Set theory and logic
... to give self-contained characterizations in turn of the system of integers, of rational numbers, and, finally, of real numbers. This is clone in the last three sections of the chapter. Finally, there is Chapter 9, which is an introductory account of relatively recent investigations of the foundation ...
... to give self-contained characterizations in turn of the system of integers, of rational numbers, and, finally, of real numbers. This is clone in the last three sections of the chapter. Finally, there is Chapter 9, which is an introductory account of relatively recent investigations of the foundation ...
Measure Quantifier in Monadic Second Order Logic
... hierarchy. However, since the family of MSO + ∀=1 definable relations is closed under Boolean operations and projections, it is not clear if every MSO + ∀=1 definable relation is Lebesgue measurable. We formulate this as Problem 2 is Section 8. In the rest of the paper we assume sufficiently strong ...
... hierarchy. However, since the family of MSO + ∀=1 definable relations is closed under Boolean operations and projections, it is not clear if every MSO + ∀=1 definable relation is Lebesgue measurable. We formulate this as Problem 2 is Section 8. In the rest of the paper we assume sufficiently strong ...
Chapter 9: Initial Theorems about Axiom System AS1
... number has a given property Ã, one proves that 0 has Ã, and one proves that if a number n has Ã, then so does its successor n+. This is known as weak induction. Recall that there is also the method of strong induction. According to this method, if one wants to prove that every number has Ã, one prov ...
... number has a given property Ã, one proves that 0 has Ã, and one proves that if a number n has Ã, then so does its successor n+. This is known as weak induction. Recall that there is also the method of strong induction. According to this method, if one wants to prove that every number has Ã, one prov ...
A Logical Expression of Reasoning
... It is the way of preference to prejudice, fanaticism, dogmatism, patriotism and other isms of the like. It is not solved by the mere change of premises, which may just switch from one ism to another. The rather radical, though appropriate, solution is to take into consideration all looking reasonabl ...
... It is the way of preference to prejudice, fanaticism, dogmatism, patriotism and other isms of the like. It is not solved by the mere change of premises, which may just switch from one ism to another. The rather radical, though appropriate, solution is to take into consideration all looking reasonabl ...
Incompleteness in the finite domain
... some sentences as conjectures. First, we believe that some basic theorems of proof theory should also hold true with suitable bounds on the lengths of proofs. The prime example is the Second Incompleteness Theorem discussed above. Second, some results in proof complexity and bounded arithmetic seem ...
... some sentences as conjectures. First, we believe that some basic theorems of proof theory should also hold true with suitable bounds on the lengths of proofs. The prime example is the Second Incompleteness Theorem discussed above. Second, some results in proof complexity and bounded arithmetic seem ...
Mathematical Logic
... Remark 1.1.15. In our study of formulas, we shall be indifferent to the question of which system of notation is actually used. The only point of interest for us is that each non-atomic formula is uniquely of the form ¬ A or AbB, where A and B are formulas and b is a binary connective. ...
... Remark 1.1.15. In our study of formulas, we shall be indifferent to the question of which system of notation is actually used. The only point of interest for us is that each non-atomic formula is uniquely of the form ¬ A or AbB, where A and B are formulas and b is a binary connective. ...
Sketch-as-proof - Norbert Preining
... This thesis results from the wish to connect two interesting parts of mathematics, proof theory and projective geometry. We applied theoretic methods of proof theory to projective geometry. This should be another try to span the gap between theoretical and applied mathematics. The gap arises from th ...
... This thesis results from the wish to connect two interesting parts of mathematics, proof theory and projective geometry. We applied theoretic methods of proof theory to projective geometry. This should be another try to span the gap between theoretical and applied mathematics. The gap arises from th ...
Higher Order Logic - Indiana University
... Higher order logics, long considered by many to be an esoteric subject, 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 comp ...
... Higher order logics, long considered by many to be an esoteric subject, 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 comp ...
Higher Order Logic - Theory and Logic Group
... Higher order logics, long considered by many to be an esoteric subject, 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 comp ...
... Higher order logics, long considered by many to be an esoteric subject, 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 comp ...
Logical Theories and Compatible Operations
... second-order theory). The second approach is also useful for classes of nite structures as not every such class has a decidable theory. In order to process structures by algorithmic means, a nite encoding of the structure is required. Such encodings are trivial when structures are nite (though on ...
... second-order theory). The second approach is also useful for classes of nite structures as not every such class has a decidable theory. In order to process structures by algorithmic means, a nite encoding of the structure is required. Such encodings are trivial when structures are nite (though on ...
The Arithmetical Hierarchy Math 503
... 2. A set A is ψ-recursively enumerable if and only if it is the projection of a ψ-recursive relation R, i.e. if and only if A = {x : ∃yR(x, y)}. Note. There is no loss of generality above in considering R a binary relation. In recursion theory, it is common to see sets of tuples as sets of encodings ...
... 2. A set A is ψ-recursively enumerable if and only if it is the projection of a ψ-recursive relation R, i.e. if and only if A = {x : ∃yR(x, y)}. Note. There is no loss of generality above in considering R a binary relation. In recursion theory, it is common to see sets of tuples as sets of encodings ...
Structural Multi-type Sequent Calculus for Inquisitive Logic
... An easy inductive proof shows that InqL-formulas have the downward closure property and the empty team property: (Downward Closure Property) If S |= φ and S ′ ⊆ S , then S ′ |= φ. (Empty Team Property) ∅ |= φ. CPL extended with the dependence atoms =(p1 , . . . , pn , q) is called propositional depe ...
... An easy inductive proof shows that InqL-formulas have the downward closure property and the empty team property: (Downward Closure Property) If S |= φ and S ′ ⊆ S , then S ′ |= φ. (Empty Team Property) ∅ |= φ. CPL extended with the dependence atoms =(p1 , . . . , pn , q) is called propositional depe ...
A New Theory of Content
... Consider the generic classical quantificational language L'. The vocabulary of L' is limited to an infinite stock of individual constants, 'a', 'a1', 'a2', ... ; an infinite stock of individual variables 'x', 'x1', 'x2, ... ; an infinite stock of predicate letters of varying degrees (one-placed, two ...
... Consider the generic classical quantificational language L'. The vocabulary of L' is limited to an infinite stock of individual constants, 'a', 'a1', 'a2', ... ; an infinite stock of individual variables 'x', 'x1', 'x2, ... ; an infinite stock of predicate letters of varying degrees (one-placed, two ...
Propositional logic - Cheriton School of Computer Science
... role. But this is not commonly done. ...
... role. But this is not commonly done. ...
Interactive Theorem Proving with Temporal Logic
... reasoning about time is important for ensuring correctness. These logics are mainly used to formalize and express properties about future or possible behaviors in such systems. For example, linear temporal logics have been successfully used to express and prove properties of concurrent and reactive ...
... reasoning about time is important for ensuring correctness. These logics are mainly used to formalize and express properties about future or possible behaviors in such systems. For example, linear temporal logics have been successfully used to express and prove properties of concurrent and reactive ...
Paper - Department of Computer Science and Information Systems
... General Terms: theory. Additional Key Words and Phrases: unification, admissible rule, description logic, hybrid logic, decidability. ...
... General Terms: theory. Additional Key Words and Phrases: unification, admissible rule, description logic, hybrid logic, decidability. ...
Here - Dorodnicyn Computing Centre of the Russian Academy of
... twice and is, thus, a disposable meta-mathematical theorem. It's already something like ... not a meta- , but a para-"mathematics". Taking into account that the set N of finite natural numbers is countable by definition, we deduce from the Corollary 1 the following quite unexpected consequence. CORO ...
... twice and is, thus, a disposable meta-mathematical theorem. It's already something like ... not a meta- , but a para-"mathematics". Taking into account that the set N of finite natural numbers is countable by definition, we deduce from the Corollary 1 the following quite unexpected consequence. CORO ...
pdf
... M = R2 and let M be the structure with universe M that has a binary relation symbol Rp for every 4-type over ∅ of R and where Rp is interpreted as {((a, b), (a0 , b0 )) : R |= p(a, b, a0 , b0 )}. Then M satis es the hypotheses of Theorem 1.2. (The same holds if R is any binary random structure and w ...
... M = R2 and let M be the structure with universe M that has a binary relation symbol Rp for every 4-type over ∅ of R and where Rp is interpreted as {((a, b), (a0 , b0 )) : R |= p(a, b, a0 , b0 )}. Then M satis es the hypotheses of Theorem 1.2. (The same holds if R is any binary random structure and w ...
Intuitionistic completeness part I
... uniform validity, an intuitively smaller collection of formulas than those constructively valid. Nevertheless, uniform validity is extremely useful in practice when thinking about purely logical formulas precisely because it corresponds exactly to proof and yet is an entirely semantic notion based o ...
... uniform validity, an intuitively smaller collection of formulas than those constructively valid. Nevertheless, uniform validity is extremely useful in practice when thinking about purely logical formulas precisely because it corresponds exactly to proof and yet is an entirely semantic notion based o ...
Incompleteness in the finite domain
... and bounded arithmetic seem to follow a general pattern. For example, as we noted above, polynomial time computations are associated with the theory S21 by a witnessing theorem. If we take S22 , which we believe is a stronger theory, then the corresponding function class is PNP ,2 which we believe i ...
... and bounded arithmetic seem to follow a general pattern. For example, as we noted above, polynomial time computations are associated with the theory S21 by a witnessing theorem. If we take S22 , which we believe is a stronger theory, then the corresponding function class is PNP ,2 which we believe i ...
Elements of Finite Model Theory
... although in recent years connections with other areas, such as formal methods and verification, and artificial intelligence, have been discovered. The birth of finite model theory is often identified with Trakhtenbrot’s result from 1950 stating that validity over finite models is not recursively enumerab ...
... although in recent years connections with other areas, such as formal methods and verification, and artificial intelligence, have been discovered. The birth of finite model theory is often identified with Trakhtenbrot’s result from 1950 stating that validity over finite models is not recursively enumerab ...
propositional logic extended with a pedagogically useful relevant
... Having described the paradoxes, textbooks and logic teachers sometimes try to reason them away. Two types of moves are invoked in this connection. The first move is legitimate but insufficient: one shows that it is correct that the paradoxes are apparent only. Indeed, there is a discrepancy between ...
... Having described the paradoxes, textbooks and logic teachers sometimes try to reason them away. Two types of moves are invoked in this connection. The first move is legitimate but insufficient: one shows that it is correct that the paradoxes are apparent only. Indeed, there is a discrepancy between ...
Proof, Sets, and Logic - Department of Mathematics
... 4/5/2013: Considering subversive language about second-order logic. Where to put it? I added a couple of sections with musings about second order logic. They are probably not in the right places, but they might be modified to fit where they are or moved to better locations. November 30, 2012: readin ...
... 4/5/2013: Considering subversive language about second-order logic. Where to put it? I added a couple of sections with musings about second order logic. They are probably not in the right places, but they might be modified to fit where they are or moved to better locations. November 30, 2012: readin ...
A Tableau Calculus for Minimal Modal Model Generation
... using hyperresolution to generate Herbrand models for modal problems [9,4] and [3,13]. In this paper we focus on the generation of minimal Herbrand models. Though minimal Herbrand models are not domain minimal, in certain applications they tend to be more natural than domain minimal models. For exam ...
... using hyperresolution to generate Herbrand models for modal problems [9,4] and [3,13]. In this paper we focus on the generation of minimal Herbrand models. Though minimal Herbrand models are not domain minimal, in certain applications they tend to be more natural than domain minimal models. For exam ...
brouwer`s intuitionism as a self-interpreted mathematical theory
... In that way, the truth of the fundamental intuitions or concepts is preserved by the defined concepts and proofs. A siT is trivially categorical, since there is only one interpretation associated to it, consistent, as long as the fundamental mental interpretations are non-contradictory, and complete ...
... In that way, the truth of the fundamental intuitions or concepts is preserved by the defined concepts and proofs. A siT is trivially categorical, since there is only one interpretation associated to it, consistent, as long as the fundamental mental interpretations are non-contradictory, and complete ...