Proof, Sets, and Logic - Boise State University
... relations work smoothly (and preserve the basic theorems) but one needs to modify the definitions of weak set picture and of the relation E on weak set pictures. July 12, 2009: I cleaned up the section on isomorphism types of wellfounded extensional relations (it is now a single section, not Old Ve ...
... relations work smoothly (and preserve the basic theorems) but one needs to modify the definitions of weak set picture and of the relation E on weak set pictures. July 12, 2009: I cleaned up the section on isomorphism types of wellfounded extensional relations (it is now a single section, not Old Ve ...
Provability as a Modal Operator with the models of PA as the Worlds
... Now if B is equivalent to PA, then {ϕ: N Bϕ} = Th(PA), then Mod(PA) = Mod(PA ∪ {ϕ: N Bϕ}) and N is central. If on the other hand we B not equivalent to PA we can first rule out the case where {ϕ: A Bϕ} ⊂ Th(PA) by the first property of B. From this we conclude that {ϕ: A Bϕ}⊆Th(PA), that is, t ...
... Now if B is equivalent to PA, then {ϕ: N Bϕ} = Th(PA), then Mod(PA) = Mod(PA ∪ {ϕ: N Bϕ}) and N is central. If on the other hand we B not equivalent to PA we can first rule out the case where {ϕ: A Bϕ} ⊂ Th(PA) by the first property of B. From this we conclude that {ϕ: A Bϕ}⊆Th(PA), that is, t ...
Deep Sequent Systems for Modal Logic
... syntax of a proof system, see for example [1]. For motivating the present work I would just like to take it as a given that it is an interesting question whether something that has been achieved using labels can also be achieved without them. My goal here is to develop proof systems with the same go ...
... syntax of a proof system, see for example [1]. For motivating the present work I would just like to take it as a given that it is an interesting question whether something that has been achieved using labels can also be achieved without them. My goal here is to develop proof systems with the same go ...
CUED PhD and MPhil Thesis Classes
... which yields a syntactic proof of the extended subformula property. This method can easily be extended to any system enriched with assumptions, if its pure version admits cut elimination. Then we provide some decision procedures for modal extensions of NL. It turns out that the modal extensions of N ...
... which yields a syntactic proof of the extended subformula property. This method can easily be extended to any system enriched with assumptions, if its pure version admits cut elimination. Then we provide some decision procedures for modal extensions of NL. It turns out that the modal extensions of N ...
LTL and CTL - UT Computer Science
... Exercise Try proving the equivalences in figure 2.(One of them has been done for you) ...
... Exercise Try proving the equivalences in figure 2.(One of them has been done for you) ...
Tableau-based decision procedure for the full
... r, r′ , and time moments n, n′ . It turns out that the presence or absence of synchrony, under no other assumptions, does not affect the outcome of our tableau procedure, and therefore, the satisfiability of formulae. The systems with (global) states represented as tuples of local states are general ...
... r, r′ , and time moments n, n′ . It turns out that the presence or absence of synchrony, under no other assumptions, does not affect the outcome of our tableau procedure, and therefore, the satisfiability of formulae. The systems with (global) states represented as tuples of local states are general ...
The Development of Categorical Logic
... Diaconescu’s result was refined by Goodman and Myhill (1978) to show that, in IZF, the law of excluded middle follows from the axiom of choice for unordered pairs. (It was later realized that the law or excluded middle in fact follows from the mere existence of a choice function on the power set of ...
... Diaconescu’s result was refined by Goodman and Myhill (1978) to show that, in IZF, the law of excluded middle follows from the axiom of choice for unordered pairs. (It was later realized that the law or excluded middle in fact follows from the mere existence of a choice function on the power set of ...
PDF
... There are two natural approaches to lift the definition of vacuity in the context of model checking to a definition of inherent vacuity. In order to see the idea behind the first approach, consider specifications that are tautologies or contradictions. One need not have a context in order to see th ...
... There are two natural approaches to lift the definition of vacuity in the context of model checking to a definition of inherent vacuity. In order to see the idea behind the first approach, consider specifications that are tautologies or contradictions. One need not have a context in order to see th ...
Multiverse Set Theory and Absolutely Undecidable Propositions
... existence of two models M1 and M2 of ZFC, one for ' and another for ¬'. This is indeed the “outside” view about a theory, such as ZFC, familiar already to Skolem [25] and von Neumann [32]. But we are trying to make sense of this from the “inside” of ZFC. A theory like ZFC is a theory of all mathema ...
... existence of two models M1 and M2 of ZFC, one for ' and another for ¬'. This is indeed the “outside” view about a theory, such as ZFC, familiar already to Skolem [25] and von Neumann [32]. But we are trying to make sense of this from the “inside” of ZFC. A theory like ZFC is a theory of all mathema ...
Proof, Sets, and Logic - Department of Mathematics
... Added material on forcing in type theory. Note that I have basic constructions for consistency and independence set up in the outline in to be in the first instance in type theory rather than in either untyped or stratified theories. My foundation for mathematics as far as possible is in TSTU here, ...
... Added material on forcing in type theory. Note that I have basic constructions for consistency and independence set up in the outline in to be in the first instance in type theory rather than in either untyped or stratified theories. My foundation for mathematics as far as possible is in TSTU here, ...
The Model-Theoretic Ordinal Analysis of Theories of Predicative
... theories bar definitions that require quantification over the full power set of N, depicting instead a universe of sets of numbers that is constructed "from the bottom up." Work of Feferman and Schfitte has established that the ordinal F7 is the least upper bound to the strength of such theories (se ...
... theories bar definitions that require quantification over the full power set of N, depicting instead a universe of sets of numbers that is constructed "from the bottom up." Work of Feferman and Schfitte has established that the ordinal F7 is the least upper bound to the strength of such theories (se ...
Compositional reasoning using intervals and time reversal
... behaviour, then their sequential composition is guaranteed to as well. The 2-to-1 formulas play a quite central role in almost all of the techniques presented here. For example, we can show that for the propositional variables p and q , the conventional temporal logic formula p ⊃ ✸q (“if p is true i ...
... behaviour, then their sequential composition is guaranteed to as well. The 2-to-1 formulas play a quite central role in almost all of the techniques presented here. For example, we can show that for the propositional variables p and q , the conventional temporal logic formula p ⊃ ✸q (“if p is true i ...
Sets, Logic, Computation
... consequence and the syntactic notion of provability give us two completely different ways to make precise the idea that a sentence may follow from some others. The soundness and completeness theorems link these two characterization. In particular, we will prove Gödel’s completeness theorem, which st ...
... consequence and the syntactic notion of provability give us two completely different ways to make precise the idea that a sentence may follow from some others. The soundness and completeness theorems link these two characterization. In particular, we will prove Gödel’s completeness theorem, which st ...
Barwise: Infinitary Logic and Admissible Sets
... relation and operation symbols with finitely many argument places. As usual, by an L-structure M, we mean a universe set M with an interpretation for each symbol of L. In cases where the vocabulary L is clear, we may just say structure. For a given vocabulary L and infinite cardinals µ ≤ κ, Lκµ is t ...
... relation and operation symbols with finitely many argument places. As usual, by an L-structure M, we mean a universe set M with an interpretation for each symbol of L. In cases where the vocabulary L is clear, we may just say structure. For a given vocabulary L and infinite cardinals µ ≤ κ, Lκµ is t ...
Insights into Modal Slash Logic and Modal Decidability
... in φ, and they are ordered by the relation of syntactic precedence given by the syntax of the logic in question. To distinguish τ -structures from modal structures, we use calligraphic symbols such as M and N for the former. If M is such a structure, M stands for its domain. For the semantics of ESO ...
... in φ, and they are ordered by the relation of syntactic precedence given by the syntax of the logic in question. To distinguish τ -structures from modal structures, we use calligraphic symbols such as M and N for the former. If M is such a structure, M stands for its domain. For the semantics of ESO ...
Sets, Logic, Computation
... consequence and the syntactic notion of provability give us two completely different ways to make precise the idea that a sentence may follow from some others. The soundness and completeness theorems link these two characterization. In particular, we will prove Gödel’s completeness theorem, which st ...
... consequence and the syntactic notion of provability give us two completely different ways to make precise the idea that a sentence may follow from some others. The soundness and completeness theorems link these two characterization. In particular, we will prove Gödel’s completeness theorem, which st ...
(pdf)
... encode sentences as numbers, so that Peano Arithmetic, being a theory of numbers, may indirectly talk about its own sentences. These issues are sidestepped in a theory of sets such as ZFC, because virtually any mathematical object, including systems of sentences, can already be thought of as sets, s ...
... encode sentences as numbers, so that Peano Arithmetic, being a theory of numbers, may indirectly talk about its own sentences. These issues are sidestepped in a theory of sets such as ZFC, because virtually any mathematical object, including systems of sentences, can already be thought of as sets, s ...
Beyond Quantifier-Free Interpolation in Extensions of Presburger
... Unfortunately, interpolation procedures for expressive first-order fragments, such as integer arithmetic with uninterpreted predicates, often generate interpolants with quantifiers, which makes subsequent calls to decision procedures involving these interpolants expensive. This is not by accident. In ...
... Unfortunately, interpolation procedures for expressive first-order fragments, such as integer arithmetic with uninterpreted predicates, often generate interpolants with quantifiers, which makes subsequent calls to decision procedures involving these interpolants expensive. This is not by accident. In ...
Introduction to Mathematical Logic lecture notes
... language such as English, Hebrew or French) such that each statement seems to follow from the previous ones (and sometimes, unfortunately, not even that). Here we will define formal proofs, or deductions, which will follow very strict rules that ensure that no mistake is possible. Throughout this cou ...
... language such as English, Hebrew or French) such that each statement seems to follow from the previous ones (and sometimes, unfortunately, not even that). Here we will define formal proofs, or deductions, which will follow very strict rules that ensure that no mistake is possible. Throughout this cou ...
On Decidability of Intuitionistic Modal Logics
... the intuitionistic and modal accessibility relations can be expressed using mso definable closure operators. We illustrate this method by showing that it works for various truth definitions for modalities and various conditions on the intuitionistic and modal accessibility occurring in the literatur ...
... the intuitionistic and modal accessibility relations can be expressed using mso definable closure operators. We illustrate this method by showing that it works for various truth definitions for modalities and various conditions on the intuitionistic and modal accessibility occurring in the literatur ...
Arithmetics in finite but potentially infinite worlds ∀ ∃ ∀ ∃
... statistical representability and weak statistical representability. We show that the relations which can be represented by these methods are the Σ2 , ∆2 and Π3 relations, respectively. Thus, the second of these methods is no more powerful than the original FM–representability concept introduced by M ...
... statistical representability and weak statistical representability. We show that the relations which can be represented by these methods are the Σ2 , ∆2 and Π3 relations, respectively. Thus, the second of these methods is no more powerful than the original FM–representability concept introduced by M ...
Notes on Modal Logic - Stanford University
... • is obliged to be • is now • will be • can do something to ensure that he is The basic modal language is a generic formal language with unary operators that have been used to reason about situations involving modal notions. This language is defined as follows: Definition 1.1 (The Basic Modal Langua ...
... • is obliged to be • is now • will be • can do something to ensure that he is The basic modal language is a generic formal language with unary operators that have been used to reason about situations involving modal notions. This language is defined as follows: Definition 1.1 (The Basic Modal Langua ...
Logic and Discrete Mathematics for Computer Scientists
... The formal approach to the presentation of material has, we believe, a number of significant advantages, especially for Computer Science students, but also, for more traditional math students who might find their way into the course. In mathematics departments proofs are typically learned by student ...
... The formal approach to the presentation of material has, we believe, a number of significant advantages, especially for Computer Science students, but also, for more traditional math students who might find their way into the course. In mathematics departments proofs are typically learned by student ...
A Complexity of Two-variable Logic on Finite Trees
... As mentioned in the introduction, satisfiability of the modal language XPath has been heavily studied, beginning with the work of Marx [Marx 2004] and Marx and de Rijke [Marx and de Rijke 2004]. A survey of some later results can be found in [Benedikt and Koch 2008]. Given that the translation from ...
... As mentioned in the introduction, satisfiability of the modal language XPath has been heavily studied, beginning with the work of Marx [Marx 2004] and Marx and de Rijke [Marx and de Rijke 2004]. A survey of some later results can be found in [Benedikt and Koch 2008]. Given that the translation from ...
On Countable Chains Having Decidable Monadic Theory.
... satisfy the criterion given in [1]. We proved in [3] that for every chain M = (A, <, P) such that (A, <) contains a sub-interval of type or −, M is not maximal with respect to MSO logic, i.e., there exists an expansion M of M by a predicate which is not MSO definable in M , and such that the MSO ...
... satisfy the criterion given in [1]. We proved in [3] that for every chain M = (A, <, P) such that (A, <) contains a sub-interval of type or −, M is not maximal with respect to MSO logic, i.e., there exists an expansion M of M by a predicate which is not MSO definable in M , and such that the MSO ...