Jean Van Heijenoort`s View of Modern Logic
... the proposition into subject and predicate had been replaced by its analysis into function and argument(s). A preliminary accomplishment was the propositional calculus, with a truth-functional definition of the connectives, including the conditional. Of cardinal importance was the realization that, ...
... the proposition into subject and predicate had been replaced by its analysis into function and argument(s). A preliminary accomplishment was the propositional calculus, with a truth-functional definition of the connectives, including the conditional. Of cardinal importance was the realization that, ...
January 12
... For Frege, requirements (A) and (B) hang together in the following way: Propositions are the (semantic) content of thought (i.e., what we think when we think something), and propositions stand essentially in inferential relations to each other. Thus it makes good sense that a language that fulfilled ...
... For Frege, requirements (A) and (B) hang together in the following way: Propositions are the (semantic) content of thought (i.e., what we think when we think something), and propositions stand essentially in inferential relations to each other. Thus it makes good sense that a language that fulfilled ...
Strict Predicativity 3
... in parallel to the situation with predicativity given the natural numbers. A worry might arise right at the beginning, however, because we are talking about formal systems such as Q or ramified type theory, and it may be thought that some impredicative notion, such as natural number or finite set or ...
... in parallel to the situation with predicativity given the natural numbers. A worry might arise right at the beginning, however, because we are talking about formal systems such as Q or ramified type theory, and it may be thought that some impredicative notion, such as natural number or finite set or ...
Exercises: Sufficiently expressive/strong
... 2. In this exercise, take ‘theory’ to mean any set of sentences equipped with deductive rules, whether or not effectively axiomatizable: (a) If a theory is effectively decidable, must it be negation complete? (b) If a theory is effectively decidable, must it be effectively axiomatizable? (c) If a th ...
... 2. In this exercise, take ‘theory’ to mean any set of sentences equipped with deductive rules, whether or not effectively axiomatizable: (a) If a theory is effectively decidable, must it be negation complete? (b) If a theory is effectively decidable, must it be effectively axiomatizable? (c) If a th ...
22.1 Representability of Functions in a Formal Theory
... There is also a successor of λ-PRL that is based on a much richer formal logic called type theory, but introducing that logic and its applications is a course by itself. However, before we do so, let us explore the theoretical consequences of the axiomatizations we have so far. The Peano axioms appe ...
... There is also a successor of λ-PRL that is based on a much richer formal logic called type theory, but introducing that logic and its applications is a course by itself. However, before we do so, let us explore the theoretical consequences of the axiomatizations we have so far. The Peano axioms appe ...
LOGIC AND PSYCHOTHERAPY
... The basic principle in the medical science is: “Each disease has an etiology, that is, mechanism of development, and therefore the logical relation to make a diagnosis and a respective treatment”.2 Steve de Shazer, one of the pioneers of the Solution-Focused Brief Therapy (SFBT), propose something d ...
... The basic principle in the medical science is: “Each disease has an etiology, that is, mechanism of development, and therefore the logical relation to make a diagnosis and a respective treatment”.2 Steve de Shazer, one of the pioneers of the Solution-Focused Brief Therapy (SFBT), propose something d ...
The Discovery of the Computer
... Frege’s “first order logic”. While we have not discussed this, it extends the logic of AND, OR, NOT, IF with “there exists” and “for all”. Hilbert showed that mathematics could be described by this new logic, and they raised three important questions. Completeness. For any valid inference, would it ...
... Frege’s “first order logic”. While we have not discussed this, it extends the logic of AND, OR, NOT, IF with “there exists” and “for all”. Hilbert showed that mathematics could be described by this new logic, and they raised three important questions. Completeness. For any valid inference, would it ...
The absolute proof!
... absolute and devoid of doubt. Pythagoras died confident in the knowledge that his theorem, which was true in 500 a.c. , would remain true for eternity. Science is operated according to the judicial system. A theory is assumed to be true if there is enough evidence to prove it “beyond all reasonable ...
... absolute and devoid of doubt. Pythagoras died confident in the knowledge that his theorem, which was true in 500 a.c. , would remain true for eternity. Science is operated according to the judicial system. A theory is assumed to be true if there is enough evidence to prove it “beyond all reasonable ...
Propositional Logic Predicate Logic
... Name of Symbols ∀ (universal quantifier), and ∃ (existential quantifier). Definition. A formula A is valid if A is true no matter how we replace the individual constants in A with concrete individuals and the predicate variables in A with concrete predicates. Note. The set of individuals must be in ...
... Name of Symbols ∀ (universal quantifier), and ∃ (existential quantifier). Definition. A formula A is valid if A is true no matter how we replace the individual constants in A with concrete individuals and the predicate variables in A with concrete predicates. Note. The set of individuals must be in ...
CHAPTER 0: WELCOME TO MATHEMATICS A Preface of Logic
... met. Thus, anyone playing the game of mathematics should agree on the truth value of a statement asserting that an object satisfies a definition. The fact that definitions and axioms are accepted without proof does not mean that anything goes. A good definition should both clarify the meaning of a t ...
... met. Thus, anyone playing the game of mathematics should agree on the truth value of a statement asserting that an object satisfies a definition. The fact that definitions and axioms are accepted without proof does not mean that anything goes. A good definition should both clarify the meaning of a t ...
A Note on Naive Set Theory in LP
... 3 What this might mean The choice of LP as the logic in which to embed a naive set theory is not without justification. As we have noticed, it is easy to work in since models are quite easy to construct. Secondly, it is perhaps the most natural paraconsistent expansion of classical predicate logic. ...
... 3 What this might mean The choice of LP as the logic in which to embed a naive set theory is not without justification. As we have noticed, it is easy to work in since models are quite easy to construct. Secondly, it is perhaps the most natural paraconsistent expansion of classical predicate logic. ...
ASSIGNMENT 3
... example, in modern Euclidean geometry the terms ‘point and ‘line’ are typically left undefined. 2. Axioms (or Postulates): An axiom (or postulate) is a logical statement about terms that is accepted without proof. For example, the statement “A straight line can be drawn from any point to any point” ...
... example, in modern Euclidean geometry the terms ‘point and ‘line’ are typically left undefined. 2. Axioms (or Postulates): An axiom (or postulate) is a logical statement about terms that is accepted without proof. For example, the statement “A straight line can be drawn from any point to any point” ...
Stephen Cook and Phuong Nguyen. Logical foundations of proof
... proofs in P. And a sort of converse to this last statement holds too since the theory T proves the soundness of P. Thus, in the language of the previous paragraph, the proof system P is not only complete, but efficiently so, with respect to the propositional translations of bounded theorems in T . A ...
... proofs in P. And a sort of converse to this last statement holds too since the theory T proves the soundness of P. Thus, in the language of the previous paragraph, the proof system P is not only complete, but efficiently so, with respect to the propositional translations of bounded theorems in T . A ...
Relative normalization
... normalization as each axiomatic theory T requires a specific notion of reduction. Thus we use an extension of predicate logic called Deduction modulo [?]. In Deduction modulo, a theory is formed is formed with a set of axioms Γ and a congruence ≡ defined on formulæ. Then, the deduction rules take th ...
... normalization as each axiomatic theory T requires a specific notion of reduction. Thus we use an extension of predicate logic called Deduction modulo [?]. In Deduction modulo, a theory is formed is formed with a set of axioms Γ and a congruence ≡ defined on formulæ. Then, the deduction rules take th ...
PDF
... about programs, integers and lists of integers. There is also a successor of λ-PRL that is based on a much richer formal logic called type theory, but introducing that logic and its applications is a course by itself. However, before we do so, let us explore the theoretical consequences of the axiom ...
... about programs, integers and lists of integers. There is also a successor of λ-PRL that is based on a much richer formal logic called type theory, but introducing that logic and its applications is a course by itself. However, before we do so, let us explore the theoretical consequences of the axiom ...
Lecture 7. Model theory. Consistency, independence, completeness
... of the sentences in ∆ hold in the model.) And if the answer is NO, usually the easiest way to show it is by deriving a contradiction, i.e. by showing that ∆ ├ ⊥. See homework problems 5-8. 2.4. Independence. The notion of independence is less crucial than some of the other notions we have studied; i ...
... of the sentences in ∆ hold in the model.) And if the answer is NO, usually the easiest way to show it is by deriving a contradiction, i.e. by showing that ∆ ├ ⊥. See homework problems 5-8. 2.4. Independence. The notion of independence is less crucial than some of the other notions we have studied; i ...
General Proof Theory - Matematički institut SANU
... general proof theory, he concentrated on the justification of the inference steps, rather than on the propositions that make the proofs. In general proof theory one looks for an algebra of proofs, and for that one should concentrate on the operations of this algebra, which come with the inference rul ...
... general proof theory, he concentrated on the justification of the inference steps, rather than on the propositions that make the proofs. In general proof theory one looks for an algebra of proofs, and for that one should concentrate on the operations of this algebra, which come with the inference rul ...
Section 2.6 Cantor`s Theorem and the ZFC Axioms
... which contradicts the our assumption that such a correspondence exists. Need for Axioms in Set Theory The reader should not entertain the belief that Cantor was the first mathematician to think about sets and their operations, such as union, intersection and compliment. The concept of a set has been ...
... which contradicts the our assumption that such a correspondence exists. Need for Axioms in Set Theory The reader should not entertain the belief that Cantor was the first mathematician to think about sets and their operations, such as union, intersection and compliment. The concept of a set has been ...
Theories.Axioms,Rules of Inference
... (toobig x)) Well, if we have not defined the function toobig, then it certainly is not a theorem and ACL2 won't even attempt to prove the proposition. If we make this definition, (defun toobig (x) (> x 1000)) then the theorem is clearly true. ACL2 proves it: (thm (implies (> x 20000) ...
... (toobig x)) Well, if we have not defined the function toobig, then it certainly is not a theorem and ACL2 won't even attempt to prove the proposition. If we make this definition, (defun toobig (x) (> x 1000)) then the theorem is clearly true. ACL2 proves it: (thm (implies (> x 20000) ...
CS 2742 (Logic in Computer Science) Lecture 6
... by cases, by contradiction, by transitivity and so on. They can be derived from the original logic identities. For example, modus ponens becomes ((p → q) ∧ p) → q. ...
... by cases, by contradiction, by transitivity and so on. They can be derived from the original logic identities. For example, modus ponens becomes ((p → q) ∧ p) → q. ...
Chapter Nine - Queen of the South
... There is much more work for Ockham's Razor. The study of selffunctioning-feedback-systems in the new discipline of Aseistics and the development of an elementary NeoCantorian Set Theory Underlying Feedback Functions, acronymically termed STUFF, rationalize conclusions which controvert many cherished ...
... There is much more work for Ockham's Razor. The study of selffunctioning-feedback-systems in the new discipline of Aseistics and the development of an elementary NeoCantorian Set Theory Underlying Feedback Functions, acronymically termed STUFF, rationalize conclusions which controvert many cherished ...
Book Review: Lorenz J. Halbeisen: “Combinatorial Set Theory.”
... It is not difficult to use AC, the Axiom of Choice, to show that there is some A ⊂ P∞ (N) which is not Ramsey. With the help of concepts from Descriptive Set Theory, we may however make sense of the following question, which then naturally arises: In the presence of AC, how “definable” can a non–Ram ...
... It is not difficult to use AC, the Axiom of Choice, to show that there is some A ⊂ P∞ (N) which is not Ramsey. With the help of concepts from Descriptive Set Theory, we may however make sense of the following question, which then naturally arises: In the presence of AC, how “definable” can a non–Ram ...
Howework 8
... The nal three lectures will review the material that we have covered so far, elaborate some of the issues a bit deeper, and discuss the philosphical implications of the results and methods used. Please prepare questions that you would like to see adressed in these lectures. ...
... The nal three lectures will review the material that we have covered so far, elaborate some of the issues a bit deeper, and discuss the philosphical implications of the results and methods used. Please prepare questions that you would like to see adressed in these lectures. ...
Kurt Gödel and His Theorems
... cannot prove its own consistency • It is not possible to formalize all of mathematics, as any attempt at such a formalism will omit some true mathematical statements • A theory such as Peano arithmetic cannot even prove its own consistency • There is no mechanical way to decide the truth (or provabi ...
... cannot prove its own consistency • It is not possible to formalize all of mathematics, as any attempt at such a formalism will omit some true mathematical statements • A theory such as Peano arithmetic cannot even prove its own consistency • There is no mechanical way to decide the truth (or provabi ...
Different notions of conuity and intensional models for λ
... and negative (i.e. useless) information: the pragmatism and goal-driveness. Tasks - as objects, and reductions of descriptable classes of tasks with the smooth abstraction type property - as continuos functions and morphisms. Let < A, ≤> be any complete and co-complete poset (partially ordered set), ...
... and negative (i.e. useless) information: the pragmatism and goal-driveness. Tasks - as objects, and reductions of descriptable classes of tasks with the smooth abstraction type property - as continuos functions and morphisms. Let < A, ≤> be any complete and co-complete poset (partially ordered set), ...