PREPOSITIONAL LOGIS
... • The 3rd sentence is entailed by the first two, but we need an explicit symbol, R, to represent an individual, Confucius, who is a member of the classes person and mortal ...
... • The 3rd sentence is entailed by the first two, but we need an explicit symbol, R, to represent an individual, Confucius, who is a member of the classes person and mortal ...
Sequentiality by Linear Implication and Universal Quantification
... / S + . ǫS is the empty sequence (of S) and at times we shall write ǫ or nothing instead of ǫS . If s ∈ S then (s) and s denote the same object. On sequences is defined a concatenation operator k, with unity ǫ. Given the (possibly infinite) sequence Q = (s1 , s2 , . . .) and given f : S → S ′ , if s ...
... / S + . ǫS is the empty sequence (of S) and at times we shall write ǫ or nothing instead of ǫS . If s ∈ S then (s) and s denote the same object. On sequences is defined a concatenation operator k, with unity ǫ. Given the (possibly infinite) sequence Q = (s1 , s2 , . . .) and given f : S → S ′ , if s ...
Semi-constr. theories - Stanford Mathematics
... indefinite totality. Thus quantification over the natural numbers is taken to be definite, but not quantification applied to variables for sets or functions of natural numbers. Most axiomatizations of set theory that have been treated metamathematically have been based either entirely on classical ...
... indefinite totality. Thus quantification over the natural numbers is taken to be definite, but not quantification applied to variables for sets or functions of natural numbers. Most axiomatizations of set theory that have been treated metamathematically have been based either entirely on classical ...
THE MODAL LOGIC OF INNER MODELS §1. Introduction. In [10, 11
... Kripke model on a finite pre-Boolean algebra. This argument is implicit in the proof of [10, Theorem 11], but since the bisimulation was not explicitly given in [10], we include the proof here. Remember from [10, p. 1802] that a partial order is called a baled tree if it has a maximal element and th ...
... Kripke model on a finite pre-Boolean algebra. This argument is implicit in the proof of [10, Theorem 11], but since the bisimulation was not explicitly given in [10], we include the proof here. Remember from [10, p. 1802] that a partial order is called a baled tree if it has a maximal element and th ...
Classical First-Order Logic Introduction
... Free and bound variables The free variables of a formula φ are those variables occurring in φ that are not quantified. FV(φ) denotes the set of free variables occurring in φ. The bound variables of a formula φ are those variables occurring in φ that do have quantifiers. BV(φ) denote the set of boun ...
... Free and bound variables The free variables of a formula φ are those variables occurring in φ that are not quantified. FV(φ) denotes the set of free variables occurring in φ. The bound variables of a formula φ are those variables occurring in φ that do have quantifiers. BV(φ) denote the set of boun ...
Co-requisite modules
... Formalism of quantum mechanics: state vectors and Dirac notation, space of states, bases, operators and observables. Recovering wave mechanics. Spin: Nature of spin in QM: matrix representation of states and operators, Stern-Gerlach experiment and measurement, angular momentum addition theorem. Inte ...
... Formalism of quantum mechanics: state vectors and Dirac notation, space of states, bases, operators and observables. Recovering wave mechanics. Spin: Nature of spin in QM: matrix representation of states and operators, Stern-Gerlach experiment and measurement, angular momentum addition theorem. Inte ...
1 Introduction 2 Formal logic
... Formal logic as we understand it in these lectures is an approach to making informal mathematical reasoning precise. It has three main ingredients: • A formal language in which to express the mathematical statements we want to reason about. • A semantics that explains the meaning of statements in ou ...
... Formal logic as we understand it in these lectures is an approach to making informal mathematical reasoning precise. It has three main ingredients: • A formal language in which to express the mathematical statements we want to reason about. • A semantics that explains the meaning of statements in ou ...
A little Big Bang
... size due to Pauli pressure. They form what is called a Fermi sea, with each atom occupying a different quantum state. It is especially striking to directly compare 40K and 41K, two isotopes of potassium that differ by a mere neutron, but which behave so differently because of their bosonic and fermi ...
... size due to Pauli pressure. They form what is called a Fermi sea, with each atom occupying a different quantum state. It is especially striking to directly compare 40K and 41K, two isotopes of potassium that differ by a mere neutron, but which behave so differently because of their bosonic and fermi ...
Introduction - Charles Ling
... Using propositional resolution alone (without other rules of inference), it is possible to build a theorem prover that is sound and complete for all of Propositional Logic. ...
... Using propositional resolution alone (without other rules of inference), it is possible to build a theorem prover that is sound and complete for all of Propositional Logic. ...
Axiomatic and constructive quantum field theory Thesis for the
... analyis, measure and integration theory, functional analysis, operator algebras, Lie groups and Lie algebras. Since I have followed several courses1 in these fields during my study, I did not bother explaining anything concerning these topics. For instance, I do not give the definition of C ∗ algebr ...
... analyis, measure and integration theory, functional analysis, operator algebras, Lie groups and Lie algebras. Since I have followed several courses1 in these fields during my study, I did not bother explaining anything concerning these topics. For instance, I do not give the definition of C ∗ algebr ...
Introduction to Artificial Intelligence
... In propositional logic there are two truth values: t for “true” and f for “false”. Is a formula, such as A ∧ B true? The answer depends on whether the variables A and B are true. Example: If A stands for “It is raining today” and B for “It is cold today” and these are both true, then A ∧ B is true. ...
... In propositional logic there are two truth values: t for “true” and f for “false”. Is a formula, such as A ∧ B true? The answer depends on whether the variables A and B are true. Example: If A stands for “It is raining today” and B for “It is cold today” and these are both true, then A ∧ B is true. ...
Unification in Propositional Logic
... From Part I, it is evident that there are two main computational problems for a given formula A: • check whether A is projective or not; • in the negative case, compute a projective approximation of A. The explicit computation of mgus or of complete sets of unifiers seems to be less important (see t ...
... From Part I, it is evident that there are two main computational problems for a given formula A: • check whether A is projective or not; • in the negative case, compute a projective approximation of A. The explicit computation of mgus or of complete sets of unifiers seems to be less important (see t ...
Action Logic and Pure Induction
... However no such list can be complete, because REG is not finitely based [Red64, Con71]. That is, there is no finite list of equations of REG from which the rest of REG may be inferred. But in addition to this syntactic problem, REG has a semantic problem. It is not strong enough to constrain a∗ to ...
... However no such list can be complete, because REG is not finitely based [Red64, Con71]. That is, there is no finite list of equations of REG from which the rest of REG may be inferred. But in addition to this syntactic problem, REG has a semantic problem. It is not strong enough to constrain a∗ to ...
Document
... 1. Binary operators and their representations Boolean algebra is the basic mathematics needed for logic design of digital systems; Boolean algebra uses Boolean (logical) variables with two values (0 or ...
... 1. Binary operators and their representations Boolean algebra is the basic mathematics needed for logic design of digital systems; Boolean algebra uses Boolean (logical) variables with two values (0 or ...
Document
... 1. Binary operators and their representations Boolean algebra is the basic mathematics needed for logic design of digital systems; Boolean algebra uses Boolean (logical) variables with two values (0 or ...
... 1. Binary operators and their representations Boolean algebra is the basic mathematics needed for logic design of digital systems; Boolean algebra uses Boolean (logical) variables with two values (0 or ...
1. Binary operators and their representations
... The duality principle is formed by replacing AND with OR, OR with AND, 0 with 1, 1 with 0, variables and complements are left unchanged. de Morgan’s laws Allow us to convert between types of gates; we can generalize ...
... The duality principle is formed by replacing AND with OR, OR with AND, 0 with 1, 1 with 0, variables and complements are left unchanged. de Morgan’s laws Allow us to convert between types of gates; we can generalize ...
Practice Problem Set 1
... • These problems will not be graded. • Mutual discussion and discussion with the instructor/TA is strongly encouraged. 1. [From HW1, Autumn 2011] Use the proof system of first order logic studied in class to prove each of the following sequents. You must indicate which proof rule you are applying at ...
... • These problems will not be graded. • Mutual discussion and discussion with the instructor/TA is strongly encouraged. 1. [From HW1, Autumn 2011] Use the proof system of first order logic studied in class to prove each of the following sequents. You must indicate which proof rule you are applying at ...
The Decision Problem for Standard Classes
... We say that a class K of formulas is decidable if both satisfiability and finite satisfiability (that is, satisfiability in a finite model) are decidable for formulas in K. K is conservative [8] if there exists an algorithm a. '> a' which associates a formula a' E K with each formula a in such a way ...
... We say that a class K of formulas is decidable if both satisfiability and finite satisfiability (that is, satisfiability in a finite model) are decidable for formulas in K. K is conservative [8] if there exists an algorithm a. '> a' which associates a formula a' E K with each formula a in such a way ...
Relation between “phases” and “distance” in quantum evolution
... C {0} is a multiplicative group of non-zero cornplex numbers. The projective Hilbert space is ...
... C {0} is a multiplicative group of non-zero cornplex numbers. The projective Hilbert space is ...
Birkhoff`s variety theorem in many sorts
... infinite cardinal λ, we call X λ-presentable provided that card s∈S Xs < λ. Definition. An equation (in the finitary logic) is a formula ∀ X : t = t where X is a finite S-sorted set and t, t are elements of FΣ X of the same sort. A Σ-algebra A satisfies the equation provided that for every S-sorted fun ...
... infinite cardinal λ, we call X λ-presentable provided that card s∈S Xs < λ. Definition. An equation (in the finitary logic) is a formula ∀ X : t = t where X is a finite S-sorted set and t, t are elements of FΣ X of the same sort. A Σ-algebra A satisfies the equation provided that for every S-sorted fun ...
THE PHILOSOPHY OF PHYSICS
... matters of fact about the world to which we can even in principle have no empirical access. The only facts there are, on the empiricist way of thinking, are the ones that can in principle be established by means of some imaginable sort of measurement. And so the only spatial facts there are, on this ...
... matters of fact about the world to which we can even in principle have no empirical access. The only facts there are, on the empiricist way of thinking, are the ones that can in principle be established by means of some imaginable sort of measurement. And so the only spatial facts there are, on this ...