B 0
... NB: Sensitivity in b3 that rivals astrophysical or atomic-physics bounds can only be attained if spectral resolution of 1 mHz is achieved. Not feasible at present in anti-H factories ...
... NB: Sensitivity in b3 that rivals astrophysical or atomic-physics bounds can only be attained if spectral resolution of 1 mHz is achieved. Not feasible at present in anti-H factories ...
Intuitionistic Type Theory - The collected works of Per Martin-Löf
... The principal problem that remained after Principia Mathematica was completed was, according to its authors, that of justifying the axiom of reducibility (or, as we would now say, the impredicative comprehension axiom). The ramified theory of types was predicative, but it was not sufficient for deri ...
... The principal problem that remained after Principia Mathematica was completed was, according to its authors, that of justifying the axiom of reducibility (or, as we would now say, the impredicative comprehension axiom). The ramified theory of types was predicative, but it was not sufficient for deri ...
MATH20302 Propositional Logic
... such as p, q, respectively s, t, not just for individual propositional variables, respectively propositional terms, but also as variables ranging over propositional variables, resp. propositional terms, (as we did just above). The definition above is an inductive one, with (0) being the base case an ...
... such as p, q, respectively s, t, not just for individual propositional variables, respectively propositional terms, but also as variables ranging over propositional variables, resp. propositional terms, (as we did just above). The definition above is an inductive one, with (0) being the base case an ...
Intuitionistic Type Theory
... The principal problem that remained after Principia Mathematica was completed was, according to its authors, that of justifying the axiom of reducibility (or, as we would now say, the impredicative comprehension axiom). The ramified theory of types was predicative, but it was not sufficient for deri ...
... The principal problem that remained after Principia Mathematica was completed was, according to its authors, that of justifying the axiom of reducibility (or, as we would now say, the impredicative comprehension axiom). The ramified theory of types was predicative, but it was not sufficient for deri ...
Towards an Epistemic Logic of Grounded Belief
... (c) If φ ∈ K , then φ must be True in the current state of affairs As a technical note I will say an ideal knower with knowledge base K is said to know a proposition φ if and only if φ ∈ K . I now turn to motivating this definition. The first part of this definition, Def. 2.1.1(a), may be motivated ...
... (c) If φ ∈ K , then φ must be True in the current state of affairs As a technical note I will say an ideal knower with knowledge base K is said to know a proposition φ if and only if φ ∈ K . I now turn to motivating this definition. The first part of this definition, Def. 2.1.1(a), may be motivated ...
The Logic of Provability
... Classical first-order arithmetic with induction; also called arithmetic or PA. More formally, we take the signature of PA to have ‘0’ as a constant and ‘+’, ‘·’, and ‘<’ as binary function symbols; PA is then the theory axiomatized by the following: • ∀x(sx 6= 0) • ∀x, y(sx = sy → x = y) • For every ...
... Classical first-order arithmetic with induction; also called arithmetic or PA. More formally, we take the signature of PA to have ‘0’ as a constant and ‘+’, ‘·’, and ‘<’ as binary function symbols; PA is then the theory axiomatized by the following: • ∀x(sx 6= 0) • ∀x, y(sx = sy → x = y) • For every ...
Higher Order Logic - Indiana University
... are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foun ...
... are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foun ...
Higher Order Logic - Theory and Logic Group
... are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foun ...
... are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of completeness. Our choice of topics is driven by an attempt to cover the foun ...
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 ...
Lecture Notes
... Input: A finite structure M = hW, π, K1 , . . . , Kn i and a formula φ ∈ L{K1 ,...,Kn ,CG } . Order subformulas(φ) as φ1 , φ2 , . . . , φk where φk = φ and subformulas(φj ) ⊆ {φ1 , . . . , φj } for 0 < j. For j = 1 . . . k, ...
... Input: A finite structure M = hW, π, K1 , . . . , Kn i and a formula φ ∈ L{K1 ,...,Kn ,CG } . Order subformulas(φ) as φ1 , φ2 , . . . , φk where φk = φ and subformulas(φj ) ⊆ {φ1 , . . . , φj } for 0 < j. For j = 1 . . . k, ...
A Logical Expression of Reasoning
... that an important feature of induction is that it is a kind of inference in which the conclusion is a more primitive, or general, statement than the data from which it is concluded. We call this an ascendant inference, an inference going from the particular to the general or to more general regulari ...
... that an important feature of induction is that it is a kind of inference in which the conclusion is a more primitive, or general, statement than the data from which it is concluded. We call this an ascendant inference, an inference going from the particular to the general or to more general regulari ...
Expressiveness of Logic Programs under the General Stable Model
... (respectively, function) on A of the same arity. For convenience, we assume that the definition of assignments extends to terms natually. Given a formula ϕ and an assignment α in A, we write A |= ϕ[α] if α satisfies ϕ in A in the standard way. In particular, if ϕ is a sentence, we simply write A |= ...
... (respectively, function) on A of the same arity. For convenience, we assume that the definition of assignments extends to terms natually. Given a formula ϕ and an assignment α in A, we write A |= ϕ[α] if α satisfies ϕ in A in the standard way. In particular, if ϕ is a sentence, we simply write A |= ...
x - Homepages | The University of Aberdeen
... 1. x (Q(x) P(x)) (true for place a below) 2. x (Q(x) P(x)) (false for places b below) 3. x (Q(x) P(x)) (false for place b below) 4. x (Q(x) P(x)) (true for place a below) One solution: a model with exactly two objects in it. One object has the property Q and the property P; the other obje ...
... 1. x (Q(x) P(x)) (true for place a below) 2. x (Q(x) P(x)) (false for places b below) 3. x (Q(x) P(x)) (false for place b below) 4. x (Q(x) P(x)) (true for place a below) One solution: a model with exactly two objects in it. One object has the property Q and the property P; the other obje ...
No Syllogisms for the Numerical Syllogistic
... numerical syllogistic seems not to have attracted the attention of logicians before the Nineteenth Century. The first systematic investigation known to the author is that of de Morgan [5] (Ch. VIII), though this work was closely followed by treatments in Boole [6] (reprinted as [7], Sec. IV) and Jev ...
... numerical syllogistic seems not to have attracted the attention of logicians before the Nineteenth Century. The first systematic investigation known to the author is that of de Morgan [5] (Ch. VIII), though this work was closely followed by treatments in Boole [6] (reprinted as [7], Sec. IV) and Jev ...
Deep Sequent Systems for Modal Logic
... The labelled approach seems to have become more prominent and according to several criteria has been more successful than the unlabelled approaches. It allows to capture a wide class of modal logics and does so systematically. In many important cases it yields systems which are natural and easy to u ...
... The labelled approach seems to have become more prominent and according to several criteria has been more successful than the unlabelled approaches. It allows to capture a wide class of modal logics and does so systematically. In many important cases it yields systems which are natural and easy to u ...
An introduction to ampleness
... as N is ω-saturated we get b̄ in N which satisfies all of them. Then āb̄ ≤ N and this gives what we want. It then follows easily that if N1 , N2 are ω-saturated models of T , then the set of isomorphisms between finite ≤-substructures of N1 and N2 is a back-and-forth system (– we also need to know ...
... as N is ω-saturated we get b̄ in N which satisfies all of them. Then āb̄ ≤ N and this gives what we want. It then follows easily that if N1 , N2 are ω-saturated models of T , then the set of isomorphisms between finite ≤-substructures of N1 and N2 is a back-and-forth system (– we also need to know ...
Using linear logic to reason about sequent systems
... Theorem 1. Let Ψ be a set of flat clauses, Γ and ∆ be multisets of flat goals, and Υ be a set of atomic formulas. Then the sequent Ψ ; ∆ −→ Γ ; Υ has a proof if and only if ! Ψ, ∆ ` Γ, ? Υ has a proof in linear logic. The following lemma holds in general for Forum proofs, but it is particularly rele ...
... Theorem 1. Let Ψ be a set of flat clauses, Γ and ∆ be multisets of flat goals, and Υ be a set of atomic formulas. Then the sequent Ψ ; ∆ −→ Γ ; Υ has a proof if and only if ! Ψ, ∆ ` Γ, ? Υ has a proof in linear logic. The following lemma holds in general for Forum proofs, but it is particularly rele ...
The Development of Categorical Logic
... to establish various independence results. In Fourman (1980) , it was shown how the construction of the usual cumulative hierarchy of sets generated by a collection of atoms can be carried out within a locally small complete topos E (in particular, any Grothendieck topos). This leads to a topos E*—i ...
... to establish various independence results. In Fourman (1980) , it was shown how the construction of the usual cumulative hierarchy of sets generated by a collection of atoms can be carried out within a locally small complete topos E (in particular, any Grothendieck topos). This leads to a topos E*—i ...
Modular Construction of Complete Coalgebraic Logics
... constant sets and composition. A recent survey of existing probabilistic models of systems [3] identified no less than eight probabilistic system types of interest, all of which can be written as such combinations. This paper derives logics and proof systems for these probabilistic system types, usi ...
... constant sets and composition. A recent survey of existing probabilistic models of systems [3] identified no less than eight probabilistic system types of interest, all of which can be written as such combinations. This paper derives logics and proof systems for these probabilistic system types, usi ...
Let me begin by reminding you of a number of passages ranging
... Four years later (1897), in the partial summary that precedes the principal text of the second and longer of the two posthumous “Logic” manuscripts, Frege writes: The word ‘true’ specifies the goal. Logic is concerned with the predicate ‘true’ in a special way. The word ‘true’ characterizes logic. ( ...
... Four years later (1897), in the partial summary that precedes the principal text of the second and longer of the two posthumous “Logic” manuscripts, Frege writes: The word ‘true’ specifies the goal. Logic is concerned with the predicate ‘true’ in a special way. The word ‘true’ characterizes logic. ( ...
Safety Metric Temporal Logic is Fully Decidable
... Our main technical contribution is to show the decidability of languageemptiness over infinite timed words for a class of timed alternating automata rich enough to capture Safety MTL formulas. A key restriction is that we only consider automata in which every state is accepting. We have recently sho ...
... Our main technical contribution is to show the decidability of languageemptiness over infinite timed words for a class of timed alternating automata rich enough to capture Safety MTL formulas. A key restriction is that we only consider automata in which every state is accepting. We have recently sho ...