VARZI - Modal Logic - Class Notes

8/8/2019 VARZI - Modal Logic - Class Notes

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Modal Logic (G4424) Achille C. Varzi

1 INTRODUCTION

1. What is modal logic?

x Basic ideas:

Modal logic is an extension of (not an alternative to) ordinary logic.

It is concerned with logical facts (e.g., inferences) that involve modalities, i.e., qualifications of

propositions.

Example: the following statement is true.

Neil is Canadian.

But we may want to qualify this truth: the statement is true, but could have been false (things

could have been otherwise).

On the other hand, the following statement is not just true: it is necessarily true (could not have

been false)

Neil is Neil.

You might want to say that this is just a logical fact: self-identities are logically true (true under

every interpretation of the non-logical constants). Then consider

Neil is a person.

Hesperus is Phosphorus.

Nothing is in two places at the same time.

These are not true in every logically possible model (i.e., under every way of interpreting our

language). Yet, arguably, they are true in every possible world. (E.g., since Hesperus and

Phosphorus pick out one and the same object, that object could not be different from itself.)

Modal logic should not be confused with model theory.

x Of course, there is plenty of room for controversy:

could I have been fatter? could I have been a woman?

could I have been that chair?

could I have been a mosquito?

could I have been the French Revolution?

could I have been the number 7?


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x The basic picture (mostly from ARISTOTLE, De Interpretatione, chapters 12-13)

true false

necessary

possibl e

impossible

contingent

x Modal logic is interested in the interrelationships between these modalities.

Not interested in their nature

logical

metaphysical

physical

sociological

etc.

Not an explanation (on pain of circularity), but a help in understanding

2. Two questions

x Is this the business of logic?

One could argue it is just a matter of semantics (explain the meaning of necessary etc.)

But the same could be said of ordinary logic (explain the meaning of not, and, all, etc.)

One could argue it is a matter of theorizing (e.g., axiomatic characterization of necessity, etc., or

axiomatic characterization of identity, parthood, and other notions)

(I would be happy to change the lable to Theory of modality, or something like that.)

x Couldnt we deal with modalities without invoking anything else than standard logic?

Four possible strategies:

1) Modalities as truth-functional connectives

However, there are only four possible truth functions

p 1 2 3 4

T T F T F

F F T T F

and none is adequate to model e.g. necessity:


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1 yields p p (determinism)

2 yields p p

3 violates p p and makes a trivial operator anyhow

4 validates p p " "

In general, from the T ofp we dont know anything about the T of p

In general, from the F ofp we dont know anything about the F of p

Besides, there are many other cases of non-truth-functional connectives

It is well-known thatp

According to Mary, p

It is surprisingly the case thatp

p because q

p, and as a consequence q (non according to Davidson)

2) Move everything to the metalanguage

Modalities as metalogical properties of sentences (necessity = validity, modulo a certain

selection of admissible models)

But then we may want to formalize the metalanguage...

3) Modalities as metalinguistic predicates of sentences (cp. truth-predicate) (Carnap 1937)

It is necessary that ... .... is necessarily true

Quine 1963: the lowest grade of modal involvement:

Necessity resides in the way in which we say things, and not in the things we talk about [p. 176]

But Montague 1963 shows this is no goodcp. Tarskis problems with truth predicate.

Only recently this strategy has been reconsidered: see Schwartz 1992.

4) Translation using quantifiers

Necessarily p w p(w)

Possibly p w p(w)

But this involves a number of complications

quantify over possible world

propositions become predicates of worlds

problems when one moves on to quantified modal logic (ariety)

(Come back to this later.)


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3. Need for a logic (Theory) of modality

x So: think of modal logic as involving non-truth-functional connectives.

Notation: , or N, M.

x What is the logic of this extended language?

We need not only laws such as p q

p________

q

but also specific principles such as (pq)p

________

q

x The choice of valid principles may be a controversial matter. It depends

on ones views on what is necessary etc.

on the specific interpretation of and .

x Necessity and possibility are alethic modalities (modifications of truth = aletheia), but there are also

Epistemic, deontic, temporal, spatial modalities.

Indeterminacy

Provability

A A is provable (relative to some one formal system, e.g., Peano Arithmetic)

Gdels second incompleteness theorem would read:

x And there are other uses, too. For example:

Intuitionism (A |/A)

MacKinsey-Tarskis map I(p) p

I() I(A) (negation as impossibility)

I(B) (I(A) I(B))

[Note: This means that intuitionisma restriction of classical logicmay be interpreted as anextension of it.]


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2 SENTENTIAL MODAL LOGIC (1)

1. Syntactic Preliminaries: the Modal Language

x Vocabulary:

atomic formulas: P0,P1,P2, . . . connectives: ,T, , , , , , , .

metavariables: A, B, C, .....

x Grammar:

Straightforward.

Only be careful to distinguish necessity of the consequence vs necessity of the consequent.

1) (A)

2) A

Obviously different:

(PP) (PP is a tautology)P P

Often ambiguous in English

If I have no money, then I can't buy a new computer this probably corresponds to 1)

If I am a man, then I can't be a number this probably corresponds to 2)

2. Semantics

x Extensional models are (intuitively) possible worlds

Each model is a way of partitioning the atomic sentences into true and false:

(P i){T,F} for all ior simply

{P0,P1,P2, . . .}


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Formally: = W,P, where

W (the possible worlds)

P a sequence P0, P1, P2, . . . W associating with each i a set of worlds (those in

whichP

i holds) Intuitively: Pi = the proposition expressed by P i

Truth conditions:

=P i iff PiM M M= A iff = A for every W

= A iff = A for some W

Notes:

Pi may be empty i Pi may not add up to W

x Second idea (Kripkes standard models): Generalize Leibnizian models by relativizing possibilities:

Model = W,R, P, where

Wand P as before

R W W

[Intuitively:R is a relation of accessibility: R means that is possible relative to ]

Truth conditions:

=P i iff PiM M M= A iff = A for all Wsuch that R

= A iff = A for some Wsuch that R

NB: If R is an equivalence relation, this is equivalent to the first account.

x Third idea (Montagues minimal models): Necessity and possibility should not be understood in

terms of truth in every/some world, but treated as primitive:

Model = W,N, P, where Wand P as before

N: WWassociates each world with the propositions that are necessary at that world

W= sets of worlds = propositionsW= set of propositions

[Intuitively:N = the propositions that are necessary at ]


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Truth conditions:

=P i iff PiM M M= A iff {W: = A} N = iffA expresses a necessary proposition at

= A iff {W: A} N = iffA does not express an impossible proposition at

Notation:||A|| for {W: = A} (the proposition expressed byA in ) (p. 38)

3. Examples

x Definitions

A is valid/true in =A iff =A for every W A is valid in C =

CA iff =A for every C

A is valid =A iff =C

A for very C

x Some principles that are valid in the semantics based on Leibnizian models:

D A A

T A A

B A A

4 A A

5 A AG A A

K (AB) ( A B)

Df A A

Df A A

RN =C

A____

=C

A

RE =C

AB__________

=C

A B

RK =C

(A1 ... An) A_____________________

=C

( A1 ... An) A


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x Examples of invalid principles:

A A W={,}, Pn = {} for all n P0 P0 (note difference from RN) Then =A but A, hence A

(AB) ( B) W={,}, P0 = {} Pn = {} for all n>0. P0,P1 P0,P1 Also, considerB=

( B) (AB) same countermodel

(AB) ( B) W={,}, P0 = {} Pn = for all n>0. P0,P1 P0,P1

( B) (AB) same countermodel

x Do some exercises (1.1 to 1.9)

4. Exercise 1.11: correspondence between and

x Define predicate language with one variable,x

MAP: (Pn) = Pn(x)(T) = T

() =

(A) = (A)

(AB) = (A)(B)

( A) = x(A)

( A) = x(A)

Example1 ( Pn Pn) = ( Pn) (Pn)= x(Pn) (Pn)= xPn(x) Pn(x)

Example2 ( (P1 P2) ( P1 P2)) = ( (P1 P2)) ( P1 P2)= ( (P1 P2)) ( P1) ( P2)= x(P1 P2) x(P1) x(P2)= x((P1) (P2)) x(P1) x(P2)= x(P1(x) P2(x)) (xP1(x) xP2(x))


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x Remark: is a specification of the truth conditions

x THEOREM: =A iff(A) is valid in quantification theory

Proof. If not =A, then there is a model =W,P and a world Wsuch that not ABut =W,P is also a model for quantificational logic.

By induction, show that for any W, A iff(A) [s], where s (x) = for all W

A atomic. Then (A) = Pn(x) for some n. Then: A Pn Pn

s(x)Pn

Pn(x) [s] (Pn) [s]

A = B. Then (A) = x(B). Then: A B

B for some W (B) [s] (by IH)

(B) [s(x|)].

(B) [s]

(A) [s]

Similar Proof


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1. Standard versus Minimal models

x The general notion of a model (see 2.2): = W, ..., P, where W (the possible worlds)

P a sequence P0, P1, P2, . . . W associating each i with a set of worlds (those in which P i holds)x Recall from lecture 2: Two main ways of filling in the dots

Standard models: =W,R, P

, whereR

W

W

R = is possible relative to = A iff= A for every Ws.t. R

= A iff= A for some Ws.t. R

Minimal models: = W,N, P, whereN: WW

N = propositions necessary at = A iff {W: = A}N i.e., iff||A|| N= A iff {W: A}N i.e., iff||A|| N

x Comparisons (Theorem 3.4 + 3.5 and Exercise 7.22 + Theorem 7.11)

schema standard minimal

K (AB) ( A B) valid valid iff

||B|| N whenever ||AB|| , ||A|| N

Df A valid valid

Df A valid valid

RN =CA

_____ =C A

valid valid iff WN

(for all in all in C )

RE =CAB

__________

=C A B

valid valid

RK =C

(A1 ... An) A_____________________

=C

( A1 ... An) A

valid valid iff

||A|| N whenever ||A1|| , ..., ||An|| N


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schema standard minimal

T AA valid iffreflexive: R

P Ps P

valid iff||A||

whenever ||A|| N

D A A valid iffserial: (R)

PsP

x P

valid iff ||A|| N

whenever ||A|| N

B A A valid iffsymmetric: RR

P P sxP

x P

(b) p. 224 Chellas

4 A A valid ifftransitive: R & RR

PPs P

Ps P

s s P

(iv) p. 224

5 A A valid iffeuclidean: R & RR

x Ps x P

Px P

PxP

(v) p. 224

G A A valid iffincestual: R & R (R & R)

s P

P

s P

xP

s x P

(g) p. 225

The notation is important: the relevant conditions onR can be expressed by means of first-order

formulas, so we are essentially looking for ordinary first order models satisfying these formulas.


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x Example of proof for the if part (from Theorem 3.5):

Scheme 5 is valid in the class of all euclidean standard models:

1. Assume = A.

2. Then = A for some Wsuch that R.

3. Suppose R.

4. Then, we have R& R and therefore, by euclideanness, R (of course, we also have R

& Rand therefore R, but this is irrelevant).

5. So, for any such that Rthere exists such that R and =A.

6. So, for any such that R, = A.

7. Thus = A.

8. By 17, if= A then = A.

9. Hence = A A.

2. General comparisons

x DEFINITION (p. 36)

Two structures = W, ..., P and '= W', ..., P' are pointwise equivalent iff there is a one-onemap :WW'such that, for every sentenceA and every W

=A iff ='

( ) A

x FACT 1 (Problem 3.12 p. 73):

Every simplified model = W, P is pointwise equivalent to a standard model, namely to the

model '= W,R, P whereR=W W.Proof: A straightforward inductive argument, taking to be the identity function, i.e., setting () =

for all W:

1. Base: =i iff Piiff =

'

i

2. Truth-functional connectives: obvious

3. Modal connectives: = A iff = A for all W iff =

'

A for all W(by Inductive Hypotesis)

iff ='

A for all Wsuch that R

iff ='

A


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x FACT 2 (using parts of Theorem 7.9 p. 221)

Every standard model s= Ws,R, Ps is pointwise equivalent to a minimal model m= Wm,N, Pm, where XN iff X contains all R-accessible worlds, i.e., iff {W: R} X.

(Intuitively: the propositions necessary at are those that include the set of all worlds accessible

from )

Proof: Again, we set () = and prove by induction that, for every sentenceA:

for every W: =s

A iff =m

AAgain, the only interesting case is modal sentences:

=s

A iff =sA for all Ws.t. R

iff {W: R} {W: =s

A}

iff {

W:

R

}

{

W: =m

A} by I.H.

iff {W: =m

A} N by def. of

m

iff =m

A by recursive clause for

3. Comments

x Morals:

The simplified models may be viewed as a special case of standard models

Standard models may be viewed as a special case of minimal models

x Note: Fact 2 can be improved by saying under what conditions a minimal model is equivalent to a

standard model (Theorem 7.9).


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1. Generalizations of the G schema

x Recall the schema

G A A

R must be incestual (offprints of a common parent

have an offprint in common) or convergent:

R & R (R & R)

x s P

P

s P

x P

s x P

x DEFINITION (p. 29).

Let be any modality (, , or ). Then we define

ifn=0 n A = A

ifn = k+1 n A = k A

x FACT 1 (see p. 86).Consider the schema

G k , l , m , n = k lA m n A

Then:

G = A A is just G1 , 1 , 1 , 1

D = A A is just G0 , 1 , 0 , 1

T = A A is just G0 , 1 , 0 , 0

B = A A is just G0 , 0 , 1 , 1

4 = A A is just G0 , 1 , 2 , 0

5 = A A is just G1 , 0 , 1 , 1

x DEFINITION 3.6 (p. 86)

ifn=0 Rn =

ifn = k+1 Rn Rfor some W such that Rk


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x DEFINITION (p. 88)

A standard model = W,R, P, is k,l,m,n-incestual iffRk & Rm (Rl&Rn)

Rk

Rm

Rl

Rn

So in particular:

is incestual iff is 1111-incestual is serial iff is 0101-incestual is reflexive iff is 0100-incestual is symmetric iff is 0011-incestual is transitive iff is 0120-incestual is euclidean iff is 1011-incestual(The proofs of these equivalences se are just derivations in first order logic with identity.)

x EXAMPLE: is serial iff is 0101-incestual

Proof: [Rk & Rm (Rl&Rn)] is 0101-incestual [R0 & R0 (R1 &R1 )] def. [= & = (R&R)] def.

= & = (R&R) elim

= (R) & idem

(R) since =

(R) intro

is serial def. is serial (R) def.

(R) elim (R&R) & idem

= & = (R&R) = laws

[= & = (R&R)] intro

[R0 & R0 (R1 &R1 )] def.

is 0101-incestual def.


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x FACT (Theorem 3.8):

The schema G k , l , m , n is valid in the class of all k,l,m,n-incestual standard models

xCOROLLARY (cp. Theorem 3.5)The schema G is valid in the class of all incestual standard models

The schema D is valid in the class of all serial standard models

The schema T is valid in the class of all reflexive standard models

The schema B is valid in the class of all symmetric standard models

The schema 4 is valid in the class of all transitive standard models

The schema 5 is valid in the class of all euclidean standard models

2. Further generalizations

x G k , l , m , n is not the most general schema.

For instance, the following are not instances ofG k , l , m , n:

Gc A A

Gr ( A A) A

x Indeed there are more general schemes with interesting propertiese.g.

Sahl n (A B) (with resrtictions on the form of A and B)

But Gr and Gc are still not covered by such a schema.

3. Characterizability (for Kripkean modal logics)

x QUESTION 1:

Does every modal formula correspond to some first-order definable R?

i.e., given a formula A, is there always a first-order sentence so that, for every = W,R, P

=A (modally) iff = (quantificationally) ?

ANSWER IS NO

G k , l , m , n YES [Rk & Rm (Rl&Rn)]

Sahl YES complicated condition

Gr NO there is a condition on R (see test), but not first-order definable

Gc NO not first-order definable (though Gc 4 is)


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x QUESTION 2:

What about the other way around? Does every R correspond to a modal formula?

ANSWER IS NO

E.g. Reflexivity (R) A A

Irreflexivity (R) no characteristic wff

i.e., if a wff is true in every irreflexive model, then it

is true in every model

Ditto for

Asymmetry (R R)

Antisymmetry (R & R =)

Intransitivity (R & R R)

Exercises 3.5657

x More material on this in HughesCresswell (Chapter 10)


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1. Systems of Modal Logicbasic definitions ( 2.4)

x DEFINITION 2.11

A system of modal logic is any set of sentences (=theory) closed under the rule:

A1,, AnRPL

__________(n 0) where A is a tautological consequence of A1 An

A

is closed under RPL means:

if A1 An , then A

x Equivalently: A system of modal logic is any set of sentences closed under the rules:

PL

__________where A is a tautology

A

A, ABMP

__________

B

x NOTE

The set of all tautologies (PC) is the smallest system of modal logic

every system closed under RPL contains all tautologies (by n=0)

The set of all sentences valid in a class of models {: |=C A}

The set of all sentences valid in a model {: |=M A}

The set of all sentences true at a world {: |=M A}

The set of all sentences is the largest system of modal logic

x DEFINITION 2.12

A theorem of a system modal logic is any member of:

A

x DEFINITION 2.14

A is deducible from in system ( A) iff there are A1,, Anso that (A1 An) A


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This is equivalent to the usual definition: A iff there is finite sequence A1,, An such that

A1

Ai or follows by RPC

An = A

x DEFINITION 2.15

A set is inconsistent in (Cn ) iff

A set is consistent in (Con ) otherwise

x THEOREM 2.16

These notions have all the expected properties

x EXERCISES

3.33

3.35 4.56 read only

3.37 (seriality done in class) 2.32 (b)(e) 4.7 (some)

3.51 4.9

3.567 read only

2. Normal Systems of Modal Logic ( 4.1)

x DEFINITION 4.1A system of modal logic is normal iff it contains every instance of

Df A A

and is closed under the rule:

(A1 An) ARK

________________________(n 0)

( A1 An) A

x The idea now is:

completeNormal Systems Standard Models

sound

Additional axioms Special conditions on R


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x Eventually:

completeClassical Systems Minimal Models

sound

x THEOREM 4.2:

Every normal system of modal logic has the following rules of inference:

ARN

____n=0

A

A BRM

________n=1

A B

(A B) CRR

___________________n=2

( A B) C

A BRE

________RM 2

A B

Every normal system of modal logic has the following theorems:

N T T PL (tautol)T RN

C ( A B) (A B) (A B) (A B) PL

( A B) (A B) RR

M (A B) ( A B) (A B) A PL

(A B) B PL

(A B) A RM

(A B) B RM

(A B) ( A B) PL

R (A B) ( A B) (by C and M)

K (A B) ( A B) ((A B) A) B PL

( (A B) A) B RR

(A B) ( A B) PL


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x THEOREM 4.6

We also have the following generalizations

(A1 An) A

RKk ________________________

( kA1

kAn)kA (A1 An) A

( k1A1 k1An)

k1A I.H.

( k1A1 k1A

n)

k1A RK

( kA1 kAn)

kA Def

Df k kA kA k1A k1A I.H.k1A k1A REk1A k1A Df

k1A k1A PL

kA kA Def

This can be extended to every other principle as well: The result of putting k and k for and

in a valid principle remains valid.

3. Alternative Characterizations

x THEOREM 4.3: The following are all equivalent

Df + RK Df + K + RNDf + N + RR

Df + N,C + RM

Df + N,C,M + RE

example, the first is proved by induction

n=0 RK = RN

n=1 RK = RM A B

(A B) RN

A B K, PL

n=k+1 (A1 An) A

(A1 An1) (An A) PL

( A1 An1) (An A) I. H.

( A1 An1) ( An A) K, PL

( A1 An) A PL


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x THEOREM 4.4:

Axiomatizations using

Df + RK Df + RKDf + N + RR

Df + N + RR

Df + N ,C + RM

Df + N ,C ,M + RE

x Many more in exercises 4.5/4.6

4. Replacement and duality ( 4.2)

x THEOREM 4.7: Every normal system of modal logic has the followingRule of Replacement:

B B'REP

________

A A' where A' is obtained from A by replacing 0 occurrences of B by B'

Proof is by induction on the complexity of A (see p. 125)

x EXAMPLE: using REP to prove (A B) ( A B)

1. (A B) (A B) PL

2. (A B) (A B) 1,PL, REP3. (A B) ( A B) 2, R , REP

4. (A B) ( A B) 3, PL, REP

5. (A B) ( A B) 4, Df , REP

x EXERCISE 4.19

Df + RK Df + K,N + REPx DEFINITION2.4, P. 30

n

*

= nT

*=

*

= T

()*

= (A*)

( )*

= A* B

*

( )*

= A* B

*


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( )*

= (A*) B

*

( )*

= A*(B

*)

( )*

= (A*)

( )*

= (A*)

x EXAMPLES

N*

( T)*

= (T*) =

C*

(( ) ( ))*

= ( )*

( ( ))*

= (( )*

( )*) ( )

*

= ( *

*) (

*

*)

x FACT (seeTHEOREM 4.8): The following equivalencies hold in any normal system :

(1) | |*

proof by induction on the complexity of A

(2) | |**

from (1)

(3) |( ) |*

*

Proof of (3):

| |*

*

from (1) by REP

|*

*

|

*

*

|

*

*

from (1) by REP |(

*)*

(*)*

from (1) by REP

|**

**

def

| from (2) by REP

x DEFINITION(P. 29): the dual of a modality is the modality * obtained by interchanging andx THEOREM 4.10

| |* *


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1. 15 Main Normal Systems

K= the smallest system

The main extensions are obtained by adding one or more of the following:

D A A

T AA

B A A4 A A

5 A A

Naming conventions:

KS1 ... Sn

is the (smallest) extension of K obtained by taking the schemas S1 ... Sn as axioms.

(The order of the Si does not matter.)

E.g., KT5 is the smallest system of modal logic obtained by adding T and 5

Etc.

x Facts:

There are 25 =32 possible combinations

Only 15 of these are distinct (proved later)

General picture: Figure 4.1 on p. 132.

x Example:

KTD = KT

Proof: Obviously

KTKTD.

So we only show that

KTD KT.


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To this end it is sufficient to show that every instance ofD is a theorem ofKT:

1. KT A A T

2. KTA A duality principle

3. KT A A 1,4 PL

x Other examples:

KT5 = KTD5 = KTB5 = KT45 = KTDB5 = KTD45 = KTB45 = KTDB45 (This is S5)

Every instance ofD is a theorem ofKT5: obvious from above

Every instance ofB is a theorem ofKT5:

1. KT5 A A 5

2.KT5

A A dual of T

3. KTA A 1,2 PL

Every instance of4 is a theorem ofKT5:

1. KT5 A A 5

2. KT5 A A 5 (duality principle Theorem 4.10)

3. KT5 A A 2, RM

4. KT5 A A B (which is a theorem of KT5)

5. KT5 A A 3,4, PL

2. Some other important Normal Systems (and their positions in the diagram)

x We have encountered the following schemas:

G A A

Gc A A

Gr ( A A) A

x Facts about G:

KG KB (EXERCISE 4.34)

1. KB A A B

2.KB

A A B (Theorem 4.10)

3. KB A A 1,2 PL


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KG K5 (EXERCISE 4.38b)

1. K5 A A 5

2. K5 A A 5 (Theorem 4.10)

3. K5 ( A A) ( A A) theorem of K (Exercise 4.7p, p. 123)

4. K5 A A 2,3, PL

5. K5 A A 1,4 PL

So: KT4 KT4G KT5

| | |

S4 S4.2 S5

x Facts about Gc:

KT4 KT4Gc/ KT5| | |

S4 S4.1 S5

x Facts about Gr:

K4 KGr

KT4 / KGr

KGr/ KT5

x Much, much more in the exercises.

3. Reduction laws for modalities ( 4.4)

x Definition: two modalities and are equivalent (in system ) iff for all sentences

A A

x Example: in KT5 there are at most 6 distinct modalities:A, A, A, A, A, A.

a)KT5 A A 1. KT5 A A T

2. KT5 A A 4

3. KT5 A A 1,2, PL

b) KT5 A A 1. KT5 A A 4

2. KT5 A A T

3. KT5 A A 1,2, PL


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4. Soundness ( 5.1)

x THEOREM 5.1:

Let S1 ... Sn be schemas valid in classes of standard models C1 ...Cn (respectively). Then thesystem KS1... Sn is sound with respect to the class C1... Cn, i.e.,

KS1 ... SnA C

1... CnA

Proof: KS1 ... SnA only ifA is either one of the Si, or an instance of Df , or follows by RK

1. Each Si is valid in C1... CnCi given

2. Df is also valid in C1... CnC (all standard models) Theorem 3.3(1) p. 69

3. Validity inC1... Cn is preserved under RK Theorem 3.3(1) p. 69

(proof by induction, p. 70)

x In general, systems are sound with rspect to models as per table 5.1 (p. 164)

x Corollary (THEOREM 5.2):

The 15 normal systems of Figure 4.1 are all distinct

Proof: give a model of one that is not a model of another

E.g., R= R symmetric and transitive, but D fails. So ....

5. Two more remarks about modalities ( 5.2)

x Lower bounds (THEOREM 5.3):

The advertised distinct modalities cannot be further reduced

Proof: Find counterexamples of the sort considered in soundness, i.e. show that A A fails in

a class of models with respect to which the system is sound.

E.g. and are distict in KT4:

Px Ps x PsP

x sP

PxP

sP

R:reflexive + transitive (+ symmmetric)so a model ofKT4

We have

not P PThus, by soundness:

not KT4 P P


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x Infinitely many modalities (THEOREM 5.4):

The 8 systems

K KD KT KB K4 KDB KD4 KTB

have infinitely many modalities

Proof: Case 1: all of K, KD, KB, KT, KDB, KTB

The following is a reflexive symmetric model, hence a model of KTB by Theorems 3.5 + 5.1.

P0

P 0

P0

0

P0

1

P0

2

. . .

n

. . .

m m+ 1

P 0

We have 0mP but not 0 nP

Therefore not CmP nP (where C is the class of reflexive, symmetric models)

Therefore not KTBmP nP (by soundness)

Since all of K, KD, KB, KT, KDB are included in KTB, this completes case 1.

Proof: Case 2: all of K, KD, K4, KD4

This is exercise 5.15


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1. Structure of completeness proof for normal systems ( 2.7 + 5.3): Overview

x Let be a system of modal logic and C a class of models: is sound with respect to C iff

A CA

is complete with respect to C iff

CA A

x We have seen how to prove soundness. Now we deal with completeness. Main strategy:1) Define canonical standard model for

2) Show: if is canonical, then for all A and all W:

M

A iffA

3) Using this, show in general: if is canonical, then

M

A iff A

4) Thus, to show that is complete with respect to a class C of models, it is sufficient to showthat C contains a canonical model for . For then we have:

CA M

A A

2. Preliminary definitions and facts


A set of sentences is -consistent (written Con) iff/

x THEOREM 2.16

Basic properties of and -consistency. Notably:

(6) A and , then A

(14) A iff{A} is not -consistent

/A iff{A} is -consistent


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A set of sentences is -maximal (written Max) iff

1) is -consistent

2) has no -consistent extensions, i.e., for every A, if{A} is -consistent, then A

x THEOREM 2.18

Basic facts about -maximal sets of sentences . Notably:

1) Aiff A

2)

M

5) Aiff A

6) ABiff both Aand B

M

x THEOREM 2.19 (Lindenbaums Lemma) (p. 55)

Every -consistent set can be extended to a -maximal set

Proof structure:

List all sentences 1, A2,, ...

Define 0 =

n = n1 {An} if this is -consistent

= n1otherwise

= n 0 n

x Important COROLLARY 2.20 (p. 57)

(1) A iff A for all -maximal

(2) A iff A for all -maximal (=special case for =)

Proof:

() Let be -maximal:

A A by THEOREM 2.16 (6)

A by maximality (=THEOREM 2.18 (1))

() by contraposition:

/ A {A} is -consistent by THEOREM 2.16 (14)

-maximal {A} by THEOREM 2.19 (Lindenbaums)

-maximal such that A by THEOREM 2.18 (5)


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3. Proving completeness

STEP 1: DEFINE CANONICAL MODELS

x Recall the idea of a Henkin model: put the language in the domain


A standard model =W,R,P is a canonical standard model for a normal system iff:

1) W = set of all -maximally consistent sets of sentences

2) Pn = {: Pn}3) A iffA for every W such that R

equivalently (by THEOREM 5.6 p. 172):

3') A iffA for some W such that R

STEP 2: FUNDAMENTALTHEOREM ABOUT CANONICALMODELS

x THEOREM 5.7 (p. 172)

If is canonical, then for all A and all W:

M

A iff A

(This generalizes what is true by definition for atomic sentences)

Proof by induction:

M

n iff Pn by def ofM

iff n by def of canonical model

M

A iff / M

A by def

iff A by I.H.

iff A by maximality (THEOREM 2.18 (5))

M

M

A iff M

A for all such that R by def

iff A for all such that R by I.H.

iff A by def of canonical model


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STEP 3: DETERMINATION WITH RESPECT TO CANONICALMODELS

x THEOREM 5.8 (p. 173)

If is canonical, thenM

A iff A

Proof:

M

A iff |=M A for all W by def.

iffA for all W by step 2 above

iff A for all W by maximality of (THEOREM 2.18(1))

iff A from COROLLARY 2.20 (2)

x NOTE: This means that if is a canonical standard model for ,then is sound and complete

with respect to the class { }

STEP 4: CONCLUSION

x To show:

is complete with respect to a class C of models,

it is sufficient to show:

C contains a canonical model for .

For then we have:

CA M

A A

x In general (though not always), one shows that C contains the proper canonical standard model:


A standard model =W,R,P is the proper canonical standard model for a normal system iff:

1) W = as any canonical model

2) Pn = as any canonical model

3) R A whenever A, i.e.

{A: A}

Equivalently (by THEOREM 5.10 p. 174):

3') R A whenever A

{ A: A}


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x Must show: proper canonical standard models are indeed canonical standard models, i.e., 3)

implies the corresponding clause in the definition of canonical model

x THEOREM 5.11

If is the proper canonical standard model for , then

A A for every W such that R

Proof (usingTHEOREM 4.30, p. 158)

Suppose A

Let W such that R

Then A (clause 3 in def of proper canonical)

Suppose A for every W such that R

i.e. A for every W such that B whenever Bi.e. A for every W such that {B: B}

We have A iff A for all -maximal COROLLARY 2.20 of Lindenbaum

hence A iff A for all W such that since W ={: is -maximal}

Thus {B: sB} A taking ={B: sB}Hence (B1 ... Bn) A for B1 ... Bn{B: sB} by DEFINITION of So (sB1 ...sBn) sA RKBut sB1, ...,sBnThus sA by DEFINITION of So sA by maximality (THEOREM 2.18(1))

4. Specific determination results (Soundness and Completeness) ( 5.4 + 5.5)

x THEOREM 5.12

Kis determined by (=sound and complete with respect to) the class C of all standard models

Soundness: trivial

1. Df is valid in C

2. validity inC is preserved by RK

Completeness: also trivial

Let = the proper canonical standard model for K

Then is a standard model

Hence C


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x LEMMA 5.13

If is the proper canonical standard model for , then

(1) is serial if contains (every instance of) D

(2) is reflexive T

(3) is symmetric B

(4) is transitive 4

(5) is euclidean 5

In general (THEOREM 5.17):

(6) is k,l,m,n-incestual Gk,l,m,n

Proof of (4)

Transitivity means: ifR and R, then R

i.e.: if {B: B} and {B: B} ,then{B: B}

Assume: 1. {B: B}

2. {B: B}

3. B

Show: B

We have 4. B B since is -maximal (THEOREM 2.18 (2))

Thus 5. B THEOREM 2.18 (2) (plus MP 3, 4)

Hence 6. B by 1Hence 7. B by 2

Thus 8. {B: sB} by 37 and generalization

x THEOREM 5.14

KD is determined by the class C = { : is serial}

KT C = { : is reflexive}

KB C = { : is symmetric}

M M

KDB C = { : is serial & symmetric}

M MIn general:

KS1... Sn is determined by the class C1... Cn

where S1 ... Sn {Gk,l,m,n: k,l,m,n 0} and C1 ...Cn are the corresponding classes of standard

models (as per LEMMA 5.13).


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5. Remarks

x Different classes can determine the same logic.

For example, the system S5 (KT5) is determined by any of the following:

C1= { }, where is the proper canonical standard model for S5

C2= { : R is reflexive and euclidean}

C3= { : R is universal}

C4= { : C2and W is finite} (using the method of filtrations)

x In general, determination for the basic systems holds also if we take only finite models.

(This is good for decidability)

x Not every determination result can be established using proper canonical models.

There exist systems that are determined by a class C even if the canonical model C.

E.g., the canonical model for KGr is not irreflexive, but Gr fails for any model in which R is not

irreflexive.


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8 NON-ALETHIC INTERPRETATIONS (1)

1. Introduction

Three interpretations of (and consequently of):

Deontic

A = It ought to be the case thatA (often writtenA)

Epistemic

A = The agent,x, believes thatA (often written BA)A = The agent,x, knows thatA (often written KA)

Temporal

A = It will always be the case thatA (often written GA)A = It has always been the case thatA (often written HA)

2. Deontic interpretation of modalities

Basic normal system of deontic logic is KD (also known asD*)

D A A (Ought implies can)

is often written (for Obligatory) and (i.e., ) is written P (for Permissible). So:

D A PA (Ought implies can)

Of course we dont want

T A A (Ought implies is)

FACT: the following are equivalent to D in any K-system:

OD (A A) (No impossible obligation)

OD* (A A) (No incompatible obligations)

Intuitively, these principles express different thoughts, so their equivalence is a defect of any K-system, hence of any modal logic which admits of a Kripke-style semantics.


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In other words, to avoid this result we must go below K, hence work with a weakerMontague-style semantics.

3. Deontic semantics la Kripke

Intuitive interpretation of the accessibility relation:

R is deontically admissible from the point of view of

Thus:

6=M

A 6=M

Afor every such that R

A is true in every deontically admissible world

It ought to be the case that A

Equivalently:

{: R} = the proposition that represents the standards of obligation for the world .

Thus:

6=M

A 6=M

Afor every such that R

{: R} {: 6=M

A}

{: R} ||A||M

the proposition expressed by A is entailed by the standard of obligation for .

Recall that D corresponds to the conditio0n that R be serial: (R).

Obligations should be non-vacuous. [If R=, then 6=M

A vacuously.]

There may be more than one deontically accessible world, due to non-deontic facts.

IfR, then need not be perfect: there may be such that R (i.e., the standards ofobligation for may be different from those of):

1

2

3

1

2

3


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4. Looking for extensions (KD systems)

T A A Every obligation is realized inacceptable

B A PA What is the case is obligatorily permissible inacceptable

4 A A

. . .

better perfect

Obligations remain such in every deontic alternative= Standards of obligations do not decrease= No fewer obligations

? acceptable?

5 PA PA

OP

0

P0

OP0

P0

OP0

P0

OP0

Permissions remain such in every deontic alternative= standards of obligations do not increase= no more obligations

? acceptable?

U (A A) Obligations ought to be realized(one of the few unconditional-principles)

R must be secondarily reflexive: RR? acceptable? Note: this means that

if6=M

A and 6=M

A, then R for no W

i.e., is one of the worst possible worlds

4c AA What is not obligatory is not obligatorily obligatory

AA R = density: R(RR) sounds good

KD

KD4 KD44 KD4U

KD45

KD4 KDU KD5c

c

(D*)

(D*1) (D*2)

(D*3)


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5. Problems with these theories (all KD systems)

There are two sorts of problems:

Correctness Adequacy

Correctness: two problems

1) Obligations always exist (however trivial they may be)

KD(A A)

Thus: There exists no world where we are absolutely free

2) Two important principles become indistinguishable

KD (A A) (A A)| |

No impossible obligation No incompatible obligations

= Ought implies can

Adequacy:

Cannot express conditional obligations

Ifyou cough, then you ought to apologize | |A B

= conditional obligation of B given A, written(B/A).

Two only options:

(a) (B/A) =df A B This is T whenever A is FIfthe earth is flat, then you ought to apologize.

(b) (B/A) =df (A B) This is T wheneverA orB is also TIfyou steal books, then you ought to eat pizza.Ifyou cough, then you ought to pay taxes.

Other problem: Chisholms paradox:

(i) John ought to go to help his neighbors(ii) If John is going to help his neighbors, he ought to tell them he is going.(iii) If John is not going to help his neighbors, he ought not to tell them he is going.(iv) John does not go to help his neighbors.


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(i)(iii) seem a reasonable and consistent set of requirements. Yet the fact that John does not goto help his neighbors, i.e., (iv), is enough to yield a contradiction. Formally:

(1) H given

(2) (HT) given(3) given(4) given(5) (HT) (HT) K(6) T (1), (2), (5), RPL(7) (3), (4), RPL(8) T (6), (7), RPL(9) (TT) Equivalent to D(10) (8), (9), RPL

The alternative symbolization of (ii) following (a):(2') HT

avoids the problem, but at the price of making (ii) a logical consequence of (iv) (by RPL).

Similarly, the alternative symbolization of (iii) following (b)

(3') ()avoids the problem, but at the price of making (iii) a logical consequence of (i) via the theorems

(5') H () PL(6') H() (5'), RE

So:

either( / ) must be assumed as a primitiveor( / ) is definable in terms of some other kind of conditional

6. A weaker system

KD could also be axiomatized as:

A BRM ________

A B

OD (A A)N (A A)C (A B) (A B)


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By correctness problem 1) (obligations always exist), we want to get rid ofN

But this is a K- theorem.

This means we need a system weaker than K, hence not complete with respect to Kripke models.

We need minimal models

By correctness problem 2), we must also get rid of the equivalence

(A A) (A A)

But this is provable even without N

1. (A A) (A A) C2. (A A) D3. (A A) 1,2 PL

So we must also get rid ofC or OD.

But OD is OK, so it is C that must go.

The resulting systemD = RM + D is not normal (= not a Ksystem).

D is determined by the class of minimal models such that

1) if XYN, then XN and YN (supplemented)

2) N

7. Even weaker?

There are problems withD, too.

Ross paradox (from Alf Ross, 1941).

RM implies that

D PA P(AB).

1. (AB) A PL

2.

(AB)

A 1, RM3. A (AB) 2, PL4. PA P(AB). Df P

But this is counterintuitive:

Peter may drink water/ Peter may drink either water or whiskey


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In fact, it seems natural to suppose that

Peter may drink either water or whiskey Peter may drink water and he may drink whiskey

This corresponds to the following, which is not a theorem ofD:

P(AB) (PA PB)

kvist puzzle.

Consider the epistemic operator Peter knows that, written K. Since knowledge implies truth,

KA

RMimplies that

DKA

But this is counterintuitive:

Peter ought to know that there is a fire/ There ought to be a fire

Conclusion:D is also too strong...


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1. Epistemic interpretation of modalities

Starting point:

as a belief operator, written B

BA =df the agent,x, believes thatA

Alternative notation: B(A,x), convenient for first-order or multi-agent extensions

A lot depends on what we mean by believes

implicit vs explicit

persuasion vs opinion

etc.

KD45 = the logic of full belief

D BABA (coherence)

4 BA BBA (positive introspection)

5 BA BBA (negative introspection)

Semantics

possible worlds = possible representations (consistent and complete) of reality

R iff is epistemically possible (= conceivable) for the agent in

6=M BA x thinks thatis ungiveupable (=a constant element of all of representations)

Determination

R is serial, transitive, euclidean. So, standard situation looks like this:


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Note:

not 6=M BAA so T fails: beliefs need not be true

6=M B(BAA) so U holds: beliefs are believed to be true

Problems

RK implies closure of beliefs under logical implication full (implicit?) belief

To avoid this, one must go for minimal models (non-normal systems)

Then we have the following:

6=M B(A) whenever N6=M B(A) whenever N

2. Adding Knowledge

Notation:

KA =df the agent,x, knows thatA

This can be defined in terms of B if we accept the principle that knowledge is true belief:

DfK KA BA A.

But one might prefer to have DfK as a theorem.

This can be obtained in the mixed system Kmix defined by:

D BABA (coherence)

TK KAA

?1 KA BA

4K KA KKA (introspection)

?2 BA KBA (introspection)

?3 BA KBA (introspection)

?4 (BA A) KA

Note: the rule RN for K is derivable in Kmix:

RN 6=KmixA____6=Kmix

KA

This means omniscience

Again, to avoid it one must go for minimal models (non-normal systems)


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Theorems:

Kmix KA BA A (=DfK)

Kmix BA BKA

Kmix BA KKA

So, the belief operator B is also definable in terms of K.

Axiomatization using only K?

option 1 is simply to replace B byKK in Kmix

option 2 is to give a better axiomatization of K:

T KAA

4 KA KKA

5

(BA A) KAA (BA KA)

A (KKA KA)

A (AA)

Fact: KD45 is equivalent to KT45 upon the obvious translations:

BA KKA or KA BA A

Other theories

1. KT4G is the same as KT4 + D-for-belief

Proof:

1. KKA KKA axiom G

2. KKA KKA DN

3. BABA subst.

Note: KT4G is the same system as Kmix, but with ?4 replaced by

BA BKA

Clearly, KT45

| KT4G

But also, KT4G| KT45

| | S4.2 S4.4


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2. KT5 is not good if BAKKA

For otherwise

1. KA KKA axiom 5

2. KA KKA DN

3. KA KKA DN

4. KA BA DfB

5. BA KA PL unacceptable


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1. Temporal logic

Modalities:

FA it will sometime be the case thatAFA A

GA it will always be the case thatA

= FA

PA it has sometime be the case thatAPAA

HA it has always be the case thatA

= PA

Minimal tense logic Kt

Axioms:

System K for G + System K for H + A GPA + A HFA Theorems: PGA A FHA A 1. A GPA ax

2. GPA A PL

3. FPA A dfF

4. FHA A dfH

More generally:

kt A kt A*, where A* is the mirror image of A(replace G/ H and F / P)

This means symmetry past/future

Semantics:

Note: a multimodal system

in general: oneR for each modality


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Determination: all standard models

providedRG RH

alternatively: sameR in two directions (the direction of time) Natural extension:

Kt + 4 GA GGA future transitivity

HA HHA past transitivity

or:

Kt + 4 FFA FA future transitivity

PPA PA past transitivity

Further extensions: Two main possibilities:

linear:

now

branching:

B B

FA FB

F(A B)

F(AFB)

F(FA )

PA PB

P(A B)

P(APB)

P(PA )

2. Linear extensions

Basic linear system CL (Cocchiarella):

RL (FA FB) (F(AB) F(A FB) F(FA)) right linearity

LL (PA PB) (P(AB) P(A PB) P(PA)) left linearity

Semantics:R must be:

transitive

right linear: R & R = or R or R

left linear: R & R = or R or R

System SL: non-ending time(Dana Scott)

CL + D GA FA seriality

HA PA "


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System PL: dense time(Prior)

SL + 4c + FA FFA R(R & R)

PA PPA

System PCk: circular time(Prior)

Kt + 4 + GAA

GA HA

(So: G and H are logically equivalent)

3. Branching extensions

System CR (Cocchiarella)

Kt + 4 (= CL minus linearity)

System Kb (Rescher + Urquhart)

CR + LL (= branching admitted only in the future)

symmetry P/F fails


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11 QUANTIFIED MODAL LOGIC (1)

1. Preliminaries

Vocabulary:

variablesx0,x1,x2, ...

names a0, a1,a2, ...

predicates 0, 1,2, ...

connectives and quantifiers, ,

Grammar:

Straightforward

But notice that the interplay between quantifiers and modalities yields de dicto/de re readings:

de dicto: xA e.g.,Necessarily, everything is spatio-temporally located.

de re: xA e.g.,Everything is necessarily spatio-temporally located.

de dicto: xA e.g.,It is possible that everybody votes for Berlusconi.

de re: xA e.g.,Anybody could vote for Berlusconi.

de dicto: xA e.g., There must necessarily be right-winged politicians.

de re: xA e.g., Somebody is necessarily a right-winged politician.

de dicto: xA e.g.,Bush might have had a sister.

de re: xA e.g., Somebody might have been Bushs sister.

Semantics:

Two main approaches.

1. The possibilist (or fixed-domain) approach: each model comes with a single domain of

quantification containing all possible objects (i.e., all objects that are possible according to

the model).

2. The actualist (or world-relative) approach: each model comes with a domain of quantifica-tion for each world in the modela domain containing only those objects that actually exist

in the given world.

We shall look at these options only from the perspective of Kripke-style semantics, but

Montague-style semantics admit of a similar distinction.


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2. Semantics # 1: The fixed-domain approach

MODELS

A model is a four-tuple M= W,R,D, V such that: Wis a non-empty set (of worlds);

R is a binary (accessibility) relation on W;

D is another non-empty set (of individuals);

Vis a function assigning

an element V(a) D to every name a.

a relation V() W Dn to every n-ary predicate . (If is 0-ary, V() is just thepropositionP expressed by , as before; otherwise V() is the intension of..)

In such a model, an assignment is a function that maps every variablex to a value (x)D, and

two assignments and ' are said to be x-alternatives iff they agree on every variable exceptpossibly forx [i.e., iff(y)='(y) for every variabley distinct fromx].

TRUTH AND VALIDITY

The truth conditions of a wffA at a world under an assignment are defined as follows, where*(t)=(t) if tis a variable, and *(t)=V(t) if tis a constant

1. 6=Mt1...tn [] iff, *(t1), ..., *(tn)V()

2. 6=MA [] iff not 6=MA []

3. 6=MA [] iff6=MA [] and 6=MB []

4. 6=MA [] iff6=M A [] for every Wsuch that R.

5. 6=MxA [] iff6=MA ['] for everyx-alternative of' of

A wffA is true at in Miff6=MA [] for every assignment .

The other semantic notions (validity in a model, validity in a classC of models, and logical

validity) are defined as before.

REMARK 1

The semantic condition on names corresponds to the idea that names are rigid designators.

REMARK 2

Concerning predicates, there is an expressiveness problem. Consider:

xRx e.g., There is a world in which everybody is rich

This formula refers to those existing at some world and says that they are all rich.

But do we mean to say that at some world everybody will be rich as they use the word

rich, or as weuse the word in the actual world?


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If we have equality, this half of the fixed domain idea is also captured by the following valid

wff:

NecEx xy(y = x)

REMARK 4

Note that we can read BF as asserting that if a necessity holds de re, then it also holds de dicto.

But we may as well read it as asserting that if a possibility holds de dicto, then it also holds de

refor BF is equivalent to:

BF xAxA

In any case, BF and BF are controversial. Consider the following instances:

If everything is necessarily spatio-temporally located, then necessarily everything is

spatio-temporally located.

If Berlusconi might have had a sister, then there is something that might have been

Berlusconis sister.

Likewise, we can read BFcas asserting that if a necessity holds de dicto, then it holds de re.

But we may as well read it as asserting that if a possibility holds de re it holds de dicto:

BFc xAxA

BFc and BFc are more plausible. But they are also controversial. Consider:

If necessarily everything exists, then everything necessarily exists.

If there is something that might not have existed, then it is possible for there to be

something that doesnt exist.

In general, the problem with the fixed domain approach is that it seems a fundamental

feature of common ideas about modality (at least: alethic modality) that the existence of many

things is contingent, and that different objects exist in different possible worlds.

However, one can avoid this objection by interpreting the quantifiers as ranging over all

possible objects and letting a designate predicate E express existence.

For all existing x: A x(Ex A)

For some existing x: A x(Ex A)

BF and BFc would then be non-problematic if interpreted unrestrictedly as holding of

possibilia, and their existentially restricted versions would not be theorems and, therefore,

would not give rise to the above objections

xAxA x(Ex A) x(Ex A)

xAxA x(Ex A) x(Ex A)


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1. Semantics # 2: The world-relative approach

MODELS

A model is a five-tuple M= W,R,D, V, Q such that: W,R,D, Vare as in Semantics #1

Q is a function assigning a non-empty set Q()D to each world W (the domain ofquantification for ), with the proviso Q() Q() whenever R.

(This proviso ensures that BFc comes out valid, for recall: BFcis derivable in quantified K).

Assignments are defined as in #1.

TRUTH AND VALIDITY

The truth conditions of a wffA at a world under an assignment are defined as follows:

14 as in #1.

5. 6=M

xA [] iff6=M

A ['] for everyx-alternative of' of such that '(x)Q()

Other semantic notions (truth, validity, etc.) are defined as before.

REMARKS 1

The expressiveness problem persists.

A (non-equivalent) alternative to clause 1 is:6=M

t1...tn [] iff*(t1), ..., *(tn)Q() and , *(t1), ..., *(tn)V()

In this semantics, BF is not valid. For instance, the following instance ofBFxxxx

is false in any world-relative modelM such that:W={,},D= {a, b}

Q() = {a}, Q() = {a, b}V() = {, a, , a}

However, ifR is symmetrical then BF holds, since the proviso on Q implies thatQ() = Q() whenever R.


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Since symmetry corresponds to principle B, this means that BF is a theorem ofKB. Proof:

1. xAA QL axiom

2. xAA 1, RK

3. AA B4. xAA 2,3, PL

M M M

9. xAxA as in Lecture Notes 11

10. xAxA 9, RK

11. xAxA B

12. xAxA 9,10, PL

This is disappointing, since symmetry is a plausible requrement on R (at least for alethic

modalities).

Moreover, the validity ofBFc, and more generally of the proviso that Q() Q() wheneverR, is still controversial. Can we give it up? We can give up BFc, but the resulting semantics does not validate classical logic. We need

supplement Kwith the axioms & rules of Free Logic. In particular, the QL axiom

xAA[t/x]

[which was used in the proof ofBFc.] must be weakened to:

y(xAA[y/x])

or, if we have equality, to:

(xAy(y = t)) A[t/x]

If we do so, then (as expected) BFcis valid in a model iff the inclusiveness condition

RQ() Q()

is explicitly imposed uponR.

If we also requireR to be symmetric, then BF is valid too.

2. Identity and Counterparthood

Axioms

If the language includes the identity predicate =, the standard identity axioms hold:

ID1 t= t

ID2 t= t' (A A[t'/t])


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As a consequence, we immediately get the following logical truths (provable regardless of

whether we rely on a Classical or Free Logic for the quantifiers):

NecId t= t' t= t'

NecDiv tt' tt'

Identity or counterparthood?

Both types of semantics assume that objects may exist in more than one world, and imply that

names are rigid designators.

But one could object to this view on philosophical grounds. One could say that when we

counterfactualize about someone, say John, we dont look at the ways John could have been

but, rather, at the way Johns counterparts arehis counterparts being those individuals that

somehow correspond to John in other worlds.

This means that a statement such as John could have been a singer would be true, not if there

is a possible world in which John is a singer, but if there is a counterpart of Johns, in some

possible world, who is a singer.

We can account for this intuition as follows (though a fuller account will be seen next week,

devoted to David Lewiss Counterpart theory):

MODELS

A model is a six-tuple M= W,R,D, V, Q, C such that: W,R,D are as in #1 and #2

Q is as in #2, but with the proviso that Q() Q() = for all distinct ,W.

Cis a binary relation (of counterpart) onD.

Vis a function assigning

a function V(a):WD to every name a, with the proviso that V(a)()Q(). (Intuitively,V(a)() is the individual denoted by a in .)

a relation V() WDn to every n-ary predicate (as before.)

TRUTH AND VALIDITY

The truth conditions of a wffA at a world under an assignment are defined as follows:

13 as in #2.

4. 6=M

A [] iff6=M

y1xn(Cx1t1 CxntnA[x1/t1 xn/tn]) [] for every Wsuchthat R where t1tn are all the names and variables occurring free inA.

5. as in #2.

Other semantic notions (truth, validity, etc.) are defined as before.


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1. Modal Logic without modalities: Lewiss Counterpart Theory

x Primitives:

Wx (x is a possible world)

Ixy (x is in possible worldy)

Ax (x is actual)

Cxy (x is a counterpart ofy)

x Axioms:

A1 Ixy Wy

(Nothing is in anything except a world)

A2 Ixy Ixz y=z

(Nothing is in two worlds)

A3 Cxy zIxz

(Whatever is a counterpart is in a world)

A4 Cxy zIyz

(Whatever has a counterpart is in a world)

A5 Ixy Izy Cxzx=z

(Nothing is a counterpart of anything else in its world)

A6 Ixy Cxx

(Anything in a world is a counterpart of itself)

A7 x(Wxy(Iyx Ay))

(Some world contains all and only actual things)

A8 xAx

(Something is actual)

2. Remarks

x Comments on the axioms

AdA1: The relation I is best interpreted as a mereological relation of parthood, so that Ixy

really means x is part ofy: possible worlds are large possible individuals with smaller pos-

sible individuals as parts. (As a special case, a world is an improper part of itself.)


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AdA2: Worlds do not overlap; thus, possible individuals in different worlds are never identical

(cross-world identity is replaced by the counterpart relation). However, the possible individuals

are not all the individuals: cross-world mereological fusions of possible individuals are

individuals too, though notpossible individuals: there is no way for the whole of it to be actual.

Ad A3A4: Only possible individuals are (and have) counterparts. My counterparts are

individualsI would have been, had the world been otherwise.

Ad A5A6: The counterpart relation is essentially a cross-world relation, with the only

exception that everything qualifies as a counterpart of itself.

AdA7A8: There exists a unique actual world. Its description can safely be used:

@ = xy(Iyx Ay)

x The following principles do notgenerally hold:

R1 Cxy Cyx

(Symmetry of the counterpart relation)

R2 Cxy Cyz Cxz

(Transitivity of the counterpart relation)

R3 Cy1x Cy2x Iy1w1 Iy2w2y1y2w1w2(Nothing in any world has more than one counterpart in any other world)

R4 Cyx1 Cyx2 Ix1w1 Ix2w2x1x2w1w2(No two things in any world have a common counterpart in any other world)

R5 Ww1 Ww2 Ixw1y(Iyw2 Cxy)

(For any two worlds, anything in one is a counterpart of something in the other)

R6 Ww1 Ww2 Ixw1y(Iyw2 Cyx)

(For any two worlds, anything in one has some counterpart in the other)

3. Comparison with Standard QML

x Translation:

T1 A@

whereAw (A holds in a world w) is defined recursively as follows:

T2a Aw

=A , ifA is atomicT2b (A)w = Aw

T2c (AB)w =AwBw

T2d (xA)w = x(IxwAw)

T2f0 ( A)w = z(WzAz)

(A holds in every worldz)


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T2f1 ( Ax)w = zy(Wz Iyz CyxAzy)

(A holds of every counterparty ofx in every worldz)

T2fn ( Ax1xn)w = zy1yn(Wz Iy1z Cy1x1 Iynz CynxnAzy1yn)

x Examples:

E1 xFx

x(Ix@ Fx)

(Everything actual is an F)

E2 xFx

w(Wwx(Ixw Fx))

(Some possible world contains an F)

E3 Fx

zy(Wz Iyz Cyx Fy)

(Every counterpart ofx, in any world, is an F)

E4 x(Fx Fx)

x(Ix@ zy(Wz Iyz Cyx Fx))

(If anything is a counterpart of an actual F, then it is an F)

E5 Fx

z1y1(Wz1Iy1z1Cy1xz2y2(Wz2Iy2z2Cy2y1Fy2)

(Every counterpart ofx has a counterpart that is an F)

x Critical principles:

B A A

Not a theorem (forA open) unless R1 (symmetry of C) is assumed4 A A

Notatheorem (forA open) unlessR2(transitivity of C) is assumedBF x Ax xAx

Not a theorem unless R5 is assumed.

BF' x Ax xAx

Not a theorem unless R6 is assumed.

BFc xAx x Ax

A theorem.

BFc' xAx x AxNot a theorem (obviously).

( )

VARZI - Modal Logic - Class Notes

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