The Closed World Assumption
... We view our program as a logical theory expressing knowledge about the world. In several situations, it is convenient to assume that the program contains complete information about certain kinds of logical statements. We can then make additional inferences about the world based on the assumed comple ...
... We view our program as a logical theory expressing knowledge about the world. In several situations, it is convenient to assume that the program contains complete information about certain kinds of logical statements. We can then make additional inferences about the world based on the assumed comple ...
Predicate logic. Formal and informal proofs
... • The steps of the proofs are not expressed in any formal language as e.g. propositional logic • Steps are argued less formally using English, mathematical formulas and so on • One must always watch the consistency of the argument made, logic and its rules can often help us to decide the soundness o ...
... • The steps of the proofs are not expressed in any formal language as e.g. propositional logic • Steps are argued less formally using English, mathematical formulas and so on • One must always watch the consistency of the argument made, logic and its rules can often help us to decide the soundness o ...
Epsilon Substitution for Transfinite Induction
... Hilbert introduced the epsilon calculus in [Hilbert, 1970] as a method for proving the consistency of arithmetic and analysis. In place of the usual quantifiers, a symbol is added, allowing terms of the form xφ[x], which are interpreted as “some x such that φ[x] holds, if such a number exists.” W ...
... Hilbert introduced the epsilon calculus in [Hilbert, 1970] as a method for proving the consistency of arithmetic and analysis. In place of the usual quantifiers, a symbol is added, allowing terms of the form xφ[x], which are interpreted as “some x such that φ[x] holds, if such a number exists.” W ...
Introduction to HyperReals
... the least upper bound of A. We claim r b. Suppose not. Thus r b and Hence r-b is positive or negative. Case r-b is positive. Since r-b is not a positive infinitesimal there is a positive real s, s < r-b which implies b < r-s so that r-s is an upper bound of A. Thus r-s r but r-s < r. Thus r-b ...
... the least upper bound of A. We claim r b. Suppose not. Thus r b and Hence r-b is positive or negative. Case r-b is positive. Since r-b is not a positive infinitesimal there is a positive real s, s < r-b which implies b < r-s so that r-s is an upper bound of A. Thus r-s r but r-s < r. Thus r-b ...
Predicate Calculus - National Taiwan University
... Example 2: S={P(x)∨Q(x),R(z),T(y)∨∼W(y)} There is no constant in S, so we let H0={a} There is no function symbol in S, hence H=H0=H1=…={a} Example 3: S={P(f(x),a,g(y),b)} H0={a,b} H1={a,b,f(a),f(b),g(a),g(b)} H2={a,b,f(a),f(b),g(a),g(b),f(f(a)),f(f(b)),f(g(a)),f(g (b)),g(f(a)),g(f(b)),g(g( ...
... Example 2: S={P(x)∨Q(x),R(z),T(y)∨∼W(y)} There is no constant in S, so we let H0={a} There is no function symbol in S, hence H=H0=H1=…={a} Example 3: S={P(f(x),a,g(y),b)} H0={a,b} H1={a,b,f(a),f(b),g(a),g(b)} H2={a,b,f(a),f(b),g(a),g(b),f(f(a)),f(f(b)),f(g(a)),f(g (b)),g(f(a)),g(f(b)),g(g( ...
Modal Logic and Model Theory
... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://dv1litvip.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an e ...
... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://dv1litvip.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an e ...
Subtraction, Summary, and Subspaces
... The vector space axioms talk about only two operations: addition and scalar multiplication. Nevertheless, the idea of subtracting two vectors is hidden inside those axioms. Among the real numbers, if you wanted to rewrite the subtraction 7 − 3 in terms of addition, you would write 7 + (−3). An analo ...
... The vector space axioms talk about only two operations: addition and scalar multiplication. Nevertheless, the idea of subtracting two vectors is hidden inside those axioms. Among the real numbers, if you wanted to rewrite the subtraction 7 − 3 in terms of addition, you would write 7 + (−3). An analo ...
COMPLETENESS OF THE RANDOM GRAPH
... Note that we do not need to include the ∀ quantifier because, following the rules of predicate logic, every statement of the form (∀y)θ is equivalent to a statement of the form (¬∃y)(φ). Also, we call repeated but finite applications of step (ii) to formulas a boolean combination of formulas. In a f ...
... Note that we do not need to include the ∀ quantifier because, following the rules of predicate logic, every statement of the form (∀y)θ is equivalent to a statement of the form (¬∃y)(φ). Also, we call repeated but finite applications of step (ii) to formulas a boolean combination of formulas. In a f ...
Chapter 1 Logic and Set Theory
... be used to prove it. Rigorous proofs are used to verify that a given statement that appears intuitively true is indeed true. Ultimately, a mathematical proof is a convincing argument that starts from some premises, and logically deduces the desired conclusion. Most proofs do not mention the logical ...
... be used to prove it. Rigorous proofs are used to verify that a given statement that appears intuitively true is indeed true. Ultimately, a mathematical proof is a convincing argument that starts from some premises, and logically deduces the desired conclusion. Most proofs do not mention the logical ...
Modal Logic
... for basic modal logic is quite general (although it can be further generalized as we will see later) and can be refined to yield the properties appropriate for the intended application. We will concentrate on three different applications: logic of necessity, temporal logic and logic of knowledge. T ...
... for basic modal logic is quite general (although it can be further generalized as we will see later) and can be refined to yield the properties appropriate for the intended application. We will concentrate on three different applications: logic of necessity, temporal logic and logic of knowledge. T ...
Chapter 1 Logic and Set Theory
... A proof in mathematics demonstrates the truth of certain statement. It is therefore natural to begin with a brief discussion of statements. A statement, or proposition, is the content of an assertion. It is either true or false, but cannot be both true and false at the same time. For example, the ex ...
... A proof in mathematics demonstrates the truth of certain statement. It is therefore natural to begin with a brief discussion of statements. A statement, or proposition, is the content of an assertion. It is either true or false, but cannot be both true and false at the same time. For example, the ex ...
A course in Mathematical Logic
... Terms and formulas are interpreted in a model. Definition 8. (Definition of a model) Let L be a language. An L-model M is given by a set M of elements (called the universe of the model) and 1. For every function symbol f ∈ L of arity n, a function f M : M n → M ; 2. For every relation symbol R ∈ L o ...
... Terms and formulas are interpreted in a model. Definition 8. (Definition of a model) Let L be a language. An L-model M is given by a set M of elements (called the universe of the model) and 1. For every function symbol f ∈ L of arity n, a function f M : M n → M ; 2. For every relation symbol R ∈ L o ...
Comparing Constructive Arithmetical Theories Based - Math
... the intuitionistic deductive closure of BASIC), coN P induction does not imply N P induction; and that assuming the polynomial hierarchy does not collapse, neither does N P induction imply coN P induction. This is in sharp contrast to the case for classical logic, in which the two principles are equ ...
... the intuitionistic deductive closure of BASIC), coN P induction does not imply N P induction; and that assuming the polynomial hierarchy does not collapse, neither does N P induction imply coN P induction. This is in sharp contrast to the case for classical logic, in which the two principles are equ ...
