1 The Easy Way to Gödel`s Proof and Related Matters Haim Gaifman
... It is customary to present Gödel’s results by proving first the fixed point theorem. This leaves the construction unmotivated and it appears as a magic trick. The route from Cantor, which is claimed here, is not explicit in Gödel’s paper, but is hinted at by his mentioning the Liar paradox and Rich ...
... It is customary to present Gödel’s results by proving first the fixed point theorem. This leaves the construction unmotivated and it appears as a magic trick. The route from Cantor, which is claimed here, is not explicit in Gödel’s paper, but is hinted at by his mentioning the Liar paradox and Rich ...
Gödel`s correspondence on proof theory and constructive mathematics
... In the final remark (a) of [1931] (p, 296-297 in [Herbrand, 1971]), Herbrand writes that “it seems to us almost certain that every intuitionistic argument can . . . be carried out in an arithmetic [of the form T (Γ)∗ ].” In particular, it would follow that mathematical induction is restricted to qua ...
... In the final remark (a) of [1931] (p, 296-297 in [Herbrand, 1971]), Herbrand writes that “it seems to us almost certain that every intuitionistic argument can . . . be carried out in an arithmetic [of the form T (Γ)∗ ].” In particular, it would follow that mathematical induction is restricted to qua ...
Euclidian Roles in Description Logics
... For example, in [2] the Description Logic RIQ is extended with several role axioms, like reflexive and irreflexive role axioms, disjoint role axioms and simple negation on roles. These extensions has motivated us to investigate possibilities of extending Description Logics with other role axioms. In ...
... For example, in [2] the Description Logic RIQ is extended with several role axioms, like reflexive and irreflexive role axioms, disjoint role axioms and simple negation on roles. These extensions has motivated us to investigate possibilities of extending Description Logics with other role axioms. In ...
Judgment and consequence relations
... In this paper I look into the standard definitions of logical consequence and show that they can be unified under a common scheme. Standardly, consequence relations are defined via the preservation of truth. Here I propose to generalize this as follows: consequence is the preservation of a (truth re ...
... In this paper I look into the standard definitions of logical consequence and show that they can be unified under a common scheme. Standardly, consequence relations are defined via the preservation of truth. Here I propose to generalize this as follows: consequence is the preservation of a (truth re ...
The Science of Proof - University of Arizona Math
... The thesis of this book is that there is a science of proof. Mathematics prides itself on making its assumptions explicit, but most mathematicians learn to construct proofs in an unsystematic way, by example. This is in spite of the known fact that there is an organized way of creating proofs using ...
... The thesis of this book is that there is a science of proof. Mathematics prides itself on making its assumptions explicit, but most mathematicians learn to construct proofs in an unsystematic way, by example. This is in spite of the known fact that there is an organized way of creating proofs using ...
Predicate Logic - Teaching-WIKI
... Anyone standing in the rain will get wet. and then use this knowledge. For example, suppose we also learn that Jan is standing in the rain. • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is ...
... Anyone standing in the rain will get wet. and then use this knowledge. For example, suppose we also learn that Jan is standing in the rain. • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is ...
Automata, Languages, and Programming
... where · on sets is given by A · B = {u · v | u ◦ A, v ◦ B}, and An is defined inductively as A0 = RM (1) and An+1 = An · A. The image of the interpretation RM together with the operations →, ·, ∗ , ∅, {1M } is the algebra of regular sets over M , denoted by Reg M . If M is the free monoid Σ ∗ , then ...
... where · on sets is given by A · B = {u · v | u ◦ A, v ◦ B}, and An is defined inductively as A0 = RM (1) and An+1 = An · A. The image of the interpretation RM together with the operations →, ·, ∗ , ∅, {1M } is the algebra of regular sets over M , denoted by Reg M . If M is the free monoid Σ ∗ , then ...
FIRST DEGREE ENTAILMENT, SYMMETRY AND PARADOX
... different. This means that the construction will work whatever we take the denotation of other constants to be. So, let’s consider a language with a countable supply of constants λ, λ1 , λ2, . . . whose denotation can be freely set however we please. So fdeT is the set of relational fde evaluations ...
... different. This means that the construction will work whatever we take the denotation of other constants to be. So, let’s consider a language with a countable supply of constants λ, λ1 , λ2, . . . whose denotation can be freely set however we please. So fdeT is the set of relational fde evaluations ...
A Critique of the Foundations of Hoare-Style Programming Logics
... Criteria for correctness of a logical system Two primary requirements are known for the ...
... Criteria for correctness of a logical system Two primary requirements are known for the ...
A Critique of the Foundations of Hoare-Style
... Criteria for correctness of a logical system Two primary requirements are known for the ...
... Criteria for correctness of a logical system Two primary requirements are known for the ...
Document
... DEF: Two compound propositions p, q are logically equivalent if their biconditional joining p q is a tautology. Logical equivalence is denoted by p q. EG: The contrapositive of a logical implication is the reversal of the implication, while negating both components. I.e. the contrapositive of p ...
... DEF: Two compound propositions p, q are logically equivalent if their biconditional joining p q is a tautology. Logical equivalence is denoted by p q. EG: The contrapositive of a logical implication is the reversal of the implication, while negating both components. I.e. the contrapositive of p ...
Normal form results for default logic
... In this paper we develop a representation theory for default logic of Reiter ([Rei80]). The question is whether one can find “normal forms” for default theories, that is, if there are syntactical constraints which can be imposed on default theories without changing extensions. In this section we int ...
... In this paper we develop a representation theory for default logic of Reiter ([Rei80]). The question is whether one can find “normal forms” for default theories, that is, if there are syntactical constraints which can be imposed on default theories without changing extensions. In this section we int ...
full text (.pdf)
... deterministic PDL, but neither the upper nor the lower bound of our PSPACE completeness result follows from theirs. Not only are PDL semantics restricted to relational models, but the arguments of Halpern and Reif 1983] depend on an additional nonalgebraic restriction: the relations interpreting at ...
... deterministic PDL, but neither the upper nor the lower bound of our PSPACE completeness result follows from theirs. Not only are PDL semantics restricted to relational models, but the arguments of Halpern and Reif 1983] depend on an additional nonalgebraic restriction: the relations interpreting at ...
Seventy-five problems for testing automatic
... ATPers in mind that the following list is offered. None of these problems will be the sort whose solution is, of itself, of any mathematical or logical interest. Such ‘open problems’ are regularly published in the Newsletter of the Association for Automated Reasoning. Most (but not all) of my proble ...
... ATPers in mind that the following list is offered. None of these problems will be the sort whose solution is, of itself, of any mathematical or logical interest. Such ‘open problems’ are regularly published in the Newsletter of the Association for Automated Reasoning. Most (but not all) of my proble ...
page 113 THE AGM THEORY AND INCONSISTENT BELIEF
... beliefs from implicit beliefs which are derived from the explicit beliefs, or separating relevant beliefs from irrelevant beliefs. Based on this approach, several formal techniques have been developed in recent years to deal with inconsistent beliefs; for example, Chopra and Parikh (2000), Hansson a ...
... beliefs from implicit beliefs which are derived from the explicit beliefs, or separating relevant beliefs from irrelevant beliefs. Based on this approach, several formal techniques have been developed in recent years to deal with inconsistent beliefs; for example, Chopra and Parikh (2000), Hansson a ...
Propositional logic - Computing Science
... Argument 1: If the program syntax is faulty or if program execution results in division by zero, then the computer will generate an error message. Therefore, if the computer does not generate, then the program syntax is correct and program execution does not result in division by zero. Argument 2: I ...
... Argument 1: If the program syntax is faulty or if program execution results in division by zero, then the computer will generate an error message. Therefore, if the computer does not generate, then the program syntax is correct and program execution does not result in division by zero. Argument 2: I ...
Definition: A proof is a system of reasoning or argument to convince
... **You can show that a conclusion is false by giving just ...
... **You can show that a conclusion is false by giving just ...
Algebraic Proof Systems
... A proof system f1 polynomially simulates a proof system f2 , if there exists a polynomial time computable function g such that for all ā ∈ {0, 1}∗ , f1 (g (ā)) = f2 (ā). Meaning: Given a proof ā of f2 (ā) in the second system, we can construct a proof g (ā) of the same tautology in the first s ...
... A proof system f1 polynomially simulates a proof system f2 , if there exists a polynomial time computable function g such that for all ā ∈ {0, 1}∗ , f1 (g (ā)) = f2 (ā). Meaning: Given a proof ā of f2 (ā) in the second system, we can construct a proof g (ā) of the same tautology in the first s ...
Lectures on Laws of Supply and Demand, Simple and Compound
... Example 2 “ If Brian and Angela are not both happy then either Brian is not happy or Angela is not happy”. This is an example of a compound proposition. Logic is not concerned with determining the truth values of simple propositions( that depends on facts outside of logic) but in determining the tru ...
... Example 2 “ If Brian and Angela are not both happy then either Brian is not happy or Angela is not happy”. This is an example of a compound proposition. Logic is not concerned with determining the truth values of simple propositions( that depends on facts outside of logic) but in determining the tru ...
Propositional and predicate logic - Computing Science
... Argument = premises (propositions or statements) + conclusion To have confidence in the conclusion in your argument, the premises should be acceptable on their own merits or follow from other statements that are known to be true. [Q] Any logical forms for valid arguments? ...
... Argument = premises (propositions or statements) + conclusion To have confidence in the conclusion in your argument, the premises should be acceptable on their own merits or follow from other statements that are known to be true. [Q] Any logical forms for valid arguments? ...
Interactive Theorem Proving in Coq and the Curry
... correctness of the proposition. To be of any use, a proof should always be finite. We often omit many small steps of reasoning, to reduce the size of a proof. In doing so, some steps of invalid reasoning might be introduced. Thus, verifying the correctness of a proof becomes, a very important task. ...
... correctness of the proposition. To be of any use, a proof should always be finite. We often omit many small steps of reasoning, to reduce the size of a proof. In doing so, some steps of invalid reasoning might be introduced. Thus, verifying the correctness of a proof becomes, a very important task. ...
CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness
... We extend v to a homomorphism v ∗ : PF −→ B in a usual way, i.e. we put v ∗ (A) = v(A) for A ∈ P , and for any A, B ∈ P F, v ∗ (A ⇒ B) = v ∗ (A) ⇒ v ∗ (B), v ∗ (A ∪ B) = v ∗ (A) ∪ v ∗ (B), v ∗ (A ∩ B) = v ∗ (A) ∩ v ∗ (B), v ∗ (¬A) = ¬v ∗ (A). Propositional Model A truth assignment v is called a prop ...
... We extend v to a homomorphism v ∗ : PF −→ B in a usual way, i.e. we put v ∗ (A) = v(A) for A ∈ P , and for any A, B ∈ P F, v ∗ (A ⇒ B) = v ∗ (A) ⇒ v ∗ (B), v ∗ (A ∪ B) = v ∗ (A) ∪ v ∗ (B), v ∗ (A ∩ B) = v ∗ (A) ∩ v ∗ (B), v ∗ (¬A) = ¬v ∗ (A). Propositional Model A truth assignment v is called a prop ...
proofs in mathematics
... by his fence. How will you prove to your neighbour that he has tried to encroach on your land? Your first step may be to seek the help of the village elders to sort out the difference in boundaries. But, suppose opinion is divided among the elders. Some feel you are right and others feel your neighb ...
... by his fence. How will you prove to your neighbour that he has tried to encroach on your land? Your first step may be to seek the help of the village elders to sort out the difference in boundaries. But, suppose opinion is divided among the elders. Some feel you are right and others feel your neighb ...
A Proof Theory for Generic Judgments: An extended abstract
... need to discover invariants. Another more intensional approach, however, involves introducing a new, generic variable, say, c : γ, that has not been introduced before in the proof, and to prove the formula B[c/x] instead. In natural deduction and sequent calculus proofs, such new variables are calle ...
... need to discover invariants. Another more intensional approach, however, involves introducing a new, generic variable, say, c : γ, that has not been introduced before in the proof, and to prove the formula B[c/x] instead. In natural deduction and sequent calculus proofs, such new variables are calle ...