Structural Proof Theory

... STRUCTURAL PROOF THEORY The idea of mathematical proof is very old, even if precise principles of proof have been laid down during only the past hundred years or so. Proof theory was first based on axiomatic systems with just one or two rules of inference. Such systems can be useful as formal repres ...

... STRUCTURAL PROOF THEORY The idea of mathematical proof is very old, even if precise principles of proof have been laid down during only the past hundred years or so. Proof theory was first based on axiomatic systems with just one or two rules of inference. Such systems can be useful as formal repres ...

Principia Logico-Metaphysica (Draft/Excerpt)

... Part I: Prophilosophy Part II: Philosophy Part III: Metaphilosophy Part IV: Technical Appendices, Bibliography, Index This excerpt was generated on October 28, 2016 and contains: • Part II: Chapter 7: The Language ...

... Part I: Prophilosophy Part II: Philosophy Part III: Metaphilosophy Part IV: Technical Appendices, Bibliography, Index This excerpt was generated on October 28, 2016 and contains: • Part II: Chapter 7: The Language ...

Model Theory of Modal Logic, Chapter in: Handbook of Modal Logic

... between the (ﬁrst-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic ﬂavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related to ...

... between the (ﬁrst-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic ﬂavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related to ...

Labeled Natural Deduction for Temporal Logics

... a great importance in computer science: applications include its use as a tool for the specification and verification of programs and protocols [18], in the study and development of temporal databases [39], as a framework within which to define the semantics of temporal expressions in natural langua ...

... a great importance in computer science: applications include its use as a tool for the specification and verification of programs and protocols [18], in the study and development of temporal databases [39], as a framework within which to define the semantics of temporal expressions in natural langua ...

5 model theory of modal logic

... between the (first-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic flavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related t ...

... between the (first-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic flavours, each with their own tradition in the model theory of modal logic. Still, these two semantics meet through the notion of a general frame (closely related t ...

A tableau-based decision procedure for LTL

... In implicit methods the accessibility relation is built-in into the structure of the tableau This is the case with tableau methods for linear and branching time point temporal logics Explicit methods keep track of the accessibility relation by means of some sort of external device This is the case w ...

... In implicit methods the accessibility relation is built-in into the structure of the tableau This is the case with tableau methods for linear and branching time point temporal logics Explicit methods keep track of the accessibility relation by means of some sort of external device This is the case w ...

Landau`s Fermi Liquid Theory

... The concept of a Fermi Liquid describes (strongly) interacting fermions using concepts that naively would only be applicable to a very weakly interacting gas. That this is at all possible makes Fermi Liquid theory a very versatile, but at the same time its very success is puzzling. Fermi Liquid theo ...

... The concept of a Fermi Liquid describes (strongly) interacting fermions using concepts that naively would only be applicable to a very weakly interacting gas. That this is at all possible makes Fermi Liquid theory a very versatile, but at the same time its very success is puzzling. Fermi Liquid theo ...

NOT EVEN WRONG tells a fascinating and complex story about

... Great things had happened and more were expected imminently from this impressive array of talent. During my college years I spent a formative summer working on a particle physics experiment at the Stanford Linear Accelerator Center, and a lot of time trying to figure out what quantum field theory wa ...

... Great things had happened and more were expected imminently from this impressive array of talent. During my college years I spent a formative summer working on a particle physics experiment at the Stanford Linear Accelerator Center, and a lot of time trying to figure out what quantum field theory wa ...

Introduction to Modal and Temporal Logic

... Proof: By induction on the length l of the derivation of Γ ⊢ ψ l = 0: So Γ ⊢ ψ because ψ ∈ Γ. But M Γ implies M ψ for all ψ ∈ Γ. l = 0: So Γ ⊢ ψ because ψ is an axiom schema instance. By Eg 1, Ex 1, Ex 2, Eg 2, we know ∅ |= ψ for every axiom schema instance ψ, hence Γ |= ψ. Ind. Hyp. : Theorem h ...

... Proof: By induction on the length l of the derivation of Γ ⊢ ψ l = 0: So Γ ⊢ ψ because ψ ∈ Γ. But M Γ implies M ψ for all ψ ∈ Γ. l = 0: So Γ ⊢ ψ because ψ is an axiom schema instance. By Eg 1, Ex 1, Ex 2, Eg 2, we know ∅ |= ψ for every axiom schema instance ψ, hence Γ |= ψ. Ind. Hyp. : Theorem h ...

Incompleteness

... @x pSpxq « 0q. @x @y pSpxq « Spyq Ñ x « yq (Order axioms) @x px ă 0q @x @ypx ă y _ x « y _ y ă xq. @x @y px ă Spyq Ø x ď yq (Addition axioms) @x px ` 0 « xq @x @y x ` Spyq « Spx ` yq. (Multiplication axioms) @x x ¨ 0 « 0. @x @y x ¨ Spyq « x ¨ y ` x (Exponentiation axioms) @x xE0 « Sp0q @x @y xESpyq ...

... @x pSpxq « 0q. @x @y pSpxq « Spyq Ñ x « yq (Order axioms) @x px ă 0q @x @ypx ă y _ x « y _ y ă xq. @x @y px ă Spyq Ø x ď yq (Addition axioms) @x px ` 0 « xq @x @y x ` Spyq « Spx ` yq. (Multiplication axioms) @x x ¨ 0 « 0. @x @y x ¨ Spyq « x ¨ y ` x (Exponentiation axioms) @x xE0 « Sp0q @x @y xESpyq ...

Word Consciousness - Brickhouse Bodymind

... a. A belief is a concept to which the mind is strongly attached. b. A belief that cannot be verified by direct seeing is always subject to attack by a counter-belief. Therefore, it must be constantly reinforced by repetition of the belief. c. Since Reality is absence of separation, It cannot be perc ...

... a. A belief is a concept to which the mind is strongly attached. b. A belief that cannot be verified by direct seeing is always subject to attack by a counter-belief. Therefore, it must be constantly reinforced by repetition of the belief. c. Since Reality is absence of separation, It cannot be perc ...

KURT GÖDEL - National Academy of Sciences

... These satisfiability results, which are coupled with the completeness theorem in Godel's treatment, have surprising consequences in certain cases when we aim to use a collection of formulas as axioms to characterize a mathematical system of objects. In doing so, the formulas are not to be logical ax ...

... These satisfiability results, which are coupled with the completeness theorem in Godel's treatment, have surprising consequences in certain cases when we aim to use a collection of formulas as axioms to characterize a mathematical system of objects. In doing so, the formulas are not to be logical ax ...

Enumerations in computable structure theory

... = A, there is a ∆0α (B) isomorphism. There are syntactical conditions that imply ∆0α categoricity, and are equivalent to relative ∆0α categoricity. The conditions involve the existence of nice “Scott families”. The notion comes from the proof of Scott’s Isomorphism Theorem ([21], [16]), which says t ...

... = A, there is a ∆0α (B) isomorphism. There are syntactical conditions that imply ∆0α categoricity, and are equivalent to relative ∆0α categoricity. The conditions involve the existence of nice “Scott families”. The notion comes from the proof of Scott’s Isomorphism Theorem ([21], [16]), which says t ...

Cut-elimination for provability logics and some results in display logic

... suggests, the formal inference rules in this system mimic (formalise) the sort of deductive reasoning that is employed in practice. In order to study the properties of this system, Gentzen then constructed yet another proof-system called the sequent calculus. Gentzen’s Hauptsatz or main theorem for ...

... suggests, the formal inference rules in this system mimic (formalise) the sort of deductive reasoning that is employed in practice. In order to study the properties of this system, Gentzen then constructed yet another proof-system called the sequent calculus. Gentzen’s Hauptsatz or main theorem for ...

Ground Nonmonotonic Modal Logics - Dipartimento di Informatica e

... 2.3 Ground logics Let us now turn the attention to the nonmonotonic modal logics that have been proposed on the basis of semantic considerations and, more precisely, on the intuition that in the models the knowledge attributed to the agent should be minimal. This principle was introduced in [11] and ...

... 2.3 Ground logics Let us now turn the attention to the nonmonotonic modal logics that have been proposed on the basis of semantic considerations and, more precisely, on the intuition that in the models the knowledge attributed to the agent should be minimal. This principle was introduced in [11] and ...

Enumerations in computable structure theory

... = A, there is a ∆0α (B) isomorphism. There are syntactical conditions that imply ∆0α categoricity, and are equivalent to relative ∆0α categoricity. The conditions involve the existence of nice “Scott families”. The notion comes from the proof of Scott’s Isomorphism Theorem, which says that for a cou ...

... = A, there is a ∆0α (B) isomorphism. There are syntactical conditions that imply ∆0α categoricity, and are equivalent to relative ∆0α categoricity. The conditions involve the existence of nice “Scott families”. The notion comes from the proof of Scott’s Isomorphism Theorem, which says that for a cou ...

Prime Implicates and Prime Implicants: From Propositional to Modal

... defined for the logic K is clearly of interest from a theoretical point of view. We argue, however, that this question is also practically relevant. To support this claim, we briefly discuss two application areas in which the study of prime implicates and prime implicants in K might prove useful. Th ...

... defined for the logic K is clearly of interest from a theoretical point of view. We argue, however, that this question is also practically relevant. To support this claim, we briefly discuss two application areas in which the study of prime implicates and prime implicants in K might prove useful. Th ...

A KE Tableau for a Logic of Formal Inconsistency - IME-USP

... pair T X and F X in DS. Second, all mCi KE rules with one or more premises (except (F ◦ ¬ n ◦) rules) preserve valuations. Note that (F ◦ ¬ n ◦) rules are taken into account by the last clause in Definition 2. That is, if we have a set of signed formulas that contains F ◦ ¬n ◦ X, every downward satu ...

... pair T X and F X in DS. Second, all mCi KE rules with one or more premises (except (F ◦ ¬ n ◦) rules) preserve valuations. Note that (F ◦ ¬ n ◦) rules are taken into account by the last clause in Definition 2. That is, if we have a set of signed formulas that contains F ◦ ¬n ◦ X, every downward satu ...

HOW TO DEFINE A MEREOLOGICAL (COLLECTIVE) SET

... or what are its essential features. First, we can see that every single one of objects a, b, c and d is part of x. Second, whatever part of x we take (any its fragment), it overlaps one of the four objects in question. On the other hand we can notice as well that any object which is exterior to ever ...

... or what are its essential features. First, we can see that every single one of objects a, b, c and d is part of x. Second, whatever part of x we take (any its fragment), it overlaps one of the four objects in question. On the other hand we can notice as well that any object which is exterior to ever ...

Combinaison des logiques temporelle et déontique pour la

... deadline, or the prohibition to execute a task for a too long period. Temporal and deontic logics seem well suited to specify such concepts. In this thesis, we study how to combine these logics. Firstly, we study the product of linear temporal logic and standard deontic logic, and deﬁne obligation w ...

... deadline, or the prohibition to execute a task for a too long period. Temporal and deontic logics seem well suited to specify such concepts. In this thesis, we study how to combine these logics. Firstly, we study the product of linear temporal logic and standard deontic logic, and deﬁne obligation w ...

A Pebble Weighted Automata and Weighted Logics

... In all these cases, however, one has to restrict the universal first-order quantification to stay within the class of recognizable series. In the present paper, we follow a different approach. Instead of restricting the logic, we define an extended automaton model that naturally reflects it. We will ...

... In all these cases, however, one has to restrict the universal first-order quantification to stay within the class of recognizable series. In the present paper, we follow a different approach. Instead of restricting the logic, we define an extended automaton model that naturally reflects it. We will ...

arXiv:1512.05177v1 [cs.LO] 16 Dec 2015

... has a fixed partition of the input alphabet into calls, returns and local actions. In contrast to traditional pushdown automata, VPAs may only push symbols onto the stack when reading calls and may only pop symbols off the stack when reading returns. Moreover, they may not inspect the topmost symbol ...

... has a fixed partition of the input alphabet into calls, returns and local actions. In contrast to traditional pushdown automata, VPAs may only push symbols onto the stack when reading calls and may only pop symbols off the stack when reading returns. Moreover, they may not inspect the topmost symbol ...

lecture notes in logic - UCLA Department of Mathematics

... τ = (Const, Rel, Funct, arity), where the sets of constant symbols Const, relation symbols Rel, and function symbols Funct have no common members and arity : Rel ∪ Funct → {1, 2, . . . }. A relation or function symbol P is n-ary if arity(P ) = n. We will often assume that these sets of names are fin ...

... τ = (Const, Rel, Funct, arity), where the sets of constant symbols Const, relation symbols Rel, and function symbols Funct have no common members and arity : Rel ∪ Funct → {1, 2, . . . }. A relation or function symbol P is n-ary if arity(P ) = n. We will often assume that these sets of names are fin ...

Measure Quantifier in Monadic Second Order Logic

... One of the first results about MSO was proved by Robinson [14] in 1958. He showed, answering a question of Tarski, that the theory MSO(ω, +, <) is undecidable. In 1962 Büchi [5] proved that the weaker theory MSO(ω, <) is decidable and in 1969 Rabin [13] extended this positive result to the MSO theo ...

... One of the first results about MSO was proved by Robinson [14] in 1958. He showed, answering a question of Tarski, that the theory MSO(ω, +, <) is undecidable. In 1962 Büchi [5] proved that the weaker theory MSO(ω, <) is decidable and in 1969 Rabin [13] extended this positive result to the MSO theo ...

Elements of Finite Model Theory

... historical comments. I taught two courses based on this book, and students in both classes provided very useful feedback; in addition to those I already thanked, I would like to acknowledge Antonina Kolokolova, Shiva Nejati, Ken Pu, Joseph Rideout, Mehrdad Sabetzadeh, Ramona Truta, and Zheng Zhang. ...

... historical comments. I taught two courses based on this book, and students in both classes provided very useful feedback; in addition to those I already thanked, I would like to acknowledge Antonina Kolokolova, Shiva Nejati, Ken Pu, Joseph Rideout, Mehrdad Sabetzadeh, Ramona Truta, and Zheng Zhang. ...