Mathematics: the divine madness
... realization of the simplest conceivable mathematical ideas. . . ” “We can discover by means of purely mathematical constructions . . . the key to understanding natural phenomena. . . ” “Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the ...
... realization of the simplest conceivable mathematical ideas. . . ” “We can discover by means of purely mathematical constructions . . . the key to understanding natural phenomena. . . ” “Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the ...
Chapter 2 - Princeton University Press
... thinking in terms of phrases such as “really exist,” but such phrases may be misleading. The “real world” may include three apples or three airplanes, but the abstract concept of the number 3 itself does not exist as a physical entity anywhere in our real world. It exists only in our minds. If some ...
... thinking in terms of phrases such as “really exist,” but such phrases may be misleading. The “real world” may include three apples or three airplanes, but the abstract concept of the number 3 itself does not exist as a physical entity anywhere in our real world. It exists only in our minds. If some ...
First-Order Logic, Second-Order Logic, and Completeness
... basically a two-sorted first-order logic. This would also explain the apparent tension between the above mentioned results and Lindström’s theorem: No logic that goes beyond the expressive power of FOL satisfies both the compactness and the Löwenheim-Skolem theorem.12 It is also worth noting that a ...
... basically a two-sorted first-order logic. This would also explain the apparent tension between the above mentioned results and Lindström’s theorem: No logic that goes beyond the expressive power of FOL satisfies both the compactness and the Löwenheim-Skolem theorem.12 It is also worth noting that a ...
Elements of Finite Model Theory
... Finite model theory studies the expressive power of logical languages over collections of finite structures. Over the past few decades, deep connections have emerged between finite model theory and various areas in combinatorics and computer science, including complexity theory, database theory, for ...
... Finite model theory studies the expressive power of logical languages over collections of finite structures. Over the past few decades, deep connections have emerged between finite model theory and various areas in combinatorics and computer science, including complexity theory, database theory, for ...
Chapter 1, Part I: Propositional Logic
... raining.” then p →q denotes “If I am at home then it is raining.” In p →q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). ...
... raining.” then p →q denotes “If I am at home then it is raining.” In p →q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). ...
Vectors and Vector Operations
... A relation is transitive if whenever one has both a b and b c then one also has a c. A relation is symmetric if whenever one has a b then one also has b a. A relation is reflexive if a a for every object a. Often it is easy to verify that a certain relation has these three properti ...
... A relation is transitive if whenever one has both a b and b c then one also has a c. A relation is symmetric if whenever one has a b then one also has b a. A relation is reflexive if a a for every object a. Often it is easy to verify that a certain relation has these three properti ...
EVERYONE KNOWS THAT SOMEONE KNOWS
... 5. π : P → P(W ) is a function that maps propositional variables into sets of epistemic worlds. In this article, we write u ∼X v if u ∼x v for each x ∈ X. The next definition introduces the update operation on an arbitrary function. This operation changes the value of the function at a single point. ...
... 5. π : P → P(W ) is a function that maps propositional variables into sets of epistemic worlds. In this article, we write u ∼X v if u ∼x v for each x ∈ X. The next definition introduces the update operation on an arbitrary function. This operation changes the value of the function at a single point. ...
A Proof of Nominalism. An Exercise in Successful
... for a first-order language in the same language, as is shown in Hintikka and Sandu (1999). It might also be at the bottom of Zermelo’s unfortunate construal of the axiom of choice as a non-logical, mathematical assumption. Systematically speaking, and even more importantly, the version of the axiom ...
... for a first-order language in the same language, as is shown in Hintikka and Sandu (1999). It might also be at the bottom of Zermelo’s unfortunate construal of the axiom of choice as a non-logical, mathematical assumption. Systematically speaking, and even more importantly, the version of the axiom ...
INTRODUCTION TO THE THEORY OF PROOFS 3A. The Gentzen
... Definition 3A.1. A sequent (in a fixed signature τ ) is an expression φ1 , . . . , φn ⇒ ψ1 , . . . , ψm where φ1 , . . . , φn , ψ1 , . . . , ψm are τ -formulas. We view the formulas on the left and the right as comprising multisets, i.e., we identify sequences which differ only in the order in which ...
... Definition 3A.1. A sequent (in a fixed signature τ ) is an expression φ1 , . . . , φn ⇒ ψ1 , . . . , ψm where φ1 , . . . , φn , ψ1 , . . . , ψm are τ -formulas. We view the formulas on the left and the right as comprising multisets, i.e., we identify sequences which differ only in the order in which ...
POSSIBLE WORLDS SEMANTICS AND THE LIAR Reflections on a
... cal languages where oblique constructions are represented by intensional operators. The direct-discourse approaches are, however, threatened by (self-referential) paradoxes. Montague (1963) showed that the syntactic treatment of necessity as a predicate of sentences in the object language leads to i ...
... cal languages where oblique constructions are represented by intensional operators. The direct-discourse approaches are, however, threatened by (self-referential) paradoxes. Montague (1963) showed that the syntactic treatment of necessity as a predicate of sentences in the object language leads to i ...
Fuzzy logic and probability Institute of Computer Science (ICS
... ing clear the basic differences. Admitting some simpli fication, we cotL'>ider that fuzzy logic is a logic of vague, imprecise notions and propositions, propositions that may be more or less true. Fuzzy logic is then a logic of partial degrees of truth. On the contrary, probabil ity deal'3 with cr ...
... ing clear the basic differences. Admitting some simpli fication, we cotL'>ider that fuzzy logic is a logic of vague, imprecise notions and propositions, propositions that may be more or less true. Fuzzy logic is then a logic of partial degrees of truth. On the contrary, probabil ity deal'3 with cr ...
The First Incompleteness Theorem
... time-travel. Talk of ‘Gödel’s Theorems’, however, typically refers to his two incompleteness theorems in an epoch-making 1931 paper, ‘On formally undecidable propositions of Principia Mathematica and related systems I’. More on that title in due course. For an overview of Gödel’s work, see http:// ...
... time-travel. Talk of ‘Gödel’s Theorems’, however, typically refers to his two incompleteness theorems in an epoch-making 1931 paper, ‘On formally undecidable propositions of Principia Mathematica and related systems I’. More on that title in due course. For an overview of Gödel’s work, see http:// ...
The Foundations: Logic and Proofs - UTH e
... Compound Expressions Connectives from propositional logic carry over to predicate ...
... Compound Expressions Connectives from propositional logic carry over to predicate ...
comments on the logic of constructible falsity (strong negation)
... Görnemann’s result suggests the conjecture that a classical model theory for the logic I have described may be obtained by allowing the domain to “grow with time”. This is in fact true. We may define a Nelson model structure as a triple (K, R, D), where K is a non-empty set of “stages of investigat ...
... Görnemann’s result suggests the conjecture that a classical model theory for the logic I have described may be obtained by allowing the domain to “grow with time”. This is in fact true. We may define a Nelson model structure as a triple (K, R, D), where K is a non-empty set of “stages of investigat ...
Chapter 1 - National Taiwan University
... Please read Section §1.7 of your textbook. It takes effort to know how to write correct proofs. When you read the text, please try to understand how the statements are proved, instead of what the statements are proving. Sometimes, we may make a statement without knowing whether it is true or not. Suc ...
... Please read Section §1.7 of your textbook. It takes effort to know how to write correct proofs. When you read the text, please try to understand how the statements are proved, instead of what the statements are proving. Sometimes, we may make a statement without knowing whether it is true or not. Suc ...
Logical nihilism - University of Notre Dame
... IPC satisfied the “disjunction property”: formulas such as φ ∨ ψ are provable only if either φ or ψ is as well. At first sight this might appear to be no more than a restatement of the intuitionist’s rejection of lem. After all, that rejection was motivated by the idea that instances of lem are unwa ...
... IPC satisfied the “disjunction property”: formulas such as φ ∨ ψ are provable only if either φ or ψ is as well. At first sight this might appear to be no more than a restatement of the intuitionist’s rejection of lem. After all, that rejection was motivated by the idea that instances of lem are unwa ...
A Question About Increasing Functions
... Now u < v, so u − v < 0. The quantity in the square brackets on the right side of (3) is positive because it is the sum of two squares which can’t both be zero. Thus, the product on the right side of (3) is negative, and from this it follows that u3 < v 3 . Our appeal to the definition shows that th ...
... Now u < v, so u − v < 0. The quantity in the square brackets on the right side of (3) is positive because it is the sum of two squares which can’t both be zero. Thus, the product on the right side of (3) is negative, and from this it follows that u3 < v 3 . Our appeal to the definition shows that th ...
Assignment MCS-013 Discrete Mathematics Q1: a) Make truth table
... Let P(n) be a predicate that involves a natural number n. Suppose we can show that (i) p(m) is true for some m belong N (ii) whenever p(m) ,p(m+1),………,p(k) are true, That p(k+1) is true, where k>=m we can conclude that p(n) is true for all natural number n>=m. In the induction step we are making mor ...
... Let P(n) be a predicate that involves a natural number n. Suppose we can show that (i) p(m) is true for some m belong N (ii) whenever p(m) ,p(m+1),………,p(k) are true, That p(k+1) is true, where k>=m we can conclude that p(n) is true for all natural number n>=m. In the induction step we are making mor ...
Beginning Deductive Logic
... brave or foolhardy (or both!) to venture an answer to such a question, unless perhaps one has set aside enough time and space to craft at least a sustained essay, or perhaps even a book. Well, we fancy ourselves brave; here goes; all done in but one sentence: Logic is the science and engineering of ...
... brave or foolhardy (or both!) to venture an answer to such a question, unless perhaps one has set aside enough time and space to craft at least a sustained essay, or perhaps even a book. Well, we fancy ourselves brave; here goes; all done in but one sentence: Logic is the science and engineering of ...
CSE 1400 Applied Discrete Mathematics Proofs
... Axioms come in two forms: One form is axioms that are universally True. These are called logical axioms. The other form is axioms that are True some model, but not in others. These are called non-logical axioms. Models can be loosely classified into one of two types: Discrete models that describe fi ...
... Axioms come in two forms: One form is axioms that are universally True. These are called logical axioms. The other form is axioms that are True some model, but not in others. These are called non-logical axioms. Models can be loosely classified into one of two types: Discrete models that describe fi ...
The unintended interpretations of intuitionistic logic
... development of intuitionistic logic: it was not until 1923 that Brouwer discovered the equivalence in intuitionistic mathematics of triple negation and single negation [Brouwer 1925]. While Brouwer may have been uncompromising with respect to his philosophy, his mathematical and philosophical talent ...
... development of intuitionistic logic: it was not until 1923 that Brouwer discovered the equivalence in intuitionistic mathematics of triple negation and single negation [Brouwer 1925]. While Brouwer may have been uncompromising with respect to his philosophy, his mathematical and philosophical talent ...
Logic, Human Logic, and Propositional Logic Human Logic
... If there is no fuel, the car will not start. If there is no spark, the car will not start. There is spark. The car will not start. Therefore, there is no fuel. What if the car is in a vacuum chamber? ...
... If there is no fuel, the car will not start. If there is no spark, the car will not start. There is spark. The car will not start. Therefore, there is no fuel. What if the car is in a vacuum chamber? ...
Lecture 14 Notes
... some yet unknown element of the universe. Since we do not know this element, a should be a new parameter – this way we make sure that we don’t make any further assumptions about a by accidentally linking it to a parameter that was introduced earlier in the proof. If we were to decompose T (∀x)P x be ...
... some yet unknown element of the universe. Since we do not know this element, a should be a new parameter – this way we make sure that we don’t make any further assumptions about a by accidentally linking it to a parameter that was introduced earlier in the proof. If we were to decompose T (∀x)P x be ...
Can Modalities Save Naive Set Theory?
... modal logics which are not contained in S5. Section 5.1 shows that it is inconsistent in any proper extension of S5. Section 5.2 considers the logic KDDc, axiomatized by ( D ) and its converse ( Dc): 3ϕ → 2ϕ. On the fictionalist interpretation of the modal operators, this expresses the natural const ...
... modal logics which are not contained in S5. Section 5.1 shows that it is inconsistent in any proper extension of S5. Section 5.2 considers the logic KDDc, axiomatized by ( D ) and its converse ( Dc): 3ϕ → 2ϕ. On the fictionalist interpretation of the modal operators, this expresses the natural const ...
Mathematics for Computer Science/Software Engineering
... Heuristic justification for this definition: if p is true and q is false, then the statement ‘if p is true then q is true’ obviously cannot be true, and therefore must be false. On the other hand, if p is false, then the statement ‘if p is true then ...’ is an empty statement—it is saying nothing at ...
... Heuristic justification for this definition: if p is true and q is false, then the statement ‘if p is true then q is true’ obviously cannot be true, and therefore must be false. On the other hand, if p is false, then the statement ‘if p is true then ...’ is an empty statement—it is saying nothing at ...