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Hybrid Interactive Theorem Proving using Nuprl and HOL?
... and then proves the formulas in the axioms/de nitions sections. When this is done, the theorems can then all be accepted immediately as Nuprl theorems. Type-checking is undecidable in Nuprl, so the well-typedness of terms must be proven explicitly. This means that in addition to the axioms/de nitio ...
... and then proves the formulas in the axioms/de nitions sections. When this is done, the theorems can then all be accepted immediately as Nuprl theorems. Type-checking is undecidable in Nuprl, so the well-typedness of terms must be proven explicitly. This means that in addition to the axioms/de nitio ...
Automated Deduction
... It is worthwhile to start these course notes by giving a brief outline of the history of concepts which are fundamental to the area of automated deduction. Even though this outline is quite superficial it nevertheless serves to illustrate how old many of the ideas and concepts at the base of automat ...
... It is worthwhile to start these course notes by giving a brief outline of the history of concepts which are fundamental to the area of automated deduction. Even though this outline is quite superficial it nevertheless serves to illustrate how old many of the ideas and concepts at the base of automat ...
Introduction to Logic
... to arrive at new correct arguments. The other two aspects are very intimately connected with this one. 2. In order to construct valid forms of arguments one has to know what such forms can be built from, that is, determine the ultimate “building blocks”. One has to identify the basic terms, their ki ...
... to arrive at new correct arguments. The other two aspects are very intimately connected with this one. 2. In order to construct valid forms of arguments one has to know what such forms can be built from, that is, determine the ultimate “building blocks”. One has to identify the basic terms, their ki ...
Introduction to Logic
... If Parmenides was not aware of general rules underlying his arguments, the same perhaps is not true for his disciple Zeno of Elea (5th century BC). Parmenides taught that there is no real change in the world and that all thing remain, eventually, the same one being. In the defense of this heavily cr ...
... If Parmenides was not aware of general rules underlying his arguments, the same perhaps is not true for his disciple Zeno of Elea (5th century BC). Parmenides taught that there is no real change in the world and that all thing remain, eventually, the same one being. In the defense of this heavily cr ...
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... the rewrite system for the Hydra battle (Moser, 2009; Fleischer, 2007), since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s further remarks, that he considers the case where one chooses at each elim ...
... the rewrite system for the Hydra battle (Moser, 2009; Fleischer, 2007), since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s further remarks, that he considers the case where one chooses at each elim ...
Automata, Languages, and Programming
... inductively as A0 = RM (1) and An+1 = An · A. The image of the interpretation RM together with the operations →, ·, ∗ , ∅, {1M } is the algebra of regular sets over M , denoted by Reg M . If M is the free monoid Σ ∗ , then RM is the standard language interpretation of regular expressions. It is know ...
... inductively as A0 = RM (1) and An+1 = An · A. The image of the interpretation RM together with the operations →, ·, ∗ , ∅, {1M } is the algebra of regular sets over M , denoted by Reg M . If M is the free monoid Σ ∗ , then RM is the standard language interpretation of regular expressions. It is know ...
Axiomatic Method Logical Cycle Starting Place Fe
... • An axiom set is said to have absolute consistency if there exists a real world model satisfying all of the axioms. • An axiom set is said to be relatively consistent if we can produce a model for the axiom set based upon another axiom set which we are willing to assume is consistent. ...
... • An axiom set is said to have absolute consistency if there exists a real world model satisfying all of the axioms. • An axiom set is said to be relatively consistent if we can produce a model for the axiom set based upon another axiom set which we are willing to assume is consistent. ...
The Perfect Set Theorem and Definable Wellorderings of the
... THEOREM. Let r be a reasonablepointclass and let M be a perfect set basis for r. If < is a wellorderingof a set of reals and < e r, then the field of < (i.e. the set {a: a < a}) is containedin M. PROOF. Without loss of generality we can assume that the field of < is contained in 20 so that we can wo ...
... THEOREM. Let r be a reasonablepointclass and let M be a perfect set basis for r. If < is a wellorderingof a set of reals and < e r, then the field of < (i.e. the set {a: a < a}) is containedin M. PROOF. Without loss of generality we can assume that the field of < is contained in 20 so that we can wo ...
From Syllogism to Common Sense Normal Modal Logic
... ‣ These systems are however mutually incompatible, and no base logic was given of which the other logics are extensions of. ‣ The modal logic K is such a base logic, named after SAUL KRIPKE, and which serves as a minimal logic for the class of all its (normal) extensions - defined next via a Hilbert ...
... ‣ These systems are however mutually incompatible, and no base logic was given of which the other logics are extensions of. ‣ The modal logic K is such a base logic, named after SAUL KRIPKE, and which serves as a minimal logic for the class of all its (normal) extensions - defined next via a Hilbert ...
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... e.g. for the rewrite system for the Hydra battle [Mos09, Fle07], since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s further remarks, that he considers the case where one chooses at each elimination ...
... e.g. for the rewrite system for the Hydra battle [Mos09, Fle07], since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s further remarks, that he considers the case where one chooses at each elimination ...
Chapter 4. Logical Notions This chapter introduces various logical
... is no point in representing more detail than is necessary to do so. If an argument is not logically valid, however, we should represent as much of the logical form as we can. ...
... is no point in representing more detail than is necessary to do so. If an argument is not logically valid, however, we should represent as much of the logical form as we can. ...
Properties of binary transitive closure logics over trees
... without recourse to an extended signature. The probably best known logic with this property is Σ11 , the extension of first-order logic by arbitrary relation variables that are existentially quantified. It is obviously possible to define order in Σ11 , because we can say there is a binary relation t ...
... without recourse to an extended signature. The probably best known logic with this property is Σ11 , the extension of first-order logic by arbitrary relation variables that are existentially quantified. It is obviously possible to define order in Σ11 , because we can say there is a binary relation t ...
Predicate Logic - Teaching-WIKI
... be quantified, but not predicates; second-order logic extends firstorder logic by adding the latter type of quantification • Other higher-order logics allow quantification over even higher types than second-order logic permits – These higher types include relations between relations, functions from ...
... be quantified, but not predicates; second-order logic extends firstorder logic by adding the latter type of quantification • Other higher-order logics allow quantification over even higher types than second-order logic permits – These higher types include relations between relations, functions from ...
MoL-2013-07 - Institute for Logic, Language and Computation
... by Abraham in his PhD thesis. In particular, in [Abr83], he had given a method for collapsing the second uncountable cardinal in any model of set theory without collapsing any other cardinals. We give an exposition of his intricate method. The organisation of the thesis is as follows: in Chapter 2 w ...
... by Abraham in his PhD thesis. In particular, in [Abr83], he had given a method for collapsing the second uncountable cardinal in any model of set theory without collapsing any other cardinals. We give an exposition of his intricate method. The organisation of the thesis is as follows: in Chapter 2 w ...
Bounded Proofs and Step Frames - Università degli Studi di Milano
... be interpreted in the so-called one-step frames. A one-step frame is a quadruple S = (W1 , W0 , f, R), where W0 , W1 are sets, f : W1 → W0 is a map and R ⊆ W1 × W0 is a relation between W1 and W0 . In the applications, we need two further requirements (called conservativity requirements) on such a o ...
... be interpreted in the so-called one-step frames. A one-step frame is a quadruple S = (W1 , W0 , f, R), where W0 , W1 are sets, f : W1 → W0 is a map and R ⊆ W1 × W0 is a relation between W1 and W0 . In the applications, we need two further requirements (called conservativity requirements) on such a o ...
1 LOGICAL CONSEQUENCE: A TURN IN STYLE KOSTA DO SEN
... reason, Gentzen’s sequents ├ A always have finite. (Gentzen was a pupil of David Hilbert and wanted to reason about sequents in a finitistic way.) Because of the compactness property, which syntactical consequence relations usually have, this is not such an essential limitation. But, in general, ...
... reason, Gentzen’s sequents ├ A always have finite. (Gentzen was a pupil of David Hilbert and wanted to reason about sequents in a finitistic way.) Because of the compactness property, which syntactical consequence relations usually have, this is not such an essential limitation. But, in general, ...
A Note on the Relation between Inflationary Fixpoints and Least
... formulas. It turns out that combining first-order logic with the ability to nest and complement fixpoint operators is powerful enough so that every formula of inflationary fixpoint logic is equivalent to a formula using least fixpoints of formulas positive in their fixpoint variable. This was first ...
... formulas. It turns out that combining first-order logic with the ability to nest and complement fixpoint operators is powerful enough so that every formula of inflationary fixpoint logic is equivalent to a formula using least fixpoints of formulas positive in their fixpoint variable. This was first ...
Label-free Modular Systems for Classical and Intuitionistic Modal
... logics but some coincide, such that there are only 15, which can be arranged in a cube as shown in Figure 2. This cube has the same shape in the classical as well as in the intuitionistic setting. However, the two papers [2] and [18] have one drawback: Although they provide cut-free systems for all ...
... logics but some coincide, such that there are only 15, which can be arranged in a cube as shown in Figure 2. This cube has the same shape in the classical as well as in the intuitionistic setting. However, the two papers [2] and [18] have one drawback: Although they provide cut-free systems for all ...
Label-free Modular Systems for Classical and Intuitionistic Modal
... logics but some coincide, such that there are only 15, which can be arranged in a cube as shown in Figure 2. This cube has the same shape in the classical as well as in the intuitionistic setting. However, the two papers [2] and [18] have one drawback: Although they provide cut-free systems for all ...
... logics but some coincide, such that there are only 15, which can be arranged in a cube as shown in Figure 2. This cube has the same shape in the classical as well as in the intuitionistic setting. However, the two papers [2] and [18] have one drawback: Although they provide cut-free systems for all ...
On Rosser sentences and proof predicates
... Löb’s theorem seem to tell the whole story of Pr . Indeed, the result on possible non-uniqueness of Rosser sentences is the first requiring more than these conditions, together with “the usual” ordering of proofs, for a settlement. It is also clear that “the usual” ordering and “the usual” proof pr ...
... Löb’s theorem seem to tell the whole story of Pr . Indeed, the result on possible non-uniqueness of Rosser sentences is the first requiring more than these conditions, together with “the usual” ordering of proofs, for a settlement. It is also clear that “the usual” ordering and “the usual” proof pr ...
Aspects of relation algebras
... Representations of boolean algebras The motivation for this definition comes from fields of sets. If U is a set, ℘U denotes the power set (set of all subsets) of U . Suppose that B ⊆ ℘U contains ∅, U and is closed under union, intersection, complement. For example, B = ℘U itself. Then hB, ∪, ∩, U \ ...
... Representations of boolean algebras The motivation for this definition comes from fields of sets. If U is a set, ℘U denotes the power set (set of all subsets) of U . Suppose that B ⊆ ℘U contains ∅, U and is closed under union, intersection, complement. For example, B = ℘U itself. Then hB, ∪, ∩, U \ ...
full text (.pdf)
... logic, foreshadowing Kripke’s [1963; 1965] formulation of similar state-based semantics for these logics (see [Artemov 2001]). Kripke models also form the basis of the standard semantics of DL (see [Harel et al. 2000]), although as mentioned, DL does not realize the intuitionistic nature of partial ...
... logic, foreshadowing Kripke’s [1963; 1965] formulation of similar state-based semantics for these logics (see [Artemov 2001]). Kripke models also form the basis of the standard semantics of DL (see [Harel et al. 2000]), although as mentioned, DL does not realize the intuitionistic nature of partial ...
axioms
... • Definition: An axiom set is said to have absolute consistency if there exists a real world model satisfying all of the axioms. • Example: The Fe-Fo Axiom Set exhibits absolute consistency because we produced a real world model for the system (i.e. actually two, the committee model and the graph mo ...
... • Definition: An axiom set is said to have absolute consistency if there exists a real world model satisfying all of the axioms. • Example: The Fe-Fo Axiom Set exhibits absolute consistency because we produced a real world model for the system (i.e. actually two, the committee model and the graph mo ...
Document
... Example: Prove that there are infinitely many prime numbers Proof: Assume there are not infinitely many prime numbers, therefore they are listable, i.e. p1,p2,…,pn Consider the number q = p1p2…pn+1. q is not divisible by any of the listed primes Therefore, q is a prime. However, it was not ...
... Example: Prove that there are infinitely many prime numbers Proof: Assume there are not infinitely many prime numbers, therefore they are listable, i.e. p1,p2,…,pn Consider the number q = p1p2…pn+1. q is not divisible by any of the listed primes Therefore, q is a prime. However, it was not ...
Sets
... Standard Symbols which denote sets of numbers N : The set of all natural numbers (i.e.,all positive integers) Z : The set of all integers Z+ : The set of all positive integers Z* : The set of all nonzero integers E : The set of all even integers Q : The set of all rational numbers Q* ...
... Standard Symbols which denote sets of numbers N : The set of all natural numbers (i.e.,all positive integers) Z : The set of all integers Z+ : The set of all positive integers Z* : The set of all nonzero integers E : The set of all even integers Q : The set of all rational numbers Q* ...