first-order extensions of classical modal logicfitelson.org/few/pacuit_slides.pdf · first-order...

First-Order Extensions ofClassical Modal Logic

Eric Pacuit

University of Maryland, College ParkTilburg Institute for Logic and Philosophy of Science

ai.stanford.edu/~epacuit

June 2, 2012

Eric Pacuit: First-Order Extensions of Classical Modal Logic 1/62

Horacio’s Neighborhood

Horacio Arlo-Costa (2002). First Order Extensions of Classical Systems of ModalLogic: The role of the Barcan Schemas. Studia Logica, 71:1, pgs. 87 - 118.

Horacio Arlo-Costa (2005). Non-Adjunctive Inference and Classical Modalities.Journal of Philosophical Logic, 34:5, pgs. 581 - 605.

Horacio Arlo-Costa and EP (2006). First-Order Classical Modal Logic. HoracioArlo-Costa and Eric Pacuit, Studia Logica, Volume 84:2, pgs. 171 - 210.

1. Background

• Neighborhood Semantics for Propositional Modal Logic

• First-Order Modal Logic

• The Barcan Formula

2. Neighborhood Models for First-Order Modal Logic

H. Arlo-Costa and E. Pacuit. First-Order Classical Modal Logic. Studia Logica,84, pgs. 171 - 210 (2006).

4. General Frames for First-Order Modal Logic

R. Goldblatt and E. Mares. A General Semantics for Quantified Modal Logic.AiML, 2006.

1. Background

• Neighborhood Semantics for Propositional Modal Logic

• First-Order Modal Logic

• The Barcan Formula

2. Neighborhood Models for First-Order Modal Logic

H. Arlo-Costa and E. Pacuit. First-Order Classical Modal Logic. Studia Logica,84, pgs. 171 - 210 (2006).

4. General Frames for First-Order Modal Logic

Brief History

w |= �ϕ if the truth set of ϕ is a neighborhood of w

neighborhood in some topology.J. McKinsey and A. Tarski. The Algebra of Topology. 1944.

contains all the immediate neighbors in some graphS. Kripke. A Semantic Analysis of Modal Logic. 1963.

an element of some distinguished collection of setsD. Scott. Advice on Modal Logic. 1970.

R. Montague. Pragmatics. 1968.

Brief History

What does it mean to be a neighborhood?

Brief History

Classical Modal Logic

To see the necessity of the more general approach, wecould consider probability operators, conditional necessity,or, to invoke an especially perspicuous example of DanaScott, the present progressive tense....Thus N mightreceive the awkward reading ‘it is being the case that’, inthe sense in which ‘it is being the case that Jones leaves’is synonymous with ‘Jones is leaving’. (Montague, 1970)

Brief History

PC Propositional Calculus

E �ϕ↔ ¬♦¬ϕM �(ϕ ∧ ψ)→ (�ϕ ∧�ψ)

C (�ϕ ∧�ψ)→ �(ϕ ∧ ψ)

N �>K �(ϕ→ ψ)→ (�ϕ→ �ψ)

REϕ↔ ψ

�ϕ↔ �ψ

MPϕ ϕ→ ψ

Brief History

E �ϕ↔ ¬♦¬ϕM �(ϕ ∧ ψ)→ (�ϕ ∧�ψ)

C (�ϕ ∧�ψ)→ �(ϕ ∧ ψ)

N �>K �(ϕ→ ψ)→ (�ϕ→ �ψ)

REϕ↔ ψ

�ϕ↔ �ψ

MPϕ ϕ→ ψ

A modal logic L is classical if itcontains all instances of E and isclosed under RE .

Brief History

E �ϕ↔ ¬♦¬ϕM �(ϕ ∧ ψ)→ (�ϕ ∧�ψ)

C (�ϕ ∧�ψ)→ �(ϕ ∧ ψ)

N �>K �(ϕ→ ψ)→ (�ϕ→ �ψ)

REϕ↔ ψ

�ϕ↔ �ψ

MPϕ ϕ→ ψ

A modal logic L is classical if itcontains all instances of E and isclosed under RE .

E is the smallest classical modallogic.

Brief History

E �ϕ↔ ¬♦¬ϕM �(ϕ ∧ ψ)→ (�ϕ ∧�ψ)

C (�ϕ ∧�ψ)→ �(ϕ ∧ ψ)

N �>K �(ϕ→ ψ)→ (�ϕ→ �ψ)

REϕ↔ ψ

�ϕ↔ �ψ

MPϕ ϕ→ ψ

In E, M is equivalent to

(Mon)ϕ→ ψ

�ϕ→ �ψ

Brief History

E �ϕ↔ ¬♦¬ϕ

Monϕ→ ψ

�ϕ→ �ψC (�ϕ ∧�ψ)→ �(ϕ ∧ ψ)

N �>K �(ϕ→ ψ)→ (�ϕ→ �ψ)

REϕ↔ ψ

�ϕ↔ �ψ

MPϕ ϕ→ ψ

EM is the logic E + Mon

Brief History

PC 6. Propositional Calculus

E �ϕ↔ ¬♦¬ϕ

Monϕ→ ψ

�ϕ→ �ψC (�ϕ ∧�ψ)→ �(ϕ ∧ ψ)

N �>K �(ϕ→ ψ)→ (�ϕ→ �ψ)

REϕ↔ ψ

�ϕ↔ �ψ

MPϕ ϕ→ ψ

EC is the logic E + C

Brief History

E �ϕ↔ ¬♦¬ϕ

Monϕ→ ψ

�ϕ→ �ψC (�ϕ ∧�ψ)→ �(ϕ ∧ ψ)

N �>K �(ϕ→ ψ)→ (�ϕ→ �ψ)

REϕ↔ ψ

�ϕ↔ �ψ

MPϕ ϕ→ ψ

EMC is the smallest regularmodal logic

Brief History

E �ϕ↔ ¬♦¬ϕ

Monϕ→ ψ

�ϕ→ �ψC (�ϕ ∧�ψ)→ �(ϕ ∧ ψ)

N �>K �(ϕ→ ψ)→ (�ϕ→ �ψ)

REϕ↔ ψ

�ϕ↔ �ψ

MPϕ ϕ→ ψ

A logic is normal if it contains allinstances of E , C and is closedunder Mon and Nec

Brief History

E �ϕ↔ ¬♦¬ϕ

Monϕ→ ψ

�ϕ→ �ψC (�ϕ ∧�ψ)→ �(ϕ ∧ ψ)

N �>K �(ϕ→ ψ)→ (�ϕ→ �ψ)

REϕ↔ ψ

�ϕ↔ �ψ

MPϕ ϕ→ ψ

K is the smallest normal modallogic

Brief History

E �ϕ↔ ¬♦¬ϕ

Monϕ→ ψ

�ϕ→ �ψC (�ϕ ∧�ψ)→ �(ϕ ∧ ψ)

N �>K �(ϕ→ ψ)→ (�ϕ→ �ψ)

REϕ↔ ψ

�ϕ↔ �ψ

MPϕ ϕ→ ψ

K = EMCN

Brief History

E �ϕ↔ ¬♦¬ϕ

Monϕ→ ψ

�ϕ→ �ψC (�ϕ ∧�ψ)→ �(ϕ ∧ ψ)

N �>K �(ϕ→ ψ)→ (�ϕ→ �ψ)

REϕ↔ ψ

�ϕ↔ �ψ

MPϕ ϕ→ ψ

K = PC (+E ) + K + Nec + MP

Brief History

R. Goldblatt. Mathematical Modal Logic: A View of its Evolution. Handbookof the History of Logic, 2005.

Motivating Examples

Logics of High Probability

�ϕ means “ϕ is assigned ‘high’ probability”, where high meansabove some threshold r ∈ (12 , 1].

Claim: Mon is a valid rule of inference.

Claim: C is not valid.

H. Kyburg and C.M. Teng. The Logic of Risky Knowledge. Proceedings ofWoLLIC (2002).

A. Herzig. Modal Probability, Belief, and Actions. Fundamenta Informaticae(2003).

Motivating Examples

Neighborhood Frames

Let W be a non-empty set of states.

Any map N : W → ℘(℘(W )) is called a neighborhood function

A pair 〈W ,N〉 is a called a neighborhood frame if W a non-emptyset and N is a neighborhood function.

A neighborhood model is a tuple 〈W ,N,V 〉 where V : At→ ℘(W )is a valuation function and 〈W ,N〉 is a neighborhood frame.

Motivating Examples

Truth in a Model

I M,w |= p iff w ∈ V (p)

I M,w |= ¬ϕ iff M,w 6|= ϕ

I M,w |= ϕ ∧ ψ iff M,w |= ϕ and M,w |= ψ

I M,w |= �ϕ iff (ϕ)M ∈ N(w)

I M,w |= ♦ϕ iff W − (ϕ)M 6∈ N(w)

where (ϕ)M = {w | M,w |= ϕ}.

Motivating Examples

Truth in a Model

I M,w |= p iff w ∈ V (p)

I M,w |= ¬ϕ iff M,w 6|= ϕ

I M,w |= ϕ ∧ ψ iff M,w |= ϕ and M,w |= ψ

I M,w |= �ϕ iff (ϕ)M ∈ N(w)

I M,w |= ♦ϕ iff W − (ϕ)M 6∈ N(w)

where (ϕ)M = {w | M,w |= ϕ}.

Motivating Examples

Validities

(Dual) �ϕ↔ ¬♦¬ϕ is valid in all neighborhood models.

(Re) If ϕ↔ ψ is valid then �ϕ↔ �ψ is valid.

Motivating Examples

Horacio’s Possibility

M,w |= �ϕ iff (ϕ)M ∈ N(w)

M,w |= ♦ϕ iff W − (ϕ)M 6∈ N(w)

M,w |= �ϕ iff there is an X ∈ N(w) such that for all v ∈W ,v ∈ X iff M, v |= ϕ

I M,w |= 〈〉ϕ iff ∃X ∈ N(w) such that ∃v ∈ X , M, v |= ϕ

I M,w |= [ ]ϕ iff ∀X ∈ N(w) such that ∀v ∈ X , M, v |= ϕ

I M,w |= 〈 ]ϕ iff ∃X ∈ N(w) such that ∀v ∈ X , M, v |= ϕ

I M,w |= [ 〉ϕ iff ∀X ∈ N(w) such that ∃v ∈ X , M, v |= ϕ

Motivating Examples

M,w |= ♦ϕ iff W − (ϕ)M 6∈ N(w)

Motivating Examples

M,w |= ♦ϕ iff W − (ϕ)M 6∈ N(w)

Motivating Examples

M,w |= ♦ϕ iff W − (ϕ)M 6∈ N(w)

Motivating Examples

M,w |= ♦ϕ iff W − (ϕ)M 6∈ N(w)

Motivating Examples

Other Examples, I

Reasoning about abilities

M. Brown. On the Logic of Ability. Journal of Philosophical Logic, 17, p. 1 -26 (1988).

Reasoning about games

R. Parikh. The Logic of Games and its Applications. Annals of Discrete Mathe-matics (1985).

Reasoning about coalitions

M. Pauly. Logic for Social Software. Ph.D. Thesis, ILLC (2001).

Motivating Examples

Other Examples, II

Epistemic Logic: the logical omniscience problem.

M. Vardi. On Epistemic Logic and Logical Omniscience. TARK (1986).

Reasoning about evidence and beliefs (and how they change overtime)

J. van Benthem and EP. Dynamics of Evidence-Based Beliefs. Studia Logica,2011.

J. van Benthem, D. Fernandez-Duque, EP. Evidence Logic: A New Look andNeighborhood Structures. Advances in Modal Logic, 2012.

Motivating Examples

Other Examples, II

Epistemic Logic: the logical omniscience problem.

M. Vardi. On Epistemic Logic and Logical Omniscience. TARK (1986).

Reasoning about evidence and beliefs (and how they change overtime)

J. van Benthem and EP. Dynamics of Evidence-Based Beliefs. Studia Logica,2011.

J. van Benthem, D. Fernandez-Duque, EP. Evidence Logic: A New Look andNeighborhood Structures. Advances in Modal Logic, 2012.

Motivating Examples

Other Examples, III

Program logics: modeling concurrent programs

D. Peleg. Concurrent Dynamic Logic. J. ACM (1987).

The Logic of deduction

P. Naumov. On modal logic of deductive closure. APAL (2005).

Deontic logics...

Motivating Examples

Constraints on neighborhood frames

I Monotonic or Supplemented: If X ∈ N(w) and X ⊆ Y ,then Y ∈ N(w)

I Closed under finite intersections: If X ∈ N(w) andY ∈ N(w), then X ∩ Y ∈ N(w)

I Contains the unit: W ∈ N(w)

I Augmented: Supplemented plus for each w ∈W ,⋂N(w) ∈ N(w)

Motivating Examples

Coherent Neighborhoods

A neighborhood of w is “perfectly coherent” provided⋂N(w) 6= ∅,

how should we measure the “level of coherence”when N(w) is not closed under conjunction?

Motivating Examples

Coherent Neighborhoods

A neighborhood of w is “perfectly coherent” provided⋂N(w) 6= ∅, how should we measure the “level of coherence”

when N(w) is not closed under conjunction?

Motivating Examples

From Kripke Frames to Neighborhood Frames

Let R ⊆W ×W , define a map R→ : W → ℘W :

for each w ∈W , let R→(w) = {v | wRv}

Motivating Examples

DefinitionGiven a relation R on a set W and a state w ∈W . A set X ⊆Wis R-necessary at w if R→(w) ⊆ X .

Motivating Examples

Let NRw be the set of sets that are R-necessary at w :

NRw = {X | R→(w) ⊆ X}

Motivating Examples

Let NRw be the set of sets that are R-necessary at w :

NRw = {X | R→(w) ⊆ X}

LemmaLet R be a relation on W . Then for each w ∈W , NR

w isaugmented.

Motivating Examples

From Neighborhood Frames to Kripke Frames

Theorem

I Let 〈W ,R〉 be a relational frame. Then there is an equivalentaugmented neighborhood frame.

I Let 〈W ,N〉 be an augmented neighborhood frame. Thenthere is an equivalent relational frame.

Theorem

for all X ⊆W , X ∈ N(w) iff X ∈ NRw .

Theorem

X Let 〈W ,R〉 be a relational frame. Then there is an equivalentaugmented neighborhood frame.

Proof.For each w ∈W , let N(w) = NR

Theorem

X Let 〈W ,N〉 be an augmented neighborhood frame. Thenthere is an equivalent relational frame.

Proof.For each w , v ∈W , wRNv iff v ∈ ∩N(w).

Motivating Examples

Theorem

Motivating Examples

Theorem

Motivating Examples

Theorem

Motivating Examples

Definability Results

1. F |= �(ϕ ∧ ψ)→ �ϕ ∧�ψ iff F is closed under supersets(monotonic frames).

2. F |= �ϕ ∧�ψ → �(ϕ ∧�ψ) iff F is closed under finiteintersections.

3. F |= �> iff F contains the unit

4. F |= EMCN iff F is a filter

5. F |= �ϕ→ ϕ iff for each w ∈W , w ∈ ∩N(w)

6. And so on...

Motivating Examples

6. And so on...

Motivating Examples

6. And so on...

Motivating Examples

Completeness Results

I E is sound and strongly complete with respect to the class ofall neighborhood frames

I EM is sound and strongly complete with respect to the classof all monotonic neighborhood frames

I EC is sound and strongly complete with respect to the classof all neighborhood frames that are closed under finiteintersections

I EN is sound and strongly complete with respect to the classof all neighborhood frames that contain the unit

I K is sound and strongly complete with respect to the class ofall neighborhood frames that are filters

I K is sound and strongly complete with respect to the class ofall augmented neighborhood frames

Motivating Examples

Some Results

I For each Kripke model 〈W ,R,V 〉, there is an pointwiseequivalent augmented neighborhood model 〈W ,N,V 〉.

I Bimodal normal modal logics can simulate non-normal modallogics (Kracht and Wolter 1999)

I There are logics which are complete with respect to a class ofneighborhood frames but not complete with respect torelational frames (D. Gabbay 1975, M. Gerson 1975, M.Gerson 1976).

I There are logics incomplete with respect to neighborhoodframes (M. Gerson, 1975; T. Litak, 2005).

I Model theory of classical modal logics: Bisimulation, vanBenthem Characterization Theorem, Interpolation, etc. (H.Hansen, C. Kupke and EP, 2010)

Motivating Examples

Some Results

Motivating Examples

Some Results

Motivating Examples

Some Results

Motivating Examples

Some Results

First-Order Classical Modal Logic

First-order extensions

First-Order Modal Language: L1

Extend the propositional modal language L with the usualfirst-order machinery (constants, terms, predicate symbols,quantifiers).

A := P(t1, . . . , tn) | ¬A | A ∧ A | �A | ∀xA

(note that equality is not in the language!)

First-Order Modal Language: L1

Extend the propositional modal language L with the usualfirst-order machinery (constants, terms, predicate symbols,quantifiers).

A := P(t1, . . . , tn) | ¬A | A ∧ A | �A | ∀xA

(note that equality is not in the language!)

State-of-the-art

T. Brauner and S. Ghilardi. First-order Modal Logic. Handbook of Modal Logic,pgs. 549 - 620 (2007).

D.Gabbay, V. Shehtman and D. Skvortsov. Quantification in Nonclassical Logic.Elsevier, 2009.

http://lpcs.math.msu.su/~shehtman/n.ps

M. Fitting and R. Mendelsohn. First-Order Modal Logic. Kluwer AcademicPublishers (1998).

First-order Modal Logic

A constant domain Kripke frame is a tuple 〈W ,R,D〉 where Wand D are sets, and R ⊆W ×W .

A constant domain Kripke model adds a valuation function V ,where for each n-ary relation symbol P and w ∈W ,V (P,w) ⊆ Dn.

A substitution is any function σ : V → D (V the set of variables).

A substitution σ′ is said to be an x-variant of σ if σ(y) = σ′(y)for all variable y except possibly x , this will be denoted by σ ∼x σ

A constant domain Kripke frame is a tuple 〈W ,R,D〉 where Wand D are sets, and R ⊆W ×W .

A constant domain Kripke model adds a valuation function V ,where for each n-ary relation symbol P and w ∈W ,V (P,w) ⊆ Dn.

Suppose that σ is a substitution.

1. M,w |=σ P(x1, . . . , xn) iff 〈σ(x1), . . . , σ(xn)〉 ∈ V (P,w)

2. M,w |=σ �A iff R(w) ⊆ (A)M,σ

3. M,w |=σ ∀xA iff for each x-variant σ′, M,w |=σ′ A

A constant domain Neighborhood frame is a tuple 〈W ,N,D〉where W and D are sets, and N : W → ℘(℘(W )).

A constant domain Neighborhood model adds a valuationfunction V , where for each n-ary relation symbol P and w ∈W ,V (P,w) ⊆ Dn.

Suppose that σ is a substitution.

1. M,w |=σ P(x1, . . . , xn) iff 〈σ(x1), . . . , σ(xn)〉 ∈ V (P,w)

2. M,w |=σ �A iff (A)M,σ ∈ N(w)

3. M,w |=σ ∀xA iff for each x-variant σ′, M,w |=σ′ A

Let S be any (classical) propositional modal logic, by FOL + S wemean the set of formulas closed under the following rules andaxioms:

(S) All instances of axioms and rules from S.

(∀) ∀xA→ Axt (where t is free for x in A)

(Gen)A→ B

A→ ∀xB, where x is not free in A.

Barcan Schemas

I Barcan formula (BF ): ∀x�A(x)→ �∀xA(x)

I converse Barcan formula (CBF ): �∀xA(x)→ ∀x�A(x)

Observation 1: CBF is provable in FOL + EM

Observation 2: BF and CBF both valid on relational frameswith constant domains

Observation 3: BF is valid in a varying domain relational frameiff the frame is anti-monotonic; CBF is valid in a varying domainrelational frame iff the frame is monotonic.

See (Fitting and Mendelsohn, 1998) for an extended discussion

Barcan Schemas

I Barcan formula (BF ): ∀x�A(x)→ �∀xA(x)

I converse Barcan formula (CBF ): �∀xA(x)→ ∀x�A(x)

Observation 1: CBF is provable in FOL + EM

Observation 2: BF and CBF both valid on relational frameswith constant domains

Observation 3: BF is valid in a varying domain relational frameiff the frame is anti-monotonic; CBF is valid in a varying domainrelational frame iff the frame is monotonic.

See (Fitting and Mendelsohn, 1998) for an extended discussion

Constant Domains without the Barcan Formula

The system EMN and seems to play a central role incharacterizing monadic operators of high probability (See Kyburgand Teng 2002, Arlo-Costa 2004).

Of course, BF should fail in this case, given that it instantiatescases of what is usually known as the ‘lottery paradox’:

For each individual x , it is highly probably that x will loose thelottery; however it is not necessarily highly probably that eachindividual will loose the lottery.

Constant Domains without the Barcan Formula

The system EMN and seems to play a central role incharacterizing monadic operators of high probability (See Kyburgand Teng 2002, Arlo-Costa 2004).

Of course, BF should fail in this case, given that it instantiatescases of what is usually known as the ‘lottery paradox’:

For each individual x , it is highly probably that x will loose thelottery; however it is not necessarily highly probably that eachindividual will loose the lottery.

Converse Barcan Formulas and Neighborhood Frames

A frame F is consistent iff for each w ∈W , N(w) 6= ∅

A first-order neighborhood frame F = 〈W ,N,D〉 is nontrivial iff|D| > 1

Lemma Let F be a consistent constant domain neighborhoodframe. The converse Barcan formula is valid on F iff either F istrivial or F is supplemented.

X ∈ N(w)

Y 6∈ N(w)

F = ∅

∀v 6∈ Y , I (F , v) = ∅

F = ∅

∀v ∈ X , I (F , v) = D = {a, b}

F = ∅

F = DF = {a}

∀v ∈ Y − X , I (F , v) = D = {a}

F = ∅

F = DF = {a}

(F [a])M = Y 6∈ N(w) hence w 6|= ∀x�F (x)

F = ∅

F = DF = {a}

(∀xF (x))M = (F [a])M ∩ (F [b])M = X ∈ N(w)

hence w |= �∀xF (x)

Barcan Formulas and Neighborhood Frames

We say that a frame closed under ≤ κ intersections if for eachstate w and each collection of sets {Xi | i ∈ I} where |I | ≤ κ,∩i∈IXi ∈ N(w).

Lemma Let F be a consistent constant domain neighborhoodframe. The Barcan formula is valid on F iff either

1. F is trivial or

2. if D is finite, then F is closed under finite intersections and ifD is infinite and of cardinality κ, then F is closed under ≤ κintersections.

Completeness Theorems

Theorem FOL + E is sound and strongly complete with respect tothe class of all frames.

Theorem FOL + EC is sound and strongly complete with respectto the class of frames that are closed under intersections.

Theorem FOL + EM is sound and strongly complete with respectto the class of supplemented frames.

Theorem FOL + E + CBF is sound and strongly complete withrespect to the class of frames that are either non-trivial andsupplemented or trivial and not supplemented.

FOL + K and FOL + K + BF

Theorem FOL + K is sound and strongly complete with respectto the class of filters.

Observation The augmentation of the smallest canonical modelfor FOL + K is not a canonical model for FOL + K. In fact, theclosure under infinite intersection of the minimal canonical modelfor FOL + K is not a canonical model for FOL + K.

Lemma The augmentation of the smallest canonical model forFOL + K + BF is a canonical for FOL + K + BF .

Theorem FOL + K + BF is sound and strongly complete withrespect to the class of augmented first-order neighborhood frames.

General Frames for First-Order Modal Logics

Is the addition of quantifiers straightforward?

1. S4M is complete for the class of all frames that are reflexive,transitive and final (every world can see an ‘end-point’).However FOL + S4M is incomplete for Kripke models basedon S4M-frames. (see Hughes and Cresswell, pg. 283).

2. S4.2 is S4 with ♦�ϕ→ �♦ϕ. This logics is complete for theclass of frames that are reflexive, transitive and convergent.However, FOL + S4M + BF is incomplete for the class ofconstant domain models based on reflexive, transitive andconvergent frames. (see Hughes and Cresswell, pg. 271)

3. The quantified extension of GL is not recursivelyaxiomatizable (Cresswell, 1997).

What is going on?

R. Goldblatt. Quantifiers, Propositions and Identity: Admissible Semantics forQuantified Modal and Substructural Logics. Lecture Notes in Logic No. 38,Cambridge University Press and the Association for Symbolic Logic, 2011.

What is going on?

Background: Incompleteness

There are (consistent) modal logics that are incomplete

A general model is a structure 〈W ,R,V ,A〉 where A is a suitableboolean algebra with an operator of propositions.

All modal logics are sound and strongly complete with respect togeneral frames.

Theorem (Goldblatt and Mares) For any canonical propositionalmodal logic S, its quantified extension QS is complete over a classof general frames for which the underlying propositional frame arejust the S-frames.

I New perspective on the Barcan formula: it corresponds toTarskian models

I There is a trade-off between having the underlying Kripkeframe validate the propositional logic in question and having aTarskian-reading of the quantifier.

Theorem (Goldblatt and Mares) For any canonical propositionalmodal logic S, its quantified extension QS is complete over a classof general frames for which the underlying propositional frame arejust the S-frames.

I New perspective on the Barcan formula: it corresponds toTarskian models

I There is a trade-off between having the underlying Kripkeframe validate the propositional logic in question and having aTarskian-reading of the quantifier.

Central Idea

Algebraic reading of the universal quantifier: ∀xϕ is true at aworld w iff there is some proposition X such that X entails everyinstantiation of ϕ and X obtains at w .

M,w |=σ ∀xA iff there is a proposition X such that w ∈ X andX ⊆ (A)Mσ(x |d) for all d ∈ D.

M,w |=σ ∀xA iff for all d ∈ D, M,w |=σ(x |d) A

Central Idea

Algebraic reading of the universal quantifier: ∀xϕ is true at aworld w iff there is some proposition X such that X entails everyinstantiation of ϕ and X obtains at w .

M,w |=σ ∀xA iff there is a proposition X such that w ∈ X andX ⊆ (A)Mσ(x |d) for all d ∈ D.

M,w |=σ ∀xA iff for all d ∈ D, M,w |=σ(x |d) A

General Frames

Let 〈W ,R〉 be a frame.

[R] : ℘W → ℘W where[R](X ) = {w ∈W | for all v ∈W , wRv implies v ∈ X}So (�α)M = [R](α)M

X ⇒ Y = (W − X ) ∪ YSo (α→ β)M = (α)M ⇒ (β)M.

Halmos Functions

ϕ : DV → ℘WLet ϕ and ψ be two such functions, we can lift [R] and ⇒ tooperations of functions: Eg., if ϕ : DV → ℘W and f ∈ DV .([R]ϕ)(f ) = [R](ϕ(f ))

Fix a set Prop ⊆ ℘W . This defines for each S ⊆ ℘W ,

uS =⋃{X ∈ Prop | X ⊆

Halmos Functions

ϕ : DV → ℘WLet ϕ and ψ be two such functions, we can lift [R] and ⇒ tooperations of functions: Eg., if ϕ : DV → ℘W and f ∈ DV .([R]ϕ)(f ) = [R](ϕ(f ))

Fix a set Prop ⊆ ℘W . This defines for each S ⊆ ℘W ,

uS =⋃{X ∈ Prop | X ⊆

General Frames for First-Order Modal Logic

Suppose Prop ⊆ ℘W and let ϕ : DV → Prop,(∀xϕ)f = ud∈Dϕ(f [x |d ])

〈W ,R,V ,Prop,PropFun〉 where

I Prop contains ∅ and is closed under ⇒ and [R]

I Contains the function ϕ∅(f ) = ∅ for all f ∈ DV

I PropFun is closed under ⇒, [R] and ∀x .

I Assume (P)M : DV → ℘W is an element of PropFun for eachatomic predicate P.

General Frames for First-Order Modal Logic

Suppose Prop ⊆ ℘W and let ϕ : DV → Prop,(∀xϕ)f = ud∈Dϕ(f [x |d ])

〈W ,R,V ,Prop,PropFun〉 where

I Prop contains ∅ and is closed under ⇒ and [R]

I Contains the function ϕ∅(f ) = ∅ for all f ∈ DV

I PropFun is closed under ⇒, [R] and ∀x .

I Assume (P)M : DV → ℘W is an element of PropFun for eachatomic predicate P.

General Completeness

Theorem For any propositional modal logic S, the quantified logicQS is complete for the class of (all validating) quantified generalframes.

Note that the canonical model construction has as worldsmaximally consistent sets that need not be ∀-complete.

Key ResultsTheorem (Goldblatt and Mares) If S is a canonical propositionallogic, then QS is characterized by the class of all QS-frames whoseunderlying propositional frames validate S.

Logics containing the Barcan formula have two characterizingcanonical general frames: one that is Tarskian and one that is not.

1. If S is canonical, then the second canonical model will havean underlying propositional frame that validates S (eg., S4.2),but may not be Tarskian.

2. On the other hand, The Tarskian canonical model may nothave an underlying propositional frame that is a frame for S(again S4.2 is an example).

Conclusions

I Characterize other “modality-quantifier interactions” in termsof properties on neighborhoods (eg. ∃x�ϕ(x)→ �∃xϕ(x))

I Equality, existence predicate, free systems, etc.

I Extensions with higher-order quantification Explain

G. Boella, D. Gabbay, V. Genovese, L. van der Torre. Higher-Order CoalitionLogic. Proceedings of ECAI, 2010.

I Different perspectives on the first-order modal language.

H. Sturm and F. Wolter. First-order Expressivity for S5-models: Modal vs.two-sorted Languages. Journal of Philosophical Logic (2000).

Conclusions

Thank you.

Extensions: Higher-Order Coalition Logic

G. Boella, D. Gabbay, V. Genovese, L. van der Torre. Higher-Order CoalitionLogic. 2010.

Strategy Logics

I Coalitional Logic: Reasoning about (local) group power.

[C ]ϕ: coalition C has a joint action to bring about ϕ.

M. Pauly. A Modal Logic for Coalition Powers in Games. Journal of Logic andComputation 12 (2002).

I Alternating-time Temporal Logic: Reasoning about (local andglobal) group power:

〈〈A〉〉�ϕ: The coalition A has a joint action to ensure that ϕwill remain true.

R. Alur, T. Henzinger and O. Kupferman. Alternating-time Temproal Logic.Jouranl of the ACM (2002).

Strategy Logics

I Coalitional Logic: Reasoning about (local) group power.

[C ]ϕ: coalition C has a joint action to bring about ϕ.

M. Pauly. A Modal Logic for Coalition Powers in Games. Journal of Logic andComputation 12 (2002).

I Alternating-time Temporal Logic: Reasoning about (local andglobal) group power:

〈〈A〉〉�ϕ: The coalition A has a joint action to ensure that ϕwill remain true.

R. Alur, T. Henzinger and O. Kupferman. Alternating-time Temproal Logic.Jouranl of the ACM (2002).

Multi-agent Transition Systems

x = 1q0

x = 2q1

〈set2, grant〉〈set1, grant〉

〈∗, deny〉

(Px=1 → [s]Px=1) ∧ (Px=2 → [s]Px=2)

x = 1q0

x = 2q1

〈∗, deny〉

Px=1 → ¬[s]Px=2

x = 1q0

x = 2q1

〈∗, deny〉

Px=1 → [s, c]Px=2

Coalition Logic: ϕ := p | ¬ϕ | ϕ ∧ ϕ | [C ]ϕ

M,w |= [C ]ϕ iff (ϕ)M ∈ N(w ,C ): “Coalition C has a jointstrategy to force the outcome to satisfy ϕ”.

Higher-Order Coalition Logic: ϕ :=F (x1, . . . , xn) | Xx | ¬ϕ | ϕ ∧ ϕ | ∀Xϕ | ∀xϕ | [{x}ϕ]ϕ | 〈{x}ϕ〉ϕ

I F (x1, . . . , xn) is a first-order atomic formula

I x is a first-order variable

I X is a set variable

I {x}ψ is a group operator representing the set of all d suchthat ψ[d/x ] holds

HCL: Expressivity

What does the added expressive power give you?

I Relationships between coalitions:∀x(super user(x)→ user(x))

I General quantification over coalitions:∀X (∀x(Xx → user(x))→ [{y}Xy ]ϕ)

Every coalition such that all of its members are users canachieve ϕ.

I Complex relationships between coalitions and agents:[{x}ϕ(x)]ψ → [{y}∃x(ϕ(x) ∧ collaborates(y , x))]ψ

If the coalition represented by ϕ can achieve ψ then so canany group that collaborates with at least one member of ϕ(x).

HCL: Expressivity

HCL: Barcan/Converse Barcan Formulas

Converse Barcan: [{x}ϕ(x)]∀yψ(y)→ ∀y [{x}ϕ(x)]ϕ(y)

Barcan: ∀y [{x}ϕ(x)]ϕ(y)→ [{x}ϕ(x)]∀yψ(y)

[{x}x = Eric]∀y(CMU(y)→ happy(y))→∀y [{x}x = Eric](CMU(y)→ happy(y))

If I can do something to make everyone happy at CMU implies foreach person at CMU, I can do something to make them happy.

∀y [{x}x = Eric](CMU(y)→ happy(y))6→[{x}x = Eric]∀y(CMU(y)→ happy(y))

For each person at CMU, I can make them happy does not implythat I can do something to make everyone at CMU happy.

Higher-Order Coalition Logic

Sound and complete axiomatization combines ideas from coalitionlogic, first-order extensions of non-normal modal logics andHenkin-style completeness for second-order logic.

first-order extensions of classical modal logicfitelson.org/few/pacuit_slides.pdf · first-order...

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