PDF - University of Kent

... examples of the classes do actually exist. This issue of existence is extremely important and we need to discuss it in more depth now. 2.1 Universals, particulars and existence. The basic question is, does a proposition such as “all s are m” imply that there are actually any s’s? One might think yes ...

... examples of the classes do actually exist. This issue of existence is extremely important and we need to discuss it in more depth now. 2.1 Universals, particulars and existence. The basic question is, does a proposition such as “all s are m” imply that there are actually any s’s? One might think yes ...

A Unified View of Induction Reasoning for First-Order Logic

... and implicit induction principles, [16, 22, 27, 30, 40, 56] being among the most notable. Other studies have been conducted to reduce the gap between them. Protzen [42] proposed a proof strategy to perform lazy induction on particular explicit induction proofs. Kapur and Subramaniam [29] devised a m ...

... and implicit induction principles, [16, 22, 27, 30, 40, 56] being among the most notable. Other studies have been conducted to reduce the gap between them. Protzen [42] proposed a proof strategy to perform lazy induction on particular explicit induction proofs. Kapur and Subramaniam [29] devised a m ...

Document

... when we already have the “operators” + and *?) • The reason is that + and * are operations on numbers, but we’re talking about strings, not numbers. In this discussion +, *, (, and ) are just symbols in the alphabet Introduction to Computation ...

... when we already have the “operators” + and *?) • The reason is that + and * are operations on numbers, but we’re talking about strings, not numbers. In this discussion +, *, (, and ) are just symbols in the alphabet Introduction to Computation ...

... when we already have the “operators” + and *?) • The reason is that + and * are operations on numbers, but we’re talking about strings, not numbers. In this discussion +, *, (, and ) are just symbols in the alphabet Introduction to Computation ...

Peano`s Arithmetic

... this. Other mathematicians, such as Hermann Grassmann and Gottlob Frege, had already been working to systemize arithmetic and its axioms. Grassmann, in fact, was the first to begin this process in his book Lehrbuch der arithmetik (1861). Frege made more progress on this and focused more on the logic ...

... this. Other mathematicians, such as Hermann Grassmann and Gottlob Frege, had already been working to systemize arithmetic and its axioms. Grassmann, in fact, was the first to begin this process in his book Lehrbuch der arithmetik (1861). Frege made more progress on this and focused more on the logic ...

Document

... …shew. But that’s not the end of the story. This was just the intuitive derivation of the formula, not the proof. LEMMA: The maximal number of intersection points of n lines in the plane is n(n-1)/2. Proof. Prove by induction. Base case: If n = 1, then there is only one line and therefore no interse ...

... …shew. But that’s not the end of the story. This was just the intuitive derivation of the formula, not the proof. LEMMA: The maximal number of intersection points of n lines in the plane is n(n-1)/2. Proof. Prove by induction. Base case: If n = 1, then there is only one line and therefore no interse ...

Building explicit induction schemas for cyclic induction reasoning

... Abstract. In the setting of classical ﬁrst-order logic with inductive predicates, two kinds of sequent-based induction reasoning are distinguished: cyclic and structural. Proving their equivalence is of great theoretical and practical interest for the automated reasoning community. In [3,4], it has ...

... Abstract. In the setting of classical ﬁrst-order logic with inductive predicates, two kinds of sequent-based induction reasoning are distinguished: cyclic and structural. Proving their equivalence is of great theoretical and practical interest for the automated reasoning community. In [3,4], it has ...

Chapter 2, Logic

... What is Philosophy Chapter 2 by Richard Thompson gave rise to a good deal of debate among logicians. For sometimes we assert universal generalisations without any commitment to existence. For instance if we explained ‘unicorn’ by saying ‘Unicorn’ means ’quadruped mammal resembling a horse but with ...

... What is Philosophy Chapter 2 by Richard Thompson gave rise to a good deal of debate among logicians. For sometimes we assert universal generalisations without any commitment to existence. For instance if we explained ‘unicorn’ by saying ‘Unicorn’ means ’quadruped mammal resembling a horse but with ...

3. Mathematical Induction 3.1. First Principle of

... The second principle of induction differs from the first only in the form of the induction hypothesis. Here we assume not just P (n), but P (j) for all the integers j between k and n (inclusive). We use this assumption to show P (n + 1). This method of induction is also called strong mathematical in ...

... The second principle of induction differs from the first only in the form of the induction hypothesis. Here we assume not just P (n), but P (j) for all the integers j between k and n (inclusive). We use this assumption to show P (n + 1). This method of induction is also called strong mathematical in ...

Running Time of Euclidean Algorithm

... But that’s not the end of the story. This was just the intuitive derivation of the formula, not the proof. LEMMA: The maximal number of intersection points of n lines in the plane is n(n-1)/2. Proof. Prove by induction. Base case: If n = 1, then there is only one line and therefore no intersections. ...

... But that’s not the end of the story. This was just the intuitive derivation of the formula, not the proof. LEMMA: The maximal number of intersection points of n lines in the plane is n(n-1)/2. Proof. Prove by induction. Base case: If n = 1, then there is only one line and therefore no intersections. ...

Lecture 09

... Example: Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps. Solution: Let P(n) be the proposition that postage of n cents can be formed using 4-cent and 5-cent stamps. – BASIS STEP: P(12), P(13), P(14), and P(15) hold. • P(12) uses three 4-cent ...

... Example: Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps. Solution: Let P(n) be the proposition that postage of n cents can be formed using 4-cent and 5-cent stamps. – BASIS STEP: P(12), P(13), P(14), and P(15) hold. • P(12) uses three 4-cent ...

Note 3

... sum of the first n odd numbers is not just a perfect square, but is equal precisely to n2 ! Motivated by this discovery, let’s try something counterintuitive: Let us try to show the following stronger claim. Theorem 3.4. For all n ≥ 1, the sum of the first n odd numbers is n2 . Proof. We proceed by ...

... sum of the first n odd numbers is not just a perfect square, but is equal precisely to n2 ! Motivated by this discovery, let’s try something counterintuitive: Let us try to show the following stronger claim. Theorem 3.4. For all n ≥ 1, the sum of the first n odd numbers is n2 . Proof. We proceed by ...

Induction and the Well-Ordering Principle Capturing All The Whole

... There are two possibilities. The first possibility is that x actually happens to lie in {0, 1, 2, . . ., n}. If this happens to be true, what can we say about B? What kind of set must B actually be?” By the definition, if B has an element x with x ∈ {0, 1, . . ., n}, B must be n-good. “Good,” Dr. Wu ...

... There are two possibilities. The first possibility is that x actually happens to lie in {0, 1, 2, . . ., n}. If this happens to be true, what can we say about B? What kind of set must B actually be?” By the definition, if B has an element x with x ∈ {0, 1, . . ., n}, B must be n-good. “Good,” Dr. Wu ...

Contents 1 The Natural Numbers

... in these notes. There will be occasional references to Stoll’s book. † c °May 21, 2007. ...

... in these notes. There will be occasional references to Stoll’s book. † c °May 21, 2007. ...

Principle of Mathematical Induction

... The Principle of Mathematical Induction is a method of proof normally used to prove that a proposition is true for all natural numbers 1,2,3,… , although there are many variations of the basic method. The method is particularly important in discrete mathematics, and one often sees theorems proven by ...

... The Principle of Mathematical Induction is a method of proof normally used to prove that a proposition is true for all natural numbers 1,2,3,… , although there are many variations of the basic method. The method is particularly important in discrete mathematics, and one often sees theorems proven by ...

Document

... For the inductive step, assume for some n ≥ 3 that P(n) holds and all convex polygons with n vertices have angles that sum to (n–2) · 180°. We prove P(n+1), that the sum of the angles in any convex polygon with n+1 vertices is (n–1) · 180°. Let A be an arbitrary convex polygon with n+1 vertices. Tak ...

... For the inductive step, assume for some n ≥ 3 that P(n) holds and all convex polygons with n vertices have angles that sum to (n–2) · 180°. We prove P(n+1), that the sum of the angles in any convex polygon with n+1 vertices is (n–1) · 180°. Let A be an arbitrary convex polygon with n+1 vertices. Tak ...

Induction

... In the middle of the last century, a colloquial expression in common use was ”that is a horse of a different color”, referring to something that is quite different from normal or common expectation. The famous mathematician George Polya (who was also a great expositor of mathematics for the lay publ ...

... In the middle of the last century, a colloquial expression in common use was ”that is a horse of a different color”, referring to something that is quite different from normal or common expectation. The famous mathematician George Polya (who was also a great expositor of mathematics for the lay publ ...

MATH 312H–FOUNDATIONS

... For more than two statements, say A,B and C, we have the logical rule that it is always true that the statement “{“A⇒B”} and {“B⇒C”}” implies the statement “A⇒C”. (transitive) Example. A barman has four customers. He knows that the first one is under 18, the second one over 18, the third is having a ...

... For more than two statements, say A,B and C, we have the logical rule that it is always true that the statement “{“A⇒B”} and {“B⇒C”}” implies the statement “A⇒C”. (transitive) Example. A barman has four customers. He knows that the first one is under 18, the second one over 18, the third is having a ...

The New Organon

... seventeenth-century European new technology, the camera obscura. It would, Wotton pointed out, be a particularly useful technical tool for covertly drawing accurate maps and harbour plans. Wotton’s letter (which Bacon’s nineteenth-century editor Spedding omitted from his definitive edition)4 shows h ...

... seventeenth-century European new technology, the camera obscura. It would, Wotton pointed out, be a particularly useful technical tool for covertly drawing accurate maps and harbour plans. Wotton’s letter (which Bacon’s nineteenth-century editor Spedding omitted from his definitive edition)4 shows h ...

Day00a-Induction-proofs - Rose

... induction? To show that p(n) is true for all n n0 : – Step 0: Believe in the "magic." • You will show that it's not really magic at all. But you have to believe. • If, when you are in the middle of an induction proof, you begin to doubt whether the principle of mathematical induction itself is tru ...

... induction? To show that p(n) is true for all n n0 : – Step 0: Believe in the "magic." • You will show that it's not really magic at all. But you have to believe. • If, when you are in the middle of an induction proof, you begin to doubt whether the principle of mathematical induction itself is tru ...

Review - Gerry O nolan

... separately from the rest of the book. In this chapter Stove abandons the thesis that either deductive or inductive logic is purely formal. In the latter case, this denial is used as the basis of a solution to Goodman's so-called new riddle of induction. As Stove points out, once the idea that induct ...

... separately from the rest of the book. In this chapter Stove abandons the thesis that either deductive or inductive logic is purely formal. In the latter case, this denial is used as the basis of a solution to Goodman's so-called new riddle of induction. As Stove points out, once the idea that induct ...

Lecture notes from 5860

... These notes summarize ideas discussed up to and including Lecture 14. This material is related to Chapter 6 of the textbook by Thompson. In particular the idea of extracting a program from a proof is examined. The ideas discussed here take us deeper into the issues behind the design of constructive ...

... These notes summarize ideas discussed up to and including Lecture 14. This material is related to Chapter 6 of the textbook by Thompson. In particular the idea of extracting a program from a proof is examined. The ideas discussed here take us deeper into the issues behind the design of constructive ...

PDF

... amounts to the same, how they are understood) because understanding a language, even a formal one, is not merely to understand its rules as rules of symbol manipulation. Believing that is the mistake of formalism.” The first challenge for understanding modern type theory is to understand these highe ...

... amounts to the same, how they are understood) because understanding a language, even a formal one, is not merely to understand its rules as rules of symbol manipulation. Believing that is the mistake of formalism.” The first challenge for understanding modern type theory is to understand these highe ...

Mathematical Induction

... Basis: The sum of the first 0 natural numbers is indeed 0. Inductive step: Assume the sum of the first k natural numbers is k(k-1)/2 (inductive hypothesis). We want to show that then the same is true for k+1 instead of k, that is, the sum of the first k+1 natural numbers is (k+1)((k+1)-1)/2, i.e. it ...

... Basis: The sum of the first 0 natural numbers is indeed 0. Inductive step: Assume the sum of the first k natural numbers is k(k-1)/2 (inductive hypothesis). We want to show that then the same is true for k+1 instead of k, that is, the sum of the first k+1 natural numbers is (k+1)((k+1)-1)/2, i.e. it ...

The Novum Organum, full original title Novum Organum Scientiarum (‘new instrument of science’), is a philosophical work by Francis Bacon, written in Latin and published in 1620. The title is a reference to Aristotle's work Organon, which was his treatise on logic and syllogism. In Novum Organum, Bacon details a new system of logic he believes to be superior to the old ways of syllogism. This is now known as the Baconian method.For Bacon, finding the essence of a thing was a simple process of reduction, and the use of inductive reasoning. In finding the cause of a ‘phenomenal nature’ such as heat, one must list all of the situations where heat is found. Then another list should be drawn up, listing situations that are similar to those of the first list except for the lack of heat. A third table lists situations where heat can vary. The ‘form nature’, or cause, of heat must be that which is common to all instances in the first table, is lacking from all instances of the second table and varies by degree in instances of the third table.The title page of Novum Organum depicts a galleon passing between the mythical Pillars of Hercules that stand either side of the Strait of Gibraltar, marking the exit from the well-charted waters of the Mediterranean into the Atlantic Ocean. The Pillars, as the boundary of the Mediterranean, have been smashed through opening a new world for exploration. Bacon hopes that empirical investigation will, similarly, smash the old scientific ideas and lead to greater understanding of the world and heavens.The Latin tag across the bottom – Multi pertransibunt & augebitur scientia – is taken from Daniel 12:4. It means: ""Many will travel and knowledge will be increased"".