary of MSL consists of:
Sentence letters of MSL A, B, C, ..., 1 , 2 , 3 , ...
Connectives of MSL: ~ , &, v, ⊃ , ≡, □, ◊
Punctuation Marks: ( , )
II. Syntax
An expression of MSL is any string of vocabulary elements.
Syntactic formation rules:
A sentence of MSL is a string of vocabulary elements assembled in accord with the following
formation rules:
1. If P is a sentence letter of MSL then P is a sentence of MSL.
2. If P is a sentence of MSL, then ~P, □P and ◊P are sentences of MSL.
3. If P and Q are (not necessarily distinct) sentences of MSL, then (P & Q), (P v Q), (P ⊃
Q), and (P ≡ Q) are sentences of MSL.
4. Nothing is a sentence of MSL unless it can be formed by repeated applications of rules
1-3.
(Note that in MSL, as in SL, punctuation is only to be used in conjunction with the two place connectives.
No string of however many one-place connectives, whether identical or different, calls for the use of
punctuation.)
Informal notational conventions:
- allow [, ] as substitutions for (,)
- allow omission of outer parentheses
Named forms:
If P is a sentence letter of MSL then it is called an atomic sentence.
If P is a sentence of the form ~Q then it is called a “negation.” Its main connective is “~” and its
immediate component is Q.
f P is a sentence of the form (Q & R) then it is called a “conjunction. Its main connective is “&”
and Q and R are its immediate components. These components are called “conjuncts.”
If P is a sentence of the form (Q v R) then it is called a “disjunction.” Its main connective is “v”
and Q and R are its immediate components. These components are called “disjuncts.”
If P is a sentence of the form (Q ⊃ R) then it is called a “conditional.” Its main connective is “⊃”
and Q and R are its immediate components. Q is called its “antecedent” and R is called its
“consequent.”
If P is a sentence of the form (Q ≡ R) then it is called a “biconditional.” Its main connective is “≡”
and Q and R are its immediate components.
If P is a sentence of the form □Q then it is called a “box sentence.” Its main connective is “□”
and its immediate component is Q. If P is a sentence of the form ◊Q then it is called a “diamond sentence.” Its main connective is
“◊” and its immediate component is Q.
The components of a sentence are that sentence itself, its immediate component(s), and
the components of its immediate components down to its atomic components.
Duality:
The dual of a conjunction, P & Q, is the result of negating everything and flipping the connective
to “v”: ~(~P v ~Q)
Similarly, the dual of P v Q is ~(~P & ~Q), the dual of □P is ~◊~P and the dual of ◊P is ~□~P
Exercise 1
1. Which of the following are sentences of MSL and which are not? For those that are, give a
demonstration by appeal to the formation rules that explains why they are. For those that are
not, explain why they can’t be formed in accord with the rules. Click on the links for answers to
unstarred questions.
a. ◊(□A)
*b. □A◊B
c. ◊~B
*d. □◊□A
e. □[◊(□A)]
*f. □~◊A
g. □□□A
*h. □A◊
2. Translate from English into MSL. (Note that all sentences but the last are true, and that the
last is not self-contradictory.) Click on the links for answers to unstarred questions.
a. If grass is green, then it is possible that it is green.
*b. If it is not possible that grass is green then it is necessary that grass is not green.
c. If it is possibly necessary that grass is green, then it is necessary that grass is green.
*d. If grass is green and snow is white, then grass being green is compatible with snow being
white.
e. Grass is necessarily green if and only if it is green but not contingently green.
*f. Necessarily, if grass is green then it is either necessarily green or contingently green.
g. If it is possible that grass is possibly green then it is possibly green.
*h. If grass is green then snow must be white, but even though grass is green it is not the
case that snow is necessarily white.
3. Translate from MSL into English, taking “G” to be “Grass is green.” (Note that all sentences
are true.) Click on the links for answers to unstarred questions.
a. □G ⊃ ~◊~G
*b. ◊□G ⊃ G
c. ◊G ⊃ □◊G
*d. ( □G ⊃ G) & (G ⊃ ~□G)
e. □G ⊃ □□G THE DERIVATION SYSTEM MSD
Once one has specified the syntax for a language, it is possible to formulate syntactic
derivation rules. We will do that now, building a derivation system, MSD, for the language MSL.
The rules of MSD are all the rules of SD plus the following ones:
Duality (D) – From either of ◊P and ~□~P derive the other; from either of □P and ~◊~P derive
the other.
K Rule (K) – From □(P ⊃ Q) derive (□P ⊃ □Q).
T Rule (T) – From □P derive P.
S4 Rule (S4) – From □P derive □□P.
Brower Rule (B) – From P derive □◊P.
Necessitation (N) – If P is a theorem of SD or MSD derive □P.
There is one feature of the application of these rules that you will find novel. This is the use
of the N rule. This rule is often used in conjunction with the K rule, so using it effectively
requires having a good sense, not only of what sentences of SL are theorems of SD, but of what
conditional sentences of SL are theorems of SD. These are sentences like “A ⊃ A,” “(A & B) ⊃
A,” and “[(A & B) ⊃ C] ⊃ [A ⊃ (B ⊃ C)].” Knowing what these sentences are puts you in a
position to know to use the N rule to enter sentences like “□(A ⊃ A),” “□[(A & B) ⊃ A],” and “□([(A
& B) ⊃ C] ⊃ [A ⊃ (B ⊃ C)]),” and then use the K rule to enter sentences like “□A ⊃ □A,” “□(A &
B) ⊃ □A,” and “□[(A & B) ⊃ C] ⊃ □[A ⊃ (B ⊃ C)].” This is important knowledge to have at your
disposal for certain derivations. In fact, the key to many derivations rests with thinking up a
theorem to use in conjunction with the N and K rules to get you from the premises of the
argument to its conclusion.
Often, what people will do is enter applications of the N rule as lemmas in their derivations
and then, off on the side, do another derivation in SD or MSD to show that the sentence they
applied the N rule to really is a theorem of MSD. (Note that since MSD is written on top of SD,
all theorems of SD are theorems of MSD.)
Here are some samples of derivations using these rules. You should study these examples
and try to derive the conclusions from the premises on your own (without looking at how it is
done here) before attempting the exercises below.
1. Show that {A} yields “◊A” in MSD.
1. A Assumption
2. □~A Assumption
3. ~A 2,
4. A 1,
5. ~□~A 2-~I 4,
6. ◊A D5,
This derivation illustrates another unique feature of derivations in MSD. In SD it is always a
good idea to assume the opposite of what one is trying to derive. But in MSD it is often better to assume the opposite of the dual of what one is trying to derive. (Recall that the dual of those
sentences that have duals is obtained by negating the whole and the parts and flipping the
signs. So the dual of “A & B” is “~(~A v ~B),” the dual of “A v B” is “~(~A & ~B),” the dual of □A
is ~◊~A, and the dual of ◊A is ~□~A.) Here, I want to derive ◊A. But assuming the opposite
would mean assuming ~◊A, which is not much help because there are no rules for deriving
anything from negated diamond sentences. However, we know that the D ru
le can be used to
derive ◊A from its dual ~□~A, and assuming the opposite of this dual, □~A, gives us something
we can work with.
2. Show that {□(A & B)} yields “□A” in MSD.
1. □(A & B) Assumption
2. □[(A & B) ⊃ A)] N
3. 2K,
□(A & B) ⊃ □A
4. □A 2, 1, ⊃E
Note that here “(A & B) ⊃ A” is assumed at line 2 to be a theorem of MSD for purposes of
application of the N rule. Strictly, this derivation should be followed by a derivation showing that
“(A & B) ⊃ A” really is a theorem of MSD. However, in this case the fact is so obvious that I
won’t bother to give that derivation. (A proof using just rules of SD suffices to show it.)
3. Show that {◊A} yields “◊(A v B)” in MSD.
1. ◊A Assumption
2. □~(A v B) Assumption
3. □[~(A v B) ⊃ ~A] N
4. □~(A v B) ⊃ □~A 3,
5. □~A 4, 2, ⊃E
6. ~□~A 1, D
7. ~□~(A v B) 2-5&6, ~I
8. ◊(A v B) 7, D
Here we adopt the strategy of assuming the opposite of the dual of the sentence we want. This
is done at line 2. Then we need to have the insight (which we ought to have gleaned from doing
trees) that ~(A v B) ⊃ ~A is a theorem of SD and so of MSD. That allows us to make use of the
N and K rules to derive □~A from □~(A v B). I leave the proof that ~(A v B) ⊃ ~A is a theorem of
SD and so of MSD to you as an exercise.
4. Show that “◊~A ⊃ ~□A” is a theorem of MSD.
1. ◊~A Assumption
2. □A Assumption
3. □(A ⊃ ~~A) N
4. □A ⊃ □~~A 3,
5. □~~A 4, 2, ⊃E
6. ~□~~A D 1,
7. ~□A 2-5&6, ~I
8. ◊~A ⊃ ~□A 1-7 ⊃I Here, we take it for granted that “A ⊃ ~~A” is a theorem of SD and so of MSD.
5. Show that “□~A ⊃ ~◊A” is a theorem of MSD.
1. □~A Assumption
2. ◊A Assumption
3. □~A 1,
4. ~□~A 2D,
5. ~◊A 2-3&4~,I
6. □~A ⊃ ~◊A 1-5, ⊃I
6. Show that {□(A ⊃ B)} yields □~B ⊃ □~A in MSD.
1. □(A ⊃ B) Assumption
2. □[(A ⊃ B) ⊃ (~B ⊃ ~A)] N
3. □(A ⊃ B) ⊃ □(~B ⊃ ~A) 2K,
4. □(~B ⊃ ~A) 2, 1, ⊃E
5. □~B ⊃ □~A 4K,
Here, we take it for granted that “(A ⊃ B) ⊃ (~B ⊃ ~A)” is a theorem of SD and so of MSD.
7. Show that {◊□A} yields “A” in MSD.
1. ◊□A Assumption
2. ~A Assumption
3. □◊~A B 2,
4. □(◊~A ⊃ ~□A) N
5. □◊~A ⊃ □~□A) 4K,
6. □~□A 5, 3, ⊃E
7. ~□~□A D 1,
8. A 2-6&7~, E
Here the theorem of MSD that we apply N to at line 4 is the one proven in example #4
immediately above.
8. Show that ◊◊A ⊃ ◊A is a theorem of MSD.
1. ◊◊A Assumption
2. □~A Assumption
3. □□~A S4 2,
4. □(□~A ⊃ ~◊A) N
5. □□~A ⊃ □~◊A 4,
6. □~◊A 5, 3, ⊃E
7. ~□~◊A D 1,
8. ~□~A 2-6&7, ~I
9. ◊A 8,D
10. ◊◊A ⊃ ◊A
Here the theorem of MSD that we apply N to at line 4 is the one proven in example #5
immediately above. 9. Show that {◊A} yields “□◊A.”
1. ◊A Assumption
2. □◊◊A B 1,
3. □(◊◊A ⊃ ◊A) N
4. □◊◊A ⊃ □◊A 3K,
5. □◊A
Here the theorem of MSD that we apply N to at line 3 is the one proven in example #8
immediately above.
Exercise 2
1. Prove that the following are theorems of MSD.
a. □P ⊃ □(P v Q)
b. □~~P ⊃ □P
c. ◊(P & Q) ⊃ ◊P
d. □Q ⊃ □(P ⊃ Q)
e. ~ ◊P ⊃ □(P ⊃ Q)
f. □P ≡ □□P
g. ~ ◊(P & ~P)
h. ( □P & □Q) ⊃ □(P & Q)
i. □P ⊃ □□□P
SEMANTICS OF MSL
Doing derivations in MSD raises the questions of whether the derivation rules are sound
and complete. Will the derivation rules never lead us from true premises to false conclusions?
Will they allow us to derive the conclusions of every argument of MSL that can’t have true
premises and a false conclusion? Might some of the rules be redundant? Might we be able to
put others in their places?
To answer these questions we need to know when sentences of MSL should b
e considered
to be true or false, that is, we need a semantics for MSL.
But it is not immediately obvious how to give a semantics for MSL within the confines of
classical (bivalent) logic.
Think of our four main alethic modal forms:
□P
P
◊P
~P If P is true and ~P is false, what do we say about □P and ◊P? Is □P somehow more than
true? Is ◊P something halfway between true and false? Pursuing this path would lead us out of
the ambit of classical logic, which recognizes only two truth values.
It is important to appreciate that there is a real philosophical problem here — one that is a
problem precisely because there is no obvious solution. We can do a certain amount of modal
logic without addressing this problem. We can write up a vocabulary and a syntax for a formal
language for modal logic, and we can formulate some derivation rules that seem logically sound
to us to help us in reasoning from modal premises to modal conclusions. But if we want to be
sure that our intuitions about what derivation rules are logically sound are correct, then we need
to formulate a semantics.
The problem is that there is no universally agreed upon semantics for modal logic. (For that
matter, there is no universally agreed upon semantics for sentential logic. The semantics we
studied last semester, classical semantics, is challenged by logicians who would like to
recognize more than two truth values, by logicians who would like to insist that the premises of
arguments be relevant to their conclusions, and by logicians proposing yet more radical
revisions to classical sentential semantics. But in modal logic there is less consensus.)
Working out a semantics could lead us to revise our list of derivation rules, dropping some and
adding others (it could correct our intuitions about what system of rules is sound and complete).
It could also put us in a position to devise a tree method for modal logic. But if we can’t come
up with an agreed-upon semantics, we will end up with different systems of modal logic.
So far, we don’t have an agreed-upon semantics. And we do have different systems of
modal logic, based on different versions of the semantics for modal logic. It remains a
philosophical question which one is the “correct” or the “best” one. But there is one proposed
semantics for modal logic that is fairly simple, consistent with bivalence (the tenet that there are
just two truth values), and agreeable to our intuitions concerning the conditions under which
modal statements are true or false. We will study this semantics, Leibnizian semantics, now.
To begin, note that one way to explain the difference between the alethic modes
consistently with bivalence might be to say that the difference comes down to th
e fact that □P is
always true whereas ◊P is just sometimes true. But then what do we mean by “always” and
“sometimes?” If we really mean at all times and at some times, then we have slipped into doing
temporal modal logic. Alternatively, if we mean “on all truth value assignments” and “on some
truth value assignments” then we have reduced “□” to a symbol for truth-functional truth, and “◊”
to a symbol for not being truth-functionally false. And we don’t need modal logic for that. The
semantic tests for SL are already adequate to pick out the t-f true sentences and the non-t-f
false sentences, and the derivation rules of SD are already adequate to pick out theorems and
antitheorems.
An alternative way of giving a robust, independent meaning to “□” and “◊,” due to Leibniz, is
to take “always” to mean “in all regions” rather than “at all times,” or, more extravagantly, to
mean “in all worlds.” Correspondingly, “sometimes” means “in some regions” or “in some
worlds.” Think of a world as another way this world could have been: this world without you in it;
this world without Canada in it; this world with blue grass; this world with different physical laws.
— Any variant on the way this world is, however minor or however radical (so
long as it
continues to contain at least one object and does not contain any logically contradictory states
of affairs), counts as a “world.” On this account, □P, P, ◊P, and ~P are not stratified along a line. Instead, □P and ◊P are
trans-world statements, whereas P and ~P are local world statements. □P and ◊P are
statements that are either true in all worlds or false in all worlds, whereas P and ~P are
statements that can be true in one world, but false in another, so that you cannot say that P is
true or false without specifying in what world it is true or false.
Leibniz’s account further suggests that if □P is true it is because its immediate component,
P, is true in all worlds, whereas if □P is false it is because there is at least one world in which P
is false. Conversely, if ◊P true it is because P is true in at least one world whereas if ◊P is false
it is because there is not even one world in which P is true.
So we get this sort of stratified picture:
P is true in all worlds □P
P is true in some worlds, false in others ◊P & ◊~P
P is false in all worlds □~P or, equivalently, ~◊P
Consider an example. Suppose we have two sentences, “A” and “B” and five worlds, w , 1
w 2 w3, w4, w5. We can represent the truth values of each sentence in each world on a table as
follows:
A B
w1 T T
w2 F T
w F F
3
w4 F T
w5 T T
On this model, □(A & B) is false. Moreover, it is not just false in some worlds2 w 4w . It is
false in every world, even w1and w 5 in which its immediate component, “A & B” is true. This is
because there are some worlds, w -2 ,4in which at least one of the conjuncts of “A & B” are
false, making “A & B” false in those worlds. According to Leibnizian semantics, if “A & B” is
false in even one world, then “□(A & B)” is false in all worlds, even those in which “A & B” is true.
In those other worlds, the truth of “A & B” is like the truth of “Grass is green and snow is white”
in this world — the sort of claim that is true without being necessarily true because we think
there are other ways our world could have been — worlds in which gras
s and snow have other
colours.
In contrast, ◊(A & B) is true in all worlds, even w -w , in which “A & B” is false. This is
2 4
because there are some worlds in which both “A” and “B” are true, making “A & B” true in those
worlds. And according to Leibnizian semantics, if “A & B” is true in even one world, then
“◊(A & B)” is true in all worlds, even those in which its immediate component, “A & B” is fal
se.
This illustrates an important point about Leibnizian semantics. The truth value that box and
diamond sentences have in a world is not just determined by the truth value that their immediate
components have in that world. Instead, one needs to consider the truth value of their
immediate components in the other worlds as well. Finding one world in which an immediate
component is false suffices to make a box sentence false in all worlds, but for a box sentence to
be true nothing less will suffice than finding its immediate component to be true in all worlds.
Similarly, finding one world in which its immediate component is true su
ffices to make a diamond sentence true in all worlds; nothing less than finding its immediate component to be
false in all worlds suffices to make it false.
Let’s continue to think about how this cashes out in the model I have given above. On that
model, “A v B” is true 1n w , but “□(A v B)” is false in all the worl1s, including w . That is
because both “A” and “B” are false 3n waking “A v B” false in that world. And if the
immediate component of a box sentence is false in any one world, then that box sentence is
false in all worlds, including those worlds in which the immediate component happens to be
true. In contrast, ◊(A v B) is true in all the worl3s, even w , in which its immediate component is
false.
In light of these reflections, we can complete the truth table given above as follows:
A B A & B □(A & B) ◊(A & B) A v B □(A v B) ◊(A v B)
w1 T T T F T T F T
w2 F T F F T T F T
w3 F F F F T F F T
w F T F F T T F T
4
w5 T T T F T T F T
(Note that box and diamond sentences will always have either a column o
f T’s or a column
of F’s under their main connectives. This is just what it means to say that they are trans-world
statements.)
In contrast to the box sentences we have looked at so far, □(A ⊃ B) is true in all worlds of
the model given above. That is because on that model there is no world in which “A ⊃ B” is
false.
This may be striking. We know that “A ⊃ B” is not t-f true and that “if A then B” is not
logically true. Yet “□(A ⊃ B)” turns out to be true in every world of the mod

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