MS-Word version
... to explore further the additive theory of infinite sets of prime numbers, both with absolute results and its links with number theory via the Schinzel's hypothesis ; to code natural numbers, complex numbers, and quadratic integers by automata accepting numbers (written in non-classical systems) ; to ...
... to explore further the additive theory of infinite sets of prime numbers, both with absolute results and its links with number theory via the Schinzel's hypothesis ; to code natural numbers, complex numbers, and quadratic integers by automata accepting numbers (written in non-classical systems) ; to ...
Beyond first order logic: From number of structures to structure of
... in a given cardinality and establishing invariants in order to classify the isomorphism types. Such invariants arise naturally in many concrete classes: the dimension of a vector space or the transcendence degree of an algebraically closed field are prototypical examples. A crucial innovation of mod ...
... in a given cardinality and establishing invariants in order to classify the isomorphism types. Such invariants arise naturally in many concrete classes: the dimension of a vector space or the transcendence degree of an algebraically closed field are prototypical examples. A crucial innovation of mod ...
1
... 1. (a) Identify the free and bound variable occurrences in the following logical formulas: • ∀x∃y(Rxz ∧ ∃zQyxz), • ∀x((∃yRxy→Ax)→Bxy), • ∀x(Ax→∃yBy) ∧ ∃z(Cxz→∃xDxyz). (b) Give the definition of a atomic formula of predicate logic and of a valuation of terms s based on a variable assignment s. (c) Pr ...
... 1. (a) Identify the free and bound variable occurrences in the following logical formulas: • ∀x∃y(Rxz ∧ ∃zQyxz), • ∀x((∃yRxy→Ax)→Bxy), • ∀x(Ax→∃yBy) ∧ ∃z(Cxz→∃xDxyz). (b) Give the definition of a atomic formula of predicate logic and of a valuation of terms s based on a variable assignment s. (c) Pr ...
Proof Theory - Andrew.cmu.edu
... Given the history of Hilbert’s program, it should not be surprising that proof theorists have also had a strong interest in formal representations of constructive and intuitionistic reasoning. From an intuitionistic standpoint, the use of the excluded middle ϕ ∨ ¬ϕ is not acceptable, since, generall ...
... Given the history of Hilbert’s program, it should not be surprising that proof theorists have also had a strong interest in formal representations of constructive and intuitionistic reasoning. From an intuitionistic standpoint, the use of the excluded middle ϕ ∨ ¬ϕ is not acceptable, since, generall ...
Part 1 - Logic Summer School
... Because the set of non-halting Turing machines is not recursively enumerable, {ϕ|¬ϕ has no finite models} is also not recursively ...
... Because the set of non-halting Turing machines is not recursively enumerable, {ϕ|¬ϕ has no finite models} is also not recursively ...
We showed on Tuesday that Every relation in the arithmetical
... A (first-order) proof system is a set of rules which allows certain formulas to be derived from other formulas. Proposition The usual proof system (for arithmetic) is computable. For those who worry about the deductive power of the “usual proof system”: Gödel’s Completeness Theorem The usual proof ...
... A (first-order) proof system is a set of rules which allows certain formulas to be derived from other formulas. Proposition The usual proof system (for arithmetic) is computable. For those who worry about the deductive power of the “usual proof system”: Gödel’s Completeness Theorem The usual proof ...
THE MODAL LOGIC OF INNER MODELS §1. Introduction. In [10, 11
... The modal theory S4.2 is the smallest class of formulas containing all substitution instances of the above axioms and closed under modus ponens and necessitation (in other words, the smallest normal modal logic containing the above axioms). As usual, a preorder is a set P with a reflexive and transi ...
... The modal theory S4.2 is the smallest class of formulas containing all substitution instances of the above axioms and closed under modus ponens and necessitation (in other words, the smallest normal modal logic containing the above axioms). As usual, a preorder is a set P with a reflexive and transi ...
Modal Logic and Model Theory
... obtained by Abstract. We propose a first order modal logic, the QS?E-logic, first order modal logic QS4 a rigidity axiom sch?mas :A ->O A, adding to the well-known entails the possibility where A denotes a basic formula. In this logic, the possibility of extending a given classical first order model ...
... obtained by Abstract. We propose a first order modal logic, the QS?E-logic, first order modal logic QS4 a rigidity axiom sch?mas :A ->O A, adding to the well-known entails the possibility where A denotes a basic formula. In this logic, the possibility of extending a given classical first order model ...
comments on the logic of constructible falsity (strong negation)
... logic I describe by adding the rule D). D, however, is just as unacceptable from Nelson’s point of view as it is from that of the intuitionists. Indeed, given the constructive derivability of excluded middle for atomic (and other decidable) formulas of arithmetic, the addition of D to either intuiti ...
... logic I describe by adding the rule D). D, however, is just as unacceptable from Nelson’s point of view as it is from that of the intuitionists. Indeed, given the constructive derivability of excluded middle for atomic (and other decidable) formulas of arithmetic, the addition of D to either intuiti ...
A preprint version is available here in pdf.
... M in some sense. An important example is the standard system X = SSy(M ) of the ground model. Another consists of the sets represented in its theory Th(M ). These constructions generalise to the expanded structure (M, ω) and we can look at the set X of sets coded or represented in (M, ω). This will ...
... M in some sense. An important example is the standard system X = SSy(M ) of the ground model. Another consists of the sets represented in its theory Th(M ). These constructions generalise to the expanded structure (M, ω) and we can look at the set X of sets coded or represented in (M, ω). This will ...
Intro to First
... be explicit or implicit. If I say “everything is greater than or equal to 0,” I may be referring to the set natural numbers implicitly. If I say “every natural number is greater than or equal to 0,” I am explicitly using the word “every” to range over natural numbers. To handle cases where the expli ...
... be explicit or implicit. If I say “everything is greater than or equal to 0,” I may be referring to the set natural numbers implicitly. If I say “every natural number is greater than or equal to 0,” I am explicitly using the word “every” to range over natural numbers. To handle cases where the expli ...
A simple proof of Parsons` theorem
... that a quantifier-free first-order consequence of a universal theory is a quasitautological consequence8 of a finite number of substitution instances of its axioms. When applied to the theory PRA, this additional feature explains why PRA is conservative over quantifier-free Skolem arithmetic, as obs ...
... that a quantifier-free first-order consequence of a universal theory is a quasitautological consequence8 of a finite number of substitution instances of its axioms. When applied to the theory PRA, this additional feature explains why PRA is conservative over quantifier-free Skolem arithmetic, as obs ...
Automated Discovery in Pure Mathematics
... Prove the conjectures (theorem proving) Disprove the conjectures (model generation) Assess all concepts w.r.t. new concept ...
... Prove the conjectures (theorem proving) Disprove the conjectures (model generation) Assess all concepts w.r.t. new concept ...
Notes Predicate Logic II
... The theorem states that every valid sequent can be proven, and every sequent that can be proven is valid. This theorem was proven by Kurt Gödel in 1929 in his doctoral dissertation. A description of his proof, as well as the proofs of the following theorems, is beyond the scope of this chapter. The ...
... The theorem states that every valid sequent can be proven, and every sequent that can be proven is valid. This theorem was proven by Kurt Gödel in 1929 in his doctoral dissertation. A description of his proof, as well as the proofs of the following theorems, is beyond the scope of this chapter. The ...
Canad. Math. Bull. Vol. 24 (2), 1981 INDEPENDENT SETS OF
... There are three possibilities for M: (i) M
... There are three possibilities for M: (i) M
Domino Theory. Domino theory refers to a
... 2. We tip the first domino: We prove the first theorem. This is called the trivial step. At this point we are finished, and consider all of the theorems proved. (Just like we believe all of the dominos fall down). ...
... 2. We tip the first domino: We prove the first theorem. This is called the trivial step. At this point we are finished, and consider all of the theorems proved. (Just like we believe all of the dominos fall down). ...
Idiosynchromatic Poetry
... Ramsey theory is generally concerned with problems of finding structures with some kind of homogeneity in superstructures. Often a structure contains an homogeneous substructure of a certain sort if it is itself large enough. In some contexts the notion of size can not only be interpreted as cardina ...
... Ramsey theory is generally concerned with problems of finding structures with some kind of homogeneity in superstructures. Often a structure contains an homogeneous substructure of a certain sort if it is itself large enough. In some contexts the notion of size can not only be interpreted as cardina ...
Finite Model Theory
... Remark. T is universal-existential. Without constant symbols T could not be made universal or existential. With constant symbols, we could let T be ...
... Remark. T is universal-existential. Without constant symbols T could not be made universal or existential. With constant symbols, we could let T be ...
PARADOX AND INTUITION
... which is independent of the specification of domain, and the juxtaposition of symbols cannot force the interpretation of any of its predicate-letters as a relation with a nondenumerable field. Some connections between the Löwenheim-Skolem theorem and problems of ontological reduction are discussed i ...
... which is independent of the specification of domain, and the juxtaposition of symbols cannot force the interpretation of any of its predicate-letters as a relation with a nondenumerable field. Some connections between the Löwenheim-Skolem theorem and problems of ontological reduction are discussed i ...
Exam-Computational_Logic-Subjects_2016
... I Propositional logic 1. Using a proof method: a) semantic method (truth table, semantic tableau, conjunctive normal form) b) syntactic method (resolution, definition of deduction, the theorem of deduction and its reverse) c) direct method (truth table, conjunctive normal form, definition of deducti ...
... I Propositional logic 1. Using a proof method: a) semantic method (truth table, semantic tableau, conjunctive normal form) b) syntactic method (resolution, definition of deduction, the theorem of deduction and its reverse) c) direct method (truth table, conjunctive normal form, definition of deducti ...
PDF
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
Lecture 39 Notes
... (asserted programs) justified by varieties of programming logics based on Hoare logic and programs that are implicit constructive proofs of assertions of the form ∀x : T1 .∃y : T2 .R(x, y). This is the ∀ ∃ pattern. There are connections between proofs and programs for all forms of assertion, e.g. ∀x ...
... (asserted programs) justified by varieties of programming logics based on Hoare logic and programs that are implicit constructive proofs of assertions of the form ∀x : T1 .∃y : T2 .R(x, y). This is the ∀ ∃ pattern. There are connections between proofs and programs for all forms of assertion, e.g. ∀x ...
Ch1 - COW :: Ceng
... circuit design There are efficient algorithms for reasoning in propositional logic Propositional logic is a foundation for most of the more expressive logics ...
... circuit design There are efficient algorithms for reasoning in propositional logic Propositional logic is a foundation for most of the more expressive logics ...
Infinitistic Rules of Proof and Their Semantics
... which implies zpc- f-.e where .e is the formula which arises from , by relativization of its set quantifiers to the formula £ (X). By conservativeness A 2 1-
... which implies zpc- f-
for every
Comparing Constructive Arithmetical Theories Based - Math
... (term) a, and also consider the formula ∀z 6 a(x + z = |a| → ∀y 6 t¬A(z, y)) as B(x). To prove P V + ¬¬N P − LIN D `i P V + coN P − LIN D, make similar changes. (iii) This is an immediate consequence of Proposition 2.2 and part (ii). Recall that the theory CP V is the classical closure of IP V an ...
... (term) a, and also consider the formula ∀z 6 a(x + z = |a| → ∀y 6 t¬A(z, y)) as B(x). To prove P V + ¬¬N P − LIN D `i P V + coN P − LIN D, make similar changes. (iii) This is an immediate consequence of Proposition 2.2 and part (ii). Recall that the theory CP V is the classical closure of IP V an ...