This document discusses modal logics and formalisms. It begins by defining classical and non-classical logics, with modal logics listed as an example of an extended logic. It then covers modal logics in more detail, defining their language and model theory using possible world semantics. Models are defined as structures consisting of possible worlds related by an accessibility relation. Truth is evaluated at possible worlds based on this relation. The document also discusses axiomatic modal logics like KT and relations between main modal systems. Finally, it notes that axioms like D, T, B, 4 and 5 are not valid in the class of all standard models.
Аналіз законопроекту №7206 "Купуй українське, плати українцям" - основні аргументи проти, ризики та світова практика. МЕРТ категорично ПРОТИ ухвалення даного законопроекту
Аналіз законопроекту №7206 "Купуй українське, плати українцям" - основні аргументи проти, ризики та світова практика. МЕРТ категорично ПРОТИ ухвалення даного законопроекту
Computing the volume of a convex body is a fundamental problem in computational geometry and optimization. In this talk we discuss the computational complexity of this problem from a theoretical as well as practical point of view. We show examples of how volume computation appear in applications ranging from combinatorics to algebraic geometry.
Next, we design the first practical algorithm for polytope volume approximation in high dimensions (few hundreds).
The algorithm utilizes uniform sampling from a convex region and efficient boundary polytope oracles.
Interestingly, our software provides a framework for exploring theoretical advances since it is believed, and our experiments provide evidence for this belief, that the current asymptotic bounds are unrealistically high.
TMPA-2015: Implementing the MetaVCG Approach in the C-light SystemIosif Itkin
Alexei Promsky, Dmitry Kondtratyev, A.P. Ershov Institute of Informatics Systems, Novosibirsk
12 - 14 November 2015
Tools and Methods of Program Analysis in St. Petersburg
A polycycle is a 2-connected plane locally finite graph G with faces partitioned
in two faces F1 and F2. The faces in F1 are combinatorial i-gons.
The faces in F2 are called holes and are pair-wise disjoint.
All vertices have degree {2,...,q} with interior vertices of degree q.
Polycycles can be decomposed into elementary polycycles. For some parameters (i,q) the elementary polycycles can be classified and this allows to solve many different combinatorial problems.
Semantic Web technologies are a set of languages standardized by the World Wide Web Consortium (W3C) and designed to create a web of data that can be processed by machines. One of the core languages of the Semantic Web is Web Ontology Language (OWL), a family of knowledge representation languages for authoring ontologies or knowledge bases. The newest OWL is based on Description Logics (DL), a family of logics that are decidable fragments of first-order logic. leanCoR is a new description logic reasoner designed for experimenting with the new connection method algorithms and optimization techniques for DL. leanCoR is an extension of leanCoP, a compact automated theorem prover for classical first-order logic.
Computing the volume of a convex body is a fundamental problem in computational geometry and optimization. In this talk we discuss the computational complexity of this problem from a theoretical as well as practical point of view. We show examples of how volume computation appear in applications ranging from combinatorics to algebraic geometry.
Next, we design the first practical algorithm for polytope volume approximation in high dimensions (few hundreds).
The algorithm utilizes uniform sampling from a convex region and efficient boundary polytope oracles.
Interestingly, our software provides a framework for exploring theoretical advances since it is believed, and our experiments provide evidence for this belief, that the current asymptotic bounds are unrealistically high.
TMPA-2015: Implementing the MetaVCG Approach in the C-light SystemIosif Itkin
Alexei Promsky, Dmitry Kondtratyev, A.P. Ershov Institute of Informatics Systems, Novosibirsk
12 - 14 November 2015
Tools and Methods of Program Analysis in St. Petersburg
A polycycle is a 2-connected plane locally finite graph G with faces partitioned
in two faces F1 and F2. The faces in F1 are combinatorial i-gons.
The faces in F2 are called holes and are pair-wise disjoint.
All vertices have degree {2,...,q} with interior vertices of degree q.
Polycycles can be decomposed into elementary polycycles. For some parameters (i,q) the elementary polycycles can be classified and this allows to solve many different combinatorial problems.
Semantic Web technologies are a set of languages standardized by the World Wide Web Consortium (W3C) and designed to create a web of data that can be processed by machines. One of the core languages of the Semantic Web is Web Ontology Language (OWL), a family of knowledge representation languages for authoring ontologies or knowledge bases. The newest OWL is based on Description Logics (DL), a family of logics that are decidable fragments of first-order logic. leanCoR is a new description logic reasoner designed for experimenting with the new connection method algorithms and optimization techniques for DL. leanCoR is an extension of leanCoP, a compact automated theorem prover for classical first-order logic.
Learning Analytics : entre Promesses et RéalitéSerge Garlatti
Université Bretagne Pays de Loire, UTICE : LES LEARNING ANALYTICS : QUAND LE BIG DATA S’INTÉRESSE À L’ÉDUCATION.
https://utice.u-bretagneloire.fr/evenement/les-learning-analytics-quand-le-big-data-sinteresse-leducation
L’usage du numérique dans l’éducation permet d’accéder aujourd’hui à une multitude de données sur le comportement des étudiants : identité, interactions entre apprenants, interactions avec les plateformes et outils d’apprentissage, résultats aux évaluations... La collecte et l’exploitation de ces données permettent de mieux comprendre les processus d’apprentissage et ainsi d’adapter les parcours pédagogiques proposés pour en renforcer l’efficacité, mais aussi de personnaliser les apprentissages ou de développer des outils de pilotage des formations. Une communauté de chercheurs et d’enseignants se développe autour de ce que l’on appelle les learning analytics, ou l’analyse des données d’apprentissage. Ce séminaire basé sur les recherches et des retours d’expérience d’enseignants-chercheurs et de jeunes entreprises permettra de cerner les enjeux et les perspectives des learning analytics.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
2. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
2 / 41 SG Models and Formalisms
3. Non-classical Logics
Classical logics: propositional logic, first-order logic
Non-classical logics can be classified in two main categories:
Extended logics
New logical constants are added
The set of well-formed formulas (wff) is a proper superset of the
set of well-formed formulas in classical logic.
The set of theorems generated is a proper superset of the set of
theorems generated by classical logic,
Added theorems are only the result of the new wff.
3 / 41 SG Models and Formalisms
4. Non-classical Logics
Classical logics: propositional logic, first-order logic
Non-classical logics can be classified in two main categories:
Deviant logics
The usual logical constants are kept, but with a different meaning
Only a subset of the theorems from the classical logic hold
A non-exclusive classification, a logic could be in both.
4 / 41 SG Models and Formalisms
7. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
7 / 41 SG Models and Formalisms
8. Modal Logics
Traditional alethic modalities or modalities of truth
Propositional logic
New logical constants
Necessity:
Possibility: ♦
Alphabet
A set of propositional symbols S = {p1, p2, ..., pn}
A set of connectors: ¬, →, ↔, ∨, ∧
A set of modalities: , ♦
8 / 41 SG Models and Formalisms
9. The Language
Well-Formed Formulae
Let pi be a propositional symbol, (pi ∈ S), pi is an Atomic
formula
If G and H are wff
(G → H), (G ↔ H), (G ∨ H), (G ∧ H), ¬G, G and ♦G
are wff .
Let P and Q be wff
( P → ♦P)
( ♦P → ♦♦Q)
(( ♦P ∨ ♦ ♦Q) ∧ P)
9 / 41 SG Models and Formalisms
10. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
10 / 41 SG Models and Formalisms
11. Model Theory
In Propositional and Predicate Logics
The Interpretation function I is truth-functional: the truth or
falsity of a well-formed formula is determined by the truth or
falsity of its components.
11 / 41 SG Models and Formalisms
12. Model Theory
In Propositional and Predicate Logics
The Interpretation function I is truth-functional: the truth or
falsity of a well-formed formula is determined by the truth or
falsity of its components.
In Modal Logics, is-it the same?
P ¬P P
V F ?
F V F
11 / 41 SG Models and Formalisms
13. Model Theory
In Propositional and Predicate Logics
The Interpretation function I is truth-functional: the truth or
falsity of a well-formed formula is determined by the truth or
falsity of its components.
In Modal Logics, is-it the same?
P ¬P P
V F ?
F V F
In Modal logics, the Interpretation function I is not
truth-functional
11 / 41 SG Models and Formalisms
14. Model Theory
A standard Model is a structure M = (W, R, P) where
W is a set of possible worlds
12 / 41 SG Models and Formalisms
15. Model Theory
A standard Model is a structure M = (W, R, P) where
W is a set of possible worlds
R is a binary relation on W (R ⊆ W × W)
12 / 41 SG Models and Formalisms
16. Model Theory
A standard Model is a structure M = (W, R, P) where
W is a set of possible worlds
R is a binary relation on W (R ⊆ W × W)
R is an accessibility relationship between possible worlds.
We can write: α R β
12 / 41 SG Models and Formalisms
17. Model Theory
A standard Model is a structure M = (W, R, P) where
W is a set of possible worlds
R is a binary relation on W (R ⊆ W × W)
R is an accessibility relationship between possible worlds.
We can write: α R β
P is a mapping from natural numbers to subsets of W
(Pn ⊆ W, for each natural number n)
12 / 41 SG Models and Formalisms
18. Model Theory
A standard Model is a structure M = (W, R, P) where
W is a set of possible worlds
R is a binary relation on W (R ⊆ W × W)
R is an accessibility relationship between possible worlds.
We can write: α R β
P is a mapping from natural numbers to subsets of W
(Pn ⊆ W, for each natural number n)
P is an assignement of truth value to atomic formulae at possible
worlds.
It is a function on the set {0, 1, 2, ..., } of natural numbers such
that for each such number n, Pn is a subset of W.
It represents an assignment of sets of possible worlds to atomic
formulae (for pi ∈ S, Pi ⊆ W, ∀α such that α ∈ Pi , pi is true),
12 / 41 SG Models and Formalisms
19. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
13 / 41 SG Models and Formalisms
20. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
For propositional symbols pi , |=M
α pi iff α ∈ Pi for
n = {1, 2, 3, ..., n}
13 / 41 SG Models and Formalisms
21. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
For propositional symbols pi , |=M
α pi iff α ∈ Pi for
n = {1, 2, 3, ..., n}
|=M
α A means that A is true for the world α in the standard
model M
13 / 41 SG Models and Formalisms
22. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
For propositional symbols pi , |=M
α pi iff α ∈ Pi for
n = {1, 2, 3, ..., n}
|=M
α A means that A is true for the world α in the standard
model M
|=M
α ¬A iff not |=M
α A
13 / 41 SG Models and Formalisms
23. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
For propositional symbols pi , |=M
α pi iff α ∈ Pi for
n = {1, 2, 3, ..., n}
|=M
α A means that A is true for the world α in the standard
model M
|=M
α ¬A iff not |=M
α A
|=M
α (A ∨ B) iff either |=M
α A or |=M
α B), or both
13 / 41 SG Models and Formalisms
24. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
For propositional symbols pi , |=M
α pi iff α ∈ Pi for
n = {1, 2, 3, ..., n}
|=M
α A means that A is true for the world α in the standard
model M
|=M
α ¬A iff not |=M
α A
|=M
α (A ∨ B) iff either |=M
α A or |=M
α B), or both
|=M
α (A ∧ B) iff both |=M
α A and |=M
α B)
13 / 41 SG Models and Formalisms
25. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
|=M
α A iff for every β in M such that αRβ, |=M
β A
14 / 41 SG Models and Formalisms
26. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
|=M
α A iff for every β in M such that αRβ, |=M
β A
|=M
α ♦A iff for some β in M such that αRβ, |=M
β A
14 / 41 SG Models and Formalisms
27. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
|=M
α A iff for every β in M such that αRβ, |=M
β A
|=M
α ♦A iff for some β in M such that αRβ, |=M
β A
Truth in a Model
|=M
A iff for every world α in M, |=M
α A
14 / 41 SG Models and Formalisms
28. Possible Worlds, Truth, Validity
Truth in a possible world, α ∈ W in M = (W, R, P)
|=M
α A iff for every β in M such that αRβ, |=M
β A
|=M
α ♦A iff for some β in M such that αRβ, |=M
β A
Truth in a Model
|=M
A iff for every world α in M, |=M
α A
Validity in a class of Models
|=C A iff for every model M in C, |=M
A
14 / 41 SG Models and Formalisms
29. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
15 / 41 SG Models and Formalisms
32. Axiomatic Theory
Modal Logic KT
Axioms
Necessity axiom, called T
( P → P)
Axiom K
( (P → Q) → ( P → Q))
16 / 41 SG Models and Formalisms
33. Axiomatic Theory
Modal Logic KT
Axioms
Necessity axiom, called T
( P → P)
Axiom K
( (P → Q) → ( P → Q))
Inference Rules
Necessity Rule
P P
16 / 41 SG Models and Formalisms
34. Axiomatic Theory
Modal Logic KT
Axioms
Necessity axiom, called T
( P → P)
Axiom K
( (P → Q) → ( P → Q))
Inference Rules
Necessity Rule
P P
Modus Ponens
P, (P → Q) Q
16 / 41 SG Models and Formalisms
35. KT System: Axiomatic Theory
Properties of the Modal Logic KT
Consistency: KT is Consistent
Soundness: KT is Sound
Completeness: KT is Complete
Consequences: Axioms K and T are Valid Formulae.
17 / 41 SG Models and Formalisms
36. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
18 / 41 SG Models and Formalisms
37. Main Modal Systems
The main modal systems are composed from a small set of axioms:
Axiom T: ( P → P)
Axiom D: ( P → ♦P)
Axiom B: (P → ♦P)
Axiom 4: ( P → P)
Axiom 5: (♦P → ♦P)
19 / 41 SG Models and Formalisms
38. Main Modal Systems
The main modal systems are composed from a small set of axioms:
Axiom T: ( P → P)
Axiom D: ( P → ♦P)
Axiom B: (P → ♦P)
Axiom 4: ( P → P)
Axiom 5: (♦P → ♦P)
From these axioms, it possible to build the following systems: KT,
KD, KB, K4, K5, K45, KB4, KD4, KTB, KT4, KT5, etc.
19 / 41 SG Models and Formalisms
39. Relations between Axiomatic Modal Systems
K
KD K4 K5 KB
KTKDB KD4KD5 K45
KTB KT4 KD45 KB4
KT5
20 / 41 SG Models and Formalisms
40. Inclusion of K Axiomatic Systems
For instance, we can demonstrate that KD is included in KT,
We have to deduce ( P → ♦P) from ( P → P)
( P → P)
(¬P → ¬ P)
(¬P → ♦¬P)
T contraposition
(P → ♦P)
( P → ♦P)
Transitivity
21 / 41 SG Models and Formalisms
41. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
22 / 41 SG Models and Formalisms
42. Axioms and Class of Models
Theorem: None of the Axioms D, T, B, 4 and 5 is Valid in the
class of all standards models
Proof: it is enough for each axiom to exhibit a counter-model that
falsifies it. For the axiom D: ( P → ♦P):
23 / 41 SG Models and Formalisms
43. Axioms and Class of Models
Theorem: None of the Axioms D, T, B, 4 and 5 is Valid in the
class of all standards models
Proof: it is enough for each axiom to exhibit a counter-model that
falsifies it. For the axiom D: ( P → ♦P):
1 Let M = (W, R, P) be a standard model in which W = {α},
R = ∅, Pn = ∅ for n 0
23 / 41 SG Models and Formalisms
44. Axioms and Class of Models
Theorem: None of the Axioms D, T, B, 4 and 5 is Valid in the
class of all standards models
Proof: it is enough for each axiom to exhibit a counter-model that
falsifies it. For the axiom D: ( P → ♦P):
1 Let M = (W, R, P) be a standard model in which W = {α},
R = ∅, Pn = ∅ for n 0
2 M contains one world to which no world is related and at which
every atomic formula is false.
23 / 41 SG Models and Formalisms
45. Axioms and Class of Models
Theorem: None of the Axioms D, T, B, 4 and 5 is Valid in the
class of all standards models
Proof: it is enough for each axiom to exhibit a counter-model that
falsifies it. For the axiom D: ( P → ♦P):
1 Let M = (W, R, P) be a standard model in which W = {α},
R = ∅, Pn = ∅ for n 0
2 M contains one world to which no world is related and at which
every atomic formula is false.
3 W is a set, R is a binary relation(the empty relation) and P is a
mapping from natural numbers to (empty) subsets of W
23 / 41 SG Models and Formalisms
46. Axioms and Class of Models
Theorem: None of the Axioms D, T, B, 4 and 5 is Valid in the
class of all standards models
Proof: it is enough for each axiom to exhibit a counter-model that
falsifies it. For the axiom D: ( P → ♦P):
1 Let M = (W, R, P) be a standard model in which W = {α},
R = ∅, Pn = ∅ for n 0
2 M contains one world to which no world is related and at which
every atomic formula is false.
3 W is a set, R is a binary relation(the empty relation) and P is a
mapping from natural numbers to (empty) subsets of W
4 Every β in M such that αRβ is such that |=M
β P and there is
not some β in M such that αRβ and |=M
β P
23 / 41 SG Models and Formalisms
47. Axioms and Class of Models
The necessity Axiom ( P → P) in the following Standard Model:
w1
¬p1
w2
p1
w3
p1
w4
p1R
R
R
R
R
R
RR
24 / 41 SG Models and Formalisms
48. Axioms and Class of Models
Under What Conditions, The necessity Axiom ( P → P) in
the KT system will be valid:
Hypothesis |=M
α P, we have to demonstrate |=M
α P
25 / 41 SG Models and Formalisms
49. Axioms and Class of Models
Under What Conditions, The necessity Axiom ( P → P) in
the KT system will be valid:
Hypothesis |=M
α P, we have to demonstrate |=M
α P
1 By definition |=M
α P iff for every β in M such that αRβ, |=M
β P
25 / 41 SG Models and Formalisms
50. Axioms and Class of Models
Under What Conditions, The necessity Axiom ( P → P) in
the KT system will be valid:
Hypothesis |=M
α P, we have to demonstrate |=M
α P
1 By definition |=M
α P iff for every β in M such that αRβ, |=M
β P
2 Consequently, |=M
α P iff R is reflexive, that is to say αRα.
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51. Axioms and Class of Models
Under What Conditions, The necessity Axiom ( P → P) in
the KT system will be valid:
Hypothesis |=M
α P, we have to demonstrate |=M
α P
1 By definition |=M
α P iff for every β in M such that αRβ, |=M
β P
2 Consequently, |=M
α P iff R is reflexive, that is to say αRα.
In the KT system, the necessity Axiom is valid under the following
condition:
|=C ( P → P), C is a class of models in which R is
Reflexive
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52. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
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53. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
Serial iff for every α in M there is a β in M such that α R β
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54. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
Serial iff for every α in M there is a β in M such that α R β
Reflexive iff for every α in M, α R α
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55. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
Serial iff for every α in M there is a β in M such that α R β
Reflexive iff for every α in M, α R α
Symmetric iff for every α and β in M, if α R β then β R α
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56. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
Serial iff for every α in M there is a β in M such that α R β
Reflexive iff for every α in M, α R α
Symmetric iff for every α and β in M, if α R β then β R α
Transitive iff for every α, β and γ in M, if α R β and β R γ
then α R γ
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57. Axioms and Class of Models
Definition: Let M = (W, R, P) be a standard model, the
relation R, with α, β, γ ∈ W is:
Serial iff for every α in M there is a β in M such that α R β
Reflexive iff for every α in M, α R α
Symmetric iff for every α and β in M, if α R β then β R α
Transitive iff for every α, β and γ in M, if α R β and β R γ
then α R γ
Euclidian iff for every α, β and γ in M, if α R β and α R γ
then β R γ
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58. Axioms and Class of Models
Theorem: the following axioms are valid respectively in the
indicated class of standard models:
D: Serial
T: Reflexive
B: Symmetric
4: Transitive
5: Euclidian
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59. Axioms and Class of Models
Systems Serial Reflexive Symmetric Transitive Euclidian
KD x
KT x
KB x
K4 x
K5 x
KDB x x
KD4 x x
KD5 x x
KD45 x x
KB4 x x
KB4 x x
KTB x x
KT4 x x
KT5 x x
KT5 x x x
KT5 x x x
KT5 x x x
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60. Progress
1 Introduction
2 Modal Logics
3 Model Theory
4 Axiomatic Theory
5 Main Modal Systems
6 Axioms and Class of Models
7 A Knowledge and Belief Logic
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61. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge,
Knowledge Properties for an agent λ:
Axioms
1 Axiom 1 (Distribution Axiom, K): (KλP ∧ Kλ(P → Q)) → KλQ
- Same as (Kλ(P → Q)) → (KλP → KλQ))
2 Axiom 2 (Knowledge Axiom or T): (KλP → P)
3 Axiom 3 (Positive Introspection Axiom, 4): (KλP → KλKλP)
4 Axiom 4 (Negative Introspection Axiom): (¬KλP → Kλ¬KλP)
- (¬ P → ¬ P) (♦¬P → ♦¬P)
- (♦¬P → ♦¬P) (♦Q → ♦Q) (axiom 5)
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62. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge,
Knowledge Properties for an agent λ:
Inference Rules
Necessitation: P KλP
Omniscience Logic: if P Q and KλP then KλQ (axiom 1 +
necessitation rule)
The axiomatic theory correspond with the KT45 modal system in
which the accessibility relation is symmetric, reflexive and
transitive.
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63. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
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64. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
Three chairs are lined up, all facing the same direction, with one
behind the other. The wise men are instructed to sit down.
32 / 41 SG Models and Formalisms
65. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
Three chairs are lined up, all facing the same direction, with one
behind the other. The wise men are instructed to sit down.
The wise man in the back (wise man 3) can see the backs of the
other two men.
32 / 41 SG Models and Formalisms
66. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
Three chairs are lined up, all facing the same direction, with one
behind the other. The wise men are instructed to sit down.
The wise man in the back (wise man 3) can see the backs of the
other two men.
The man in the middle (wise man 2) can only see the one wise
man in front of him (wise man 1).
32 / 41 SG Models and Formalisms
67. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
Three chairs are lined up, all facing the same direction, with one
behind the other. The wise men are instructed to sit down.
The wise man in the back (wise man 3) can see the backs of the
other two men.
The man in the middle (wise man 2) can only see the one wise
man in front of him (wise man 1).
The wise man in front (wise man 1) can see neither wise man 3
nor wise man 2.
32 / 41 SG Models and Formalisms
68. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
33 / 41 SG Models and Formalisms
69. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
The king informs the wise men that he has three cards, all of
which are either black or white, at least one of which is
white. He places one card, face up, behind each of the three
wise men.
33 / 41 SG Models and Formalisms
70. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
The king informs the wise men that he has three cards, all of
which are either black or white, at least one of which is
white. He places one card, face up, behind each of the three
wise men.
Each wise man must determine the color of his own card and
announce what it is as soon as he knows. The first to correctly
announce the color of his own card will be aptly rewarded. All
know that this will happen. The room is silent; then, after
several minutes, wise man 1 says: My card is white!.
33 / 41 SG Models and Formalisms
71. Three-Wise-Men Problem
A king wishes to know whether his three advisors are as wise as
they claim to be.
We assume in this puzzle that the wise men do not lie, that they
all have the same reasoning capabilities, and that they can all think
at the same speed. We then can postulate that the following
reasoning took place.
34 / 41 SG Models and Formalisms
73. Three-Wise-Men Problem
Each wise man knows there is at least one white card. If the
cards of wise man 2 and wise man 1 were black, then wise man
3 would have been able to announce immediately that his card
was white.
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74. Three-Wise-Men Problem
Each wise man knows there is at least one white card. If the
cards of wise man 2 and wise man 1 were black, then wise man
3 would have been able to announce immediately that his card
was white.
They all realize this (they are all truly wise). Since wise man 3
kept silent, either wise man 2’s card is white, or wise man 1’s is.
35 / 41 SG Models and Formalisms
75. Three-Wise-Men Problem
Each wise man knows there is at least one white card. If the
cards of wise man 2 and wise man 1 were black, then wise man
3 would have been able to announce immediately that his card
was white.
They all realize this (they are all truly wise). Since wise man 3
kept silent, either wise man 2’s card is white, or wise man 1’s is.
At this point wise man 2 would be able to determine, if wise man
1’s were black, that his card was white. They all realize this.
35 / 41 SG Models and Formalisms
76. Three-Wise-Men Problem
Each wise man knows there is at least one white card. If the
cards of wise man 2 and wise man 1 were black, then wise man
3 would have been able to announce immediately that his card
was white.
They all realize this (they are all truly wise). Since wise man 3
kept silent, either wise man 2’s card is white, or wise man 1’s is.
At this point wise man 2 would be able to determine, if wise man
1’s were black, that his card was white. They all realize this.
Since wise man 2 also remains silent, wise man 1 knows his card
must be white.
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77. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
36 / 41 SG Models and Formalisms
78. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
1 A and B know that each one can see the card of the other one
36 / 41 SG Models and Formalisms
79. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
1 A and B know that each one can see the card of the other one
If A does not have a white card, B knows that A does not have a
white card
- (¬Blanc(A) → KB¬Blanc(A))
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80. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
1 A and B know that each one can see the card of the other one
If A does not have a white card, B knows that A does not have a
white card
- (¬Blanc(A) → KB¬Blanc(A))
A knows that if A does not have a white card, B knows that A
does not have a white card
36 / 41 SG Models and Formalisms
81. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
1 A and B know that each one can see the card of the other one
If A does not have a white card, B knows that A does not have a
white card
- (¬Blanc(A) → KB¬Blanc(A))
A knows that if A does not have a white card, B knows that A
does not have a white card
- KA(¬Blanc(A) → KB¬Blanc(A))
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82. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
2 A and B know that at least one of them has a white card and
each one knows that the other knows it.
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83. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
2 A and B know that at least one of them has a white card and
each one knows that the other knows it.
A knows that B knows that if A does not have a white card then
B has a white card
37 / 41 SG Models and Formalisms
84. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
2 A and B know that at least one of them has a white card and
each one knows that the other knows it.
A knows that B knows that if A does not have a white card then
B has a white card
- KAKB(¬Blanc(A) → Blanc(B))
37 / 41 SG Models and Formalisms
85. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
2 A and B know that at least one of them has a white card and
each one knows that the other knows it.
A knows that B knows that if A does not have a white card then
B has a white card
- KAKB(¬Blanc(A) → Blanc(B))
B declare that he does not say if he has a white card, then A
knows that B does not knows the color of his card.
37 / 41 SG Models and Formalisms
86. Three-Wise-Men Problem
If we reduce the problem with two advisors A and B and have the
following axioms:
2 A and B know that at least one of them has a white card and
each one knows that the other knows it.
A knows that B knows that if A does not have a white card then
B has a white card
- KAKB(¬Blanc(A) → Blanc(B))
B declare that he does not say if he has a white card, then A
knows that B does not knows the color of his card.
- KA¬KBBlanc(B)
37 / 41 SG Models and Formalisms
87. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
38 / 41 SG Models and Formalisms
88. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
KA(¬Blanc(A) → KB¬Blanc(A)) [1]
38 / 41 SG Models and Formalisms
89. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
KA(¬Blanc(A) → KB¬Blanc(A)) [1]
KAKB(¬Blanc(A) → Blanc(B)) [2]
KA¬KBBlanc(B) [3]
38 / 41 SG Models and Formalisms
90. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
KA(¬Blanc(A) → KB¬Blanc(A)) [1]
KAKB(¬Blanc(A) → Blanc(B)) [2]
KA¬KBBlanc(B) [3]
38 / 41 SG Models and Formalisms
91. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
KA(¬Blanc(A) → KB¬Blanc(A)) [1]
KAKB(¬Blanc(A) → Blanc(B)) [2]
KA¬KBBlanc(B) [3]
[1] (KA(¬Blanc(A) → KB¬Blanc(A)) (KλP → P) [Axiom2]
[4] (¬Blanc(A) → KB¬Blanc(A))
38 / 41 SG Models and Formalisms
92. Three-Wise-Men Problem
If we can reduce the problem with two advisors and have the
following axioms:
KA(¬Blanc(A) → KB¬Blanc(A)) [1]
KAKB(¬Blanc(A) → Blanc(B)) [2]
KA¬KBBlanc(B) [3]
[1] (KA(¬Blanc(A) → KB¬Blanc(A)) (KλP → P) [Axiom2]
[4] (¬Blanc(A) → KB¬Blanc(A))
[2] KAKB(¬Blanc(A) → Blanc(B)) (KλP → P) [Axiom2]
[5] (KB(¬Blanc(A) → Blanc(B))
(Kλ(P → Q) → (KλP → KλQ)) [Axiom1]
[6] (KB¬Blanc(A) → KBBlanc(B))
38 / 41 SG Models and Formalisms
94. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge, Belief
Properties for an agent λ:
Axioms
40 / 41 SG Models and Formalisms
95. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge, Belief
Properties for an agent λ:
Axioms
An agent cannot belief a contradiction: ¬Bλ(False)
40 / 41 SG Models and Formalisms
96. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge, Belief
Properties for an agent λ:
Axioms
An agent cannot belief a contradiction: ¬Bλ(False)
Positive Introspection Axiom (axiom 4): (BλP → BλBλP)
an agent believes what he believes
40 / 41 SG Models and Formalisms
97. Knowledge and Belief Logic
Two modal operators, B for Belief and K for Knowledge, Belief
Properties for an agent λ:
Axioms
An agent cannot belief a contradiction: ¬Bλ(False)
Positive Introspection Axiom (axiom 4): (BλP → BλBλP)
an agent believes what he believes
(BλBλP → BλP) (converse of the previous axiom)
(BλBαP → BλP)
an agent can belief what another agent believes
40 / 41 SG Models and Formalisms
98. Modal Logic Theorem Prover
Some Modal Logic Theorem Provers
MOLTAP, a Modal Logic Tableau Prover :
http://twan.home.fmf.nl/moltap/index.html
AiML.NET Advances in Modal Logic :
http://www.cs.man.ac.uk/ schmidt/tools/
MOLLE : http://molle.sourceforge.net/
LoTREC: possible worlds finally made accessible :
http://www.irit.fr/Lotrec/
A Theorem Prover for Intuitionistic Modal Logic S5 (IS5):
http://pl.postech.ac.kr/IS5/
MleanTAP: A Modal Theorem Prover (in Prolog) :
http://www.leancop.de/mleantap/
41 / 41 SG Models and Formalisms