Lecture 10: A Digression on Absoluteness
... Moreover, it is the case that ϕ is not absolute for transitive universes, demonstrating (by Lemma 7.12) that it is not possible to find any ∆0 formula expressing the same property. We will spend the rest of the lecture exploring why. Definition 8.1. B is an elementary substructure (or elementary sub ...
... Moreover, it is the case that ϕ is not absolute for transitive universes, demonstrating (by Lemma 7.12) that it is not possible to find any ∆0 formula expressing the same property. We will spend the rest of the lecture exploring why. Definition 8.1. B is an elementary substructure (or elementary sub ...
CHAPTER 1 INTRODUCTION 1 Mathematical Paradoxes
... reality- or on the properties of these notions expressed in systems of axioms. The historical development of mathematics has shown that it is not sufficient to base theories on an intuitive understanding of their notions only. This fact became especially obvious in set theory. The basic concept of t ...
... reality- or on the properties of these notions expressed in systems of axioms. The historical development of mathematics has shown that it is not sufficient to base theories on an intuitive understanding of their notions only. This fact became especially obvious in set theory. The basic concept of t ...
Knowledge representation 1
... Logic is used by computer scientists when they are engaged in Formal Methods: describing the performance of a program precisely, so that they can prove that it does (or doesn’t) perform the task that it is supposed to. In other words, establishing the validity of a program. ...
... Logic is used by computer scientists when they are engaged in Formal Methods: describing the performance of a program precisely, so that they can prove that it does (or doesn’t) perform the task that it is supposed to. In other words, establishing the validity of a program. ...
x, y, x
... If an interpretation makes ∀xA(x) false, then we get F → F which is true. If an interpretation with structure S and valuation σ makes ∀xA(x) true? then certainly A(1) is also true. • ∃xA(x) → A(1) ...
... If an interpretation makes ∀xA(x) false, then we get F → F which is true. If an interpretation with structure S and valuation σ makes ∀xA(x) true? then certainly A(1) is also true. • ∃xA(x) → A(1) ...
Predicate Logic
... rhs is false This suggests that formulas in PNF are very sensitive to the order of the quantifiers. Q: Is the following formula valid? 8x9y(M (x) ^ F (y)) $ 9y8x(M (x) ^ F (y)) ...
... rhs is false This suggests that formulas in PNF are very sensitive to the order of the quantifiers. Q: Is the following formula valid? 8x9y(M (x) ^ F (y)) $ 9y8x(M (x) ^ F (y)) ...
Many-Valued Models
... This new way of looking to logico-philosophical scenario was not free of discussion, however. Stanisław Lesniewski argued that a third logical value never appears in scientific argumentation, and considered the third value as no sense, because “no one had been able until now to give to the symbol 2 ...
... This new way of looking to logico-philosophical scenario was not free of discussion, however. Stanisław Lesniewski argued that a third logical value never appears in scientific argumentation, and considered the third value as no sense, because “no one had been able until now to give to the symbol 2 ...
Document
... logically from the truth of the premises. Logic is the foundation for expressing formal ...
... logically from the truth of the premises. Logic is the foundation for expressing formal ...
Quantified Equilibrium Logic and the First Order Logic of Here
... introduced in [25, 26], and its monotonic base logic, here-and-there. We present a slightly modified version of QEL where the so-called unique name assumption or UNA is not assumed from the outset but may be added as a special requirement for specific applications. We also consider here an alternati ...
... introduced in [25, 26], and its monotonic base logic, here-and-there. We present a slightly modified version of QEL where the so-called unique name assumption or UNA is not assumed from the outset but may be added as a special requirement for specific applications. We also consider here an alternati ...
The semantics of predicate logic
... Readings: Section 2.4, 2.5, 2.6. In this module, we will precisely define the semantic interpretation of formulas in our predicate logic. In propositional logic, every formula had a fixed, finite number of models (interpretations); this is not the case in predicate logic. As a consequence, we must t ...
... Readings: Section 2.4, 2.5, 2.6. In this module, we will precisely define the semantic interpretation of formulas in our predicate logic. In propositional logic, every formula had a fixed, finite number of models (interpretations); this is not the case in predicate logic. As a consequence, we must t ...
Homework 3
... Problem 4: Consider a set K of numbers having all eight factors of 30, that is, {1, 2, 3, 5, 6, 10, 15, 30}. Let two binary operations be LCM (least common multiple) denoted as + and GCF (greatest common factor) denoted as · . Show that: (a) The identity element for + (LCM) is 1 and that for · (GCF) ...
... Problem 4: Consider a set K of numbers having all eight factors of 30, that is, {1, 2, 3, 5, 6, 10, 15, 30}. Let two binary operations be LCM (least common multiple) denoted as + and GCF (greatest common factor) denoted as · . Show that: (a) The identity element for + (LCM) is 1 and that for · (GCF) ...
Decidable fragments of first-order logic Decidable fragments of first
... For r = 1, 2, let Tr be the set of all r -tables that are realized in A. Both sets are finite because Lϕ is finite. b n with universe Bn = {1, . . . , n} is The random Lϕ -structure B obtained as follows. (i) To each b in Bn , assign a 1-table Tb that is chosen uniformly at random from T1 . (ii) To ...
... For r = 1, 2, let Tr be the set of all r -tables that are realized in A. Both sets are finite because Lϕ is finite. b n with universe Bn = {1, . . . , n} is The random Lϕ -structure B obtained as follows. (i) To each b in Bn , assign a 1-table Tb that is chosen uniformly at random from T1 . (ii) To ...
Least and greatest fixed points in linear logic
... Focusing is not restricted to linear logic. It has been extended to intuitionistic and classical logics. There are two approaches for doing so: either start from scratch, or use an encoding. ⊢ [F] o ...
... Focusing is not restricted to linear logic. It has been extended to intuitionistic and classical logics. There are two approaches for doing so: either start from scratch, or use an encoding. ⊢ [F] o ...
Propositional Dynamic Logic of Regular Programs*+
... or the basic programs. In this section, we show that the complexity of the validity problem for PDL is in co-NTIME(cn) f or some c, where 71is the size of the formula being tested. (That is, the complement of the validity problem for PDL is recognizable by a nondeterministic Turing machine in time < ...
... or the basic programs. In this section, we show that the complexity of the validity problem for PDL is in co-NTIME(cn) f or some c, where 71is the size of the formula being tested. (That is, the complement of the validity problem for PDL is recognizable by a nondeterministic Turing machine in time < ...
pdf
... the Φ if it is clear from context or does not play a significant role.) As usual, we define ϕ∨ψ and ϕ ⇒ ψ as abbreviations of ¬(¬ϕ ∧ ¬ψ) and ¬ϕ ∨ ψ, respectively. The intended interpretation of Kϕ varies depending on the context. It typically has been interpreted as knowledge, as belief, or as neces ...
... the Φ if it is clear from context or does not play a significant role.) As usual, we define ϕ∨ψ and ϕ ⇒ ψ as abbreviations of ¬(¬ϕ ∧ ¬ψ) and ¬ϕ ∨ ψ, respectively. The intended interpretation of Kϕ varies depending on the context. It typically has been interpreted as knowledge, as belief, or as neces ...
Logic seminar
... • The symbols, such as P, Q, and R, that are used to denote propositions are called atomic formulas, or atoms. • From propositions, we can build compound propositions by using logical connectives. • Examples of compound propositions: – “Snow is white and the sky is clear” – “If John is not at home, ...
... • The symbols, such as P, Q, and R, that are used to denote propositions are called atomic formulas, or atoms. • From propositions, we can build compound propositions by using logical connectives. • Examples of compound propositions: – “Snow is white and the sky is clear” – “If John is not at home, ...
Discrete Structure
... Applications of Predicate Logic • It is the formal notation for writing perfectly clear, concise, and unambiguous mathematical definitions, axioms, and theorems (more on these in module 2) for any branch of mathematics. • Predicate logic with function symbols, the “=” operator, and a few proof-buil ...
... Applications of Predicate Logic • It is the formal notation for writing perfectly clear, concise, and unambiguous mathematical definitions, axioms, and theorems (more on these in module 2) for any branch of mathematics. • Predicate logic with function symbols, the “=” operator, and a few proof-buil ...
Lesson 3
... Complete elementary disjunction (CED) of a given set S of elementary propositional symbols is an elementary disjunction in which each symbol (element of S) occurs just once: Ex.: p q Disjunctive normal form (DNF) of a formula F is a formula F’ such that F’ is equivalent to F and F’ has the form o ...
... Complete elementary disjunction (CED) of a given set S of elementary propositional symbols is an elementary disjunction in which each symbol (element of S) occurs just once: Ex.: p q Disjunctive normal form (DNF) of a formula F is a formula F’ such that F’ is equivalent to F and F’ has the form o ...