A survey on Interactive Theorem Proving
... Invent a new way of doing mathematics “in front of a computer”. Wang - Towards mechanical mathematics, 1960 ... the writer believes that perhaps machines may more quickly become of pracical use in mathematical research, not by proving new theorems, but by formalizing and checking outlines of proofs, ...
... Invent a new way of doing mathematics “in front of a computer”. Wang - Towards mechanical mathematics, 1960 ... the writer believes that perhaps machines may more quickly become of pracical use in mathematical research, not by proving new theorems, but by formalizing and checking outlines of proofs, ...
chapter9
... • Herbrand 1930: if a sentence is entailed by the original FO KB, then there is a proof using a finite subset of the propositionalized KB • Since any subset has a maximum depth of nesting in terms, we can find the subset by generating all instantiations with constant symbols, then all with depth 1, ...
... • Herbrand 1930: if a sentence is entailed by the original FO KB, then there is a proof using a finite subset of the propositionalized KB • Since any subset has a maximum depth of nesting in terms, we can find the subset by generating all instantiations with constant symbols, then all with depth 1, ...
The Bang-Bang Funnel Controller (long version)
... A similar approach with a switched controller was proposed earlier in [6]; however, there the freedom to choose the time-varying bounds was more restricted and the gain was monotone. These results yielded universal controllers which were able to control all systems with some qualitative property (fo ...
... A similar approach with a switched controller was proposed earlier in [6]; however, there the freedom to choose the time-varying bounds was more restricted and the gain was monotone. These results yielded universal controllers which were able to control all systems with some qualitative property (fo ...
Chapter 5 - Stanford Lagunita
... from a conjunction of any number of sentences, any one of its conjuncts. This inference pattern is sometimes called conjunction elimination or simplification, when it is presented in the context of a formal system of deduction. When it is used in informal proofs, however, it usually goes by without ...
... from a conjunction of any number of sentences, any one of its conjuncts. This inference pattern is sometimes called conjunction elimination or simplification, when it is presented in the context of a formal system of deduction. When it is used in informal proofs, however, it usually goes by without ...
AN INTRODUCTION TO LOGIC
... Books. There are two kinds of books: those you read and those you work from. For background reading, I recommend [2, 5, 6]. The remaining books are work books. Of those, you might want to start with Zegarelli [17]; don’t be put off by the Dummies tag, since it isn’t a bad introduction to logic. For ...
... Books. There are two kinds of books: those you read and those you work from. For background reading, I recommend [2, 5, 6]. The remaining books are work books. Of those, you might want to start with Zegarelli [17]; don’t be put off by the Dummies tag, since it isn’t a bad introduction to logic. For ...
Chapter 5.5
... 3) The measure of any obtuse angle is greater than 90. The sum of any two obtuse angles is greater than 180. This is a CONTRADICTION to the theorem that states that the sum of the angles of any triangle = 180. 4) Therefore, my assumption is false. A triangle cannot have two obtuse angles. ...
... 3) The measure of any obtuse angle is greater than 90. The sum of any two obtuse angles is greater than 180. This is a CONTRADICTION to the theorem that states that the sum of the angles of any triangle = 180. 4) Therefore, my assumption is false. A triangle cannot have two obtuse angles. ...
Constructing Cut Free Sequent Systems With Context Restrictions
... in particular in computer science, both generic methods with efficient tools for designing cut-free calculi, and meta-theorems that guarantee cut-elimination, decidability, and complexity bounds are therefore increasingly important. This paper explores the method of cut elimination by saturation and ...
... in particular in computer science, both generic methods with efficient tools for designing cut-free calculi, and meta-theorems that guarantee cut-elimination, decidability, and complexity bounds are therefore increasingly important. This paper explores the method of cut elimination by saturation and ...
Argumentative Approaches to Reasoning with Maximal Consistency
... Dung-style semantics for abstract argumentation, and a computational approach that is based on extensions of Gentzentype proofs systems. This brings about a new perspective on reasoning with MCS which shows a strong link between the latter and argumentation systems, and which can be extended to rela ...
... Dung-style semantics for abstract argumentation, and a computational approach that is based on extensions of Gentzentype proofs systems. This brings about a new perspective on reasoning with MCS which shows a strong link between the latter and argumentation systems, and which can be extended to rela ...
Large cardinals and the Continuum Hypothesis
... The question regarding the size of the continuum – i.e. the number of the reals – is probably the most famous question in set theory. Its appeal comes from the fact that, apparently, everyone knows what a real number is and so the question concerning their quantity seems easy to understand. While th ...
... The question regarding the size of the continuum – i.e. the number of the reals – is probably the most famous question in set theory. Its appeal comes from the fact that, apparently, everyone knows what a real number is and so the question concerning their quantity seems easy to understand. While th ...
On Cantor`s diagonal argument
... formalist concludes:’ the power of the continuum, i.e., the power of the set of real numbers between 0 and 1, is greater than aleph-null,’ a proposition which is without meaning for the intuitionist.” ([2]P74) In Cantor’s proof, Cantor’s number is defined based on the set M in a sense of actual and ...
... formalist concludes:’ the power of the continuum, i.e., the power of the set of real numbers between 0 and 1, is greater than aleph-null,’ a proposition which is without meaning for the intuitionist.” ([2]P74) In Cantor’s proof, Cantor’s number is defined based on the set M in a sense of actual and ...
Notes on Modal Logic - Stanford University
... Fact 3.10 Let M = hW, R, V i be a relational structure. Define the “exists two” operator 32 ϕ as follows: M, w |= 32 ϕ iff there is v1 , v2 ∈ W such that v1 6= v2 , M, v1 |= ϕ and M, v2 |= ϕ The exist two 32 operator is not definable in the basic modal language. Proof. Suppose that the 32 is definab ...
... Fact 3.10 Let M = hW, R, V i be a relational structure. Define the “exists two” operator 32 ϕ as follows: M, w |= 32 ϕ iff there is v1 , v2 ∈ W such that v1 6= v2 , M, v1 |= ϕ and M, v2 |= ϕ The exist two 32 operator is not definable in the basic modal language. Proof. Suppose that the 32 is definab ...
Equivalence for the G3'-stable models semantics
... which two programs are strongly G03 -equivalent also guarantee that two disjunctive programs are strongly equivalent in the p-stable semantics. We present two main results that guarantee G03 strong equivalence, one for two arbitrary programs and another one for a couple of programs of the form P , P ...
... which two programs are strongly G03 -equivalent also guarantee that two disjunctive programs are strongly equivalent in the p-stable semantics. We present two main results that guarantee G03 strong equivalence, one for two arbitrary programs and another one for a couple of programs of the form P , P ...
Intermediate Logic
... will introduce you to the concepts, results, and methods of formal logic necessary to understand and appreciate these applications as well as the limitations of formal logic. It will be mathematical in that you will be required to master abstract formal concepts and to prove theorems about logic (no ...
... will introduce you to the concepts, results, and methods of formal logic necessary to understand and appreciate these applications as well as the limitations of formal logic. It will be mathematical in that you will be required to master abstract formal concepts and to prove theorems about logic (no ...
Non-Classical Logic
... B.3 Deductive Validity We might here present a traditional deductive system for classical propositional logic. However, I assume you already familiar with at least one such system, whether it is a natural deduction system or axiom system. All such standard systems are equivalent and yield the same r ...
... B.3 Deductive Validity We might here present a traditional deductive system for classical propositional logic. However, I assume you already familiar with at least one such system, whether it is a natural deduction system or axiom system. All such standard systems are equivalent and yield the same r ...
Die Grundlagen der Arithmetik §§82–83
... 6. Every number except 0 immediately follows a number in the natural sequence of numbers. It is clear from §44 of Grundgesetze6 that Frege did not take (6) to imply that 0 does not immediately follow a number, that ¬xP0. This proposition is proved separately in Grundgesetze, as Theorem 108, and will ...
... 6. Every number except 0 immediately follows a number in the natural sequence of numbers. It is clear from §44 of Grundgesetze6 that Frege did not take (6) to imply that 0 does not immediately follow a number, that ¬xP0. This proposition is proved separately in Grundgesetze, as Theorem 108, and will ...
The Foundations
... their truth or falsity (Semantics and Axiomatics). It is the foundation for expressing formal proofs in all branches of mathematics. ...
... their truth or falsity (Semantics and Axiomatics). It is the foundation for expressing formal proofs in all branches of mathematics. ...