Hilbert`s Program Then and Now
... these objects must be capable of being completely surveyed in all their parts, and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something which cannot be reduced to something else.4 The objects in questions are si ...
... these objects must be capable of being completely surveyed in all their parts, and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something which cannot be reduced to something else.4 The objects in questions are si ...
Inductive Types in Constructive Languages
... preserving validity, with notations or sublanguages for special applications, like program correctness. The proposed notations should therefore not be regarded as immutable. Perhaps the only typical language element is the notation for (and the consistent use of) families of objects. As a preparatio ...
... preserving validity, with notations or sublanguages for special applications, like program correctness. The proposed notations should therefore not be regarded as immutable. Perhaps the only typical language element is the notation for (and the consistent use of) families of objects. As a preparatio ...
Self-Referential Probability
... on a supervaluational evaluation scheme. This variation is particularly interesting because it bears a close relationship to imprecise probabilities where agents’ credal states are taken to be sets of probability functions. In this chapter, we will also consider how to use this language to describe ...
... on a supervaluational evaluation scheme. This variation is particularly interesting because it bears a close relationship to imprecise probabilities where agents’ credal states are taken to be sets of probability functions. In this chapter, we will also consider how to use this language to describe ...
Chapter 13 BOOLEAN ALGEBRA
... (3) L = 81, 2, 3, 6< and r is the relation | (divides). We remind the reader that the pair Ha, bL as an element of the relation r can be expressed as Ha, bL œ r, or a r b, depending on convenience and readability. The posets we will concentrate on in this chapter will be those which have maxima and ...
... (3) L = 81, 2, 3, 6< and r is the relation | (divides). We remind the reader that the pair Ha, bL as an element of the relation r can be expressed as Ha, bL œ r, or a r b, depending on convenience and readability. The posets we will concentrate on in this chapter will be those which have maxima and ...
PhD Thesis First-Order Logic Investigation of Relativity Theory with
... of didactic convenience and visualization. There are many reasons for using observers (or coordinate systems, or reference frames) instead of a single observer-independent spacetime structure. One is that it helps to weed unnecessary axioms from our theories. Nevertheless, we state and emphasize the ...
... of didactic convenience and visualization. There are many reasons for using observers (or coordinate systems, or reference frames) instead of a single observer-independent spacetime structure. One is that it helps to weed unnecessary axioms from our theories. Nevertheless, we state and emphasize the ...
Enumerations in computable structure theory
... is Σ0α (A) if ϕ(x) is computable Σα , and Π0α (A) if ϕ(x) is computable Πα . Moreover, this holds with all imaginable uniformity, over structures and formulas. It is easy to see that if A has a formally c.e. Scott family, then it is relatively computably categorical, so it is computably categorical ...
... is Σ0α (A) if ϕ(x) is computable Σα , and Π0α (A) if ϕ(x) is computable Πα . Moreover, this holds with all imaginable uniformity, over structures and formulas. It is easy to see that if A has a formally c.e. Scott family, then it is relatively computably categorical, so it is computably categorical ...
JUXTAPOSITION - Brown University
... §1. Introduction. Methods of combining logics are of great interest.1 Formal systems that result from the combination of multiple logical systems into a single system have applications in mathematics, linguistics, and computer science. For example, there are many applications for logics with multipl ...
... §1. Introduction. Methods of combining logics are of great interest.1 Formal systems that result from the combination of multiple logical systems into a single system have applications in mathematics, linguistics, and computer science. For example, there are many applications for logics with multipl ...
Notions of Computability at Higher Type
... objects in connection with systems for explicit mathematics and theories of finite type (Feferman [1975], [1977b]). These systems are typically intended to reflect “semi-constructivist” standpoints that suffice for most of mathematical practice. For other recent applications of this kind, see Kohlen ...
... objects in connection with systems for explicit mathematics and theories of finite type (Feferman [1975], [1977b]). These systems are typically intended to reflect “semi-constructivist” standpoints that suffice for most of mathematical practice. For other recent applications of this kind, see Kohlen ...
A tableau-based decision procedure for LTL
... that (i) Aj satisfies all the requirements of an atom and (ii) the pair (Aj , Aj+1 ) satisfies the condition on edges. Hence, πσ : A0 , A1 , . . . is an infinite path in Tϕ induced by σ. ...
... that (i) Aj satisfies all the requirements of an atom and (ii) the pair (Aj , Aj+1 ) satisfies the condition on edges. Hence, πσ : A0 , A1 , . . . is an infinite path in Tϕ induced by σ. ...
Enumerations in computable structure theory
... It is easy to see that if A has a formally c.e. Scott family, then it is relatively computably categorical, so it is computably categorical. More generally, if A has a formally Σ0α Scott family, then we can see, using Theorem 1.1, that it is relatively ∆0α categorical, so it is ∆0α categorical. Gon ...
... It is easy to see that if A has a formally c.e. Scott family, then it is relatively computably categorical, so it is computably categorical. More generally, if A has a formally Σ0α Scott family, then we can see, using Theorem 1.1, that it is relatively ∆0α categorical, so it is ∆0α categorical. Gon ...
Understanding SPKI/SDSI Using First-Order Logic
... two efforts were merged, leading to a system called SPKI/SDSI. The standard reference on SPKI/SDSI is RFC 2693 [10], with a later paper [7], whose authors include several designers of SPKI/SDSI, providing certificate chain reduction algorithms for SPKI/SDSI, clearer descriptions of many features, an ...
... two efforts were merged, leading to a system called SPKI/SDSI. The standard reference on SPKI/SDSI is RFC 2693 [10], with a later paper [7], whose authors include several designers of SPKI/SDSI, providing certificate chain reduction algorithms for SPKI/SDSI, clearer descriptions of many features, an ...
Hilbert`s Program Then and Now - Philsci
... their meta-logical investigation, such as the proof (by Paul Bernays in 1918) of the completeness of the propositional calculus and the work in Hilbert’s school from 1921 onward on the decision problem. The investigation of the metalogical properties of logical systems led directly to some of the mo ...
... their meta-logical investigation, such as the proof (by Paul Bernays in 1918) of the completeness of the propositional calculus and the work in Hilbert’s school from 1921 onward on the decision problem. The investigation of the metalogical properties of logical systems led directly to some of the mo ...
Curry-Howard Isomorphism - Department of information engineering
... on the basis of Boolean algebras—and the soundness and completeness results are then proved. An informal proof semantics, the so-called BHKinterpretation, is also presented. Chapter 3 presents the simply typed λ-calculus and its most fundamental properties up to the subject reduction property and th ...
... on the basis of Boolean algebras—and the soundness and completeness results are then proved. An informal proof semantics, the so-called BHKinterpretation, is also presented. Chapter 3 presents the simply typed λ-calculus and its most fundamental properties up to the subject reduction property and th ...
Relevant Logic A Philosophical Examination of Inference Stephen Read February 21, 2012
... they must be (logically) relevant to it. Recognising the Relevant Account of Validity allows us to see how the formal semantics of chapter 5 retained in its response to the classical challenge one disastrous classical belief, namely, acceptance of the Classical Account of Validity, or consequence. I ...
... they must be (logically) relevant to it. Recognising the Relevant Account of Validity allows us to see how the formal semantics of chapter 5 retained in its response to the classical challenge one disastrous classical belief, namely, acceptance of the Classical Account of Validity, or consequence. I ...
some results on locally finitely presentable categories
... X.6.3, p. 243]. For (iv), see [SGA4,Exp. I, proof of 8.3.3 (i)=>(ii),p. 77]. Since for M G (C, S), and G € C, ((G, -), M) = M(C) (Yoneda), the fact that each representable functor in LEX(C, 5) is f.p. follows from (i) and the fact that (filtered) colimits in (G, S) are computed pointwise. Conversely ...
... X.6.3, p. 243]. For (iv), see [SGA4,Exp. I, proof of 8.3.3 (i)=>(ii),p. 77]. Since for M G (C, S), and G € C, ((G, -), M) = M(C) (Yoneda), the fact that each representable functor in LEX(C, 5) is f.p. follows from (i) and the fact that (filtered) colimits in (G, S) are computed pointwise. Conversely ...
On the Complexity of Qualitative Spatial Reasoning: A Maximal
... to two spatial variables XL and YL and a relation R = Rt U R; , where Rt n Rf = 0 and XL RY L . L is true iff XLRfYL holds and false iff XLRfYL holds. Additional "polarity" constraints have to be introduced to assure that for the spatial variables X,L and Y-L, corresponding to the negation of L, X,L ...
... to two spatial variables XL and YL and a relation R = Rt U R; , where Rt n Rf = 0 and XL RY L . L is true iff XLRfYL holds and false iff XLRfYL holds. Additional "polarity" constraints have to be introduced to assure that for the spatial variables X,L and Y-L, corresponding to the negation of L, X,L ...
Constraint Logic Programming with Hereditary Harrop Formula
... The notation Cσ used above means application to a constraint C of a substitution σ = [t1 /x1 , . . . , tn /xn ], using proper renaming of the variables bound in C to avoid capturing free variables from the terms ti , 1 ≤ i ≤ n. Γσ represents the application of σ to every constraint of the set Γ. In ...
... The notation Cσ used above means application to a constraint C of a substitution σ = [t1 /x1 , . . . , tn /xn ], using proper renaming of the variables bound in C to avoid capturing free variables from the terms ti , 1 ≤ i ≤ n. Γσ represents the application of σ to every constraint of the set Γ. In ...
Proofs in theories
... proposition and proof, kept separate in chapter 8 and 9, are unified into a single syntactic notion. This leads to introduce the λΠ-calculus, also known as λ-calculus with dependent types, and its extension the λΠ-calculus modulo. We then present two other formalisms: Martin-Löf’s type theory and th ...
... proposition and proof, kept separate in chapter 8 and 9, are unified into a single syntactic notion. This leads to introduce the λΠ-calculus, also known as λ-calculus with dependent types, and its extension the λΠ-calculus modulo. We then present two other formalisms: Martin-Löf’s type theory and th ...
a semantic perspective - Institute for Logic, Language and
... to both first- and second-order classical logic. This immediately raises interesting questions. How does modal logic compare with these logics as a tool for talking about graphs? Can modal expressivity over graphs be characterised in terms of classical logic? We shall ask (and answer) such questions ...
... to both first- and second-order classical logic. This immediately raises interesting questions. How does modal logic compare with these logics as a tool for talking about graphs? Can modal expressivity over graphs be characterised in terms of classical logic? We shall ask (and answer) such questions ...
Interpretability formalized
... and for different purposes. A famous and well known example is an interpretation of hyperbolic geometry in Euclidean geometry (e.g., the Beltrami-Klein model, see, for example, [Gre96]) to show the relative consistency of non-Euclidean geometry. Another example, no less famous, is Gödel’s interpret ...
... and for different purposes. A famous and well known example is an interpretation of hyperbolic geometry in Euclidean geometry (e.g., the Beltrami-Klein model, see, for example, [Gre96]) to show the relative consistency of non-Euclidean geometry. Another example, no less famous, is Gödel’s interpret ...
... [Ha187], then we identify a protocol with a certain type of Kripke structure. If we are interested in studying a class of protocols which are characterized by some properties (intuitively, these properties hold in each Kripke structure corresponding to a protocol in the class), then A! inference is ...
1 Non-deterministic Phase Semantics and the Undecidability of
... directly into BBI and Kripke semantics, exactly as this was later done for Classical BI in [LarcheyWendling 2010]. But then, the intuition behind the encoding is arguably much more difficult to grasp. We also feel that the existence of the elementary fragment of ILL is important in itself, and in pa ...
... directly into BBI and Kripke semantics, exactly as this was later done for Classical BI in [LarcheyWendling 2010]. But then, the intuition behind the encoding is arguably much more difficult to grasp. We also feel that the existence of the elementary fragment of ILL is important in itself, and in pa ...
Introduction to Linear Logic
... The main concern of this report is to give an introduction to Linear Logic. For pedagogical purposes we shall also have a look at Classical Logic as well as Intuitionistic Logic. Linear Logic was introduced by J.-Y. Girard in 1987 and it has attracted much attention from computer scientists, as it i ...
... The main concern of this report is to give an introduction to Linear Logic. For pedagogical purposes we shall also have a look at Classical Logic as well as Intuitionistic Logic. Linear Logic was introduced by J.-Y. Girard in 1987 and it has attracted much attention from computer scientists, as it i ...
The Computer Modelling of Mathematical Reasoning Alan Bundy
... This book started as notes for a postgraduate course in Mathematical Reasoning given in the Department of Artificial Intelligence at Edinburgh from 1979 onwards. Students on the course are drawn from a wide range of backgrounds: Psychology, Computer Science, Mathematics, Education, etc. The first dr ...
... This book started as notes for a postgraduate course in Mathematical Reasoning given in the Department of Artificial Intelligence at Edinburgh from 1979 onwards. Students on the course are drawn from a wide range of backgrounds: Psychology, Computer Science, Mathematics, Education, etc. The first dr ...
Everything Else Being Equal: A Modal Logic for Ceteris Paribus
... preference based on the “all other things being equal” reading of ceteris paribus. We discuss the expressive power of the new language, and show how it is a natural extension of the basic preference logic. The complete axiomatization supports this point by building on the axiomatization of the basic ...
... preference based on the “all other things being equal” reading of ceteris paribus. We discuss the expressive power of the new language, and show how it is a natural extension of the basic preference logic. The complete axiomatization supports this point by building on the axiomatization of the basic ...