full text (.pdf)
... datatypes such as infinite streams or trees, but it is not as widespread or as well understood. In this paper, we illustrate through several examples the use of coinduction in informal mathematical arguments. Our aim is to promote the principle as a useful tool for the working mathematician and to br ...
... datatypes such as infinite streams or trees, but it is not as widespread or as well understood. In this paper, we illustrate through several examples the use of coinduction in informal mathematical arguments. Our aim is to promote the principle as a useful tool for the working mathematician and to br ...
A KE Tableau for a Logic of Formal Inconsistency - IME-USP
... F ◦ X can be in DS, but only if ◦X is a subformula of some other formula in DS. If ◦X is not a subformula of some other formula in DS, neither T ◦ X nor F ◦ X are in DS; 4. if a signed formula S X is in DS, then for any sign S, for any formula X, for all subformulas Y of X and for all n ≥ 0, the sig ...
... F ◦ X can be in DS, but only if ◦X is a subformula of some other formula in DS. If ◦X is not a subformula of some other formula in DS, neither T ◦ X nor F ◦ X are in DS; 4. if a signed formula S X is in DS, then for any sign S, for any formula X, for all subformulas Y of X and for all n ≥ 0, the sig ...
PPT - UBC Department of CPSC Undergraduates
... Every logical equivalence that we’ve learned applies to predicate logic statements. For example, to prove ~x D, P(x), you can prove x D, ~P(x) and then convert it back with generalized De Morgan’s. To prove x D, P(x) Q(x), you can prove x D, ~Q(x) ~P(x) and convert it back using the ...
... Every logical equivalence that we’ve learned applies to predicate logic statements. For example, to prove ~x D, P(x), you can prove x D, ~P(x) and then convert it back with generalized De Morgan’s. To prove x D, P(x) Q(x), you can prove x D, ~Q(x) ~P(x) and convert it back using the ...
Sequent Combinators: A Hilbert System for the Lambda
... and easier implementations, this provides a basis for believing that Hilbert systems could be used to underlie type-theory based proof assistants and even functional programming language implementations. In general we shall omit proofs in this paper, because they are similar to well-known proofs of ...
... and easier implementations, this provides a basis for believing that Hilbert systems could be used to underlie type-theory based proof assistants and even functional programming language implementations. In general we shall omit proofs in this paper, because they are similar to well-known proofs of ...
Implication - Abstractmath.org
... Some of them flatly refuse to believe me when I tell them the correct interpretation. This is a classic example of semantic contamination, a form of cognitive dissonance - two sources of information appear to contradict each other, in this case the professor and a lifetime of intimate experience wit ...
... Some of them flatly refuse to believe me when I tell them the correct interpretation. This is a classic example of semantic contamination, a form of cognitive dissonance - two sources of information appear to contradict each other, in this case the professor and a lifetime of intimate experience wit ...
Notes on the Science of Logic
... without overwhelming you is that whenever a definition is justified, we feel free to employ it; but the justification itself, which is indeed an important matter for logic, is left for another course. We are furthermore compelled to say that there are some topics that we do not treat in these notes ...
... without overwhelming you is that whenever a definition is justified, we feel free to employ it; but the justification itself, which is indeed an important matter for logic, is left for another course. We are furthermore compelled to say that there are some topics that we do not treat in these notes ...
CH8B
... high true. • In Mixed logic polarity, we can have both high true signals, and low true signals. – Low true signal names are followed by ‘(L)’ to indicate low true – High true signal names are followed by ‘(H)’ to indicate low true ...
... high true. • In Mixed logic polarity, we can have both high true signals, and low true signals. – Low true signal names are followed by ‘(L)’ to indicate low true – High true signal names are followed by ‘(H)’ to indicate low true ...
Prolog 1 - Department of Computer Science
... soundness refers to logical systems, which means that if some formula can be proven in a system, then it is true in the relevant model/structure (if A is a theorem, it is true). This is the converse of completeness. Unsoundness usually violates our innate notion of Excluded Middle – but so do so man ...
... soundness refers to logical systems, which means that if some formula can be proven in a system, then it is true in the relevant model/structure (if A is a theorem, it is true). This is the converse of completeness. Unsoundness usually violates our innate notion of Excluded Middle – but so do so man ...
Completeness theorems and lambda
... It is remarkable that, in the case of λ-calculus, the use of Kripke models, which seems so crucial for the Ω-rule, is not necessary Theorem: (Hindley, 1983) M ∈ ∩A T (A) iff ` M : T (X) The proof however involves a non canonical enumeration of types and variables In effect it builds an infinite cont ...
... It is remarkable that, in the case of λ-calculus, the use of Kripke models, which seems so crucial for the Ω-rule, is not necessary Theorem: (Hindley, 1983) M ∈ ∩A T (A) iff ` M : T (X) The proof however involves a non canonical enumeration of types and variables In effect it builds an infinite cont ...
A Conditional Logical Framework *
... open up promising generalizations of the proposition-as-types paradigm. The idea underlying the Conditional Logical Framework LFK is inspired by the Honsell-Lenisa-Liquori’s General Logical Framework GLF see [HLL07], where we proposed a uniform methodology for extending LF, which allows to deal with ...
... open up promising generalizations of the proposition-as-types paradigm. The idea underlying the Conditional Logical Framework LFK is inspired by the Honsell-Lenisa-Liquori’s General Logical Framework GLF see [HLL07], where we proposed a uniform methodology for extending LF, which allows to deal with ...
Sequent Calculus in Natural Deduction Style
... work the same way. Discharge in natural deduction corresponds to the application of a sequent calculus rule that has an active formula in the antecedent of a premiss. These are the left rules and the right implication rule. In sequent calculus, ever since Gentzen, weakening and contraction have been ...
... work the same way. Discharge in natural deduction corresponds to the application of a sequent calculus rule that has an active formula in the antecedent of a premiss. These are the left rules and the right implication rule. In sequent calculus, ever since Gentzen, weakening and contraction have been ...
Horn Belief Contraction: Remainders, Envelopes and Complexity
... We assume a fixed finite set of propositional variables. We use 0 and 1 for representing truth values. The set of truth assignments satisfying (resp., falsifying) a propositional formula ψ is denoted by T (ψ) (resp., F (ψ)). For formulas ψ, ϕ it holds that ψ |= ϕ (i.e., ϕ is a consequence of ψ) iff ...
... We assume a fixed finite set of propositional variables. We use 0 and 1 for representing truth values. The set of truth assignments satisfying (resp., falsifying) a propositional formula ψ is denoted by T (ψ) (resp., F (ψ)). For formulas ψ, ϕ it holds that ψ |= ϕ (i.e., ϕ is a consequence of ψ) iff ...
Chapter 2
... prove that the product of two even integers is also even, we can use knowledge about number theory. In particular, we could use the fact that an even integer is divisible by 2, or that an even integer m can be rewritten as 2k for some integer k. In this example, P(x, y) : x is an even integer y is ...
... prove that the product of two even integers is also even, we can use knowledge about number theory. In particular, we could use the fact that an even integer is divisible by 2, or that an even integer m can be rewritten as 2k for some integer k. In this example, P(x, y) : x is an even integer y is ...
Indirect Proofs - Stanford University
... Theorem: For any n ∈ ℤ, if n2 is even, then n is even. Proof: By contrapositive; we prove that if n is odd, then n2 is odd. Since n is odd, there is some integer k such that n = 2k + 1. Squaring both sides of this equality and simplifying gives the following: n2 = (2k + 1)2 n2 = 4k2 + 4k + 1 n2 = 2 ...
... Theorem: For any n ∈ ℤ, if n2 is even, then n is even. Proof: By contrapositive; we prove that if n is odd, then n2 is odd. Since n is odd, there is some integer k such that n = 2k + 1. Squaring both sides of this equality and simplifying gives the following: n2 = (2k + 1)2 n2 = 4k2 + 4k + 1 n2 = 2 ...
tbmk5ictk6
... two examples of valid arguments; one of them was sound and the other was not. Since both examples were valid, the one with true premises was the one that was sound. We also saw two examples of invalid arguments. Both of those are unsound simply because they are invalid. Sound arguments have to be v ...
... two examples of valid arguments; one of them was sound and the other was not. Since both examples were valid, the one with true premises was the one that was sound. We also saw two examples of invalid arguments. Both of those are unsound simply because they are invalid. Sound arguments have to be v ...
full text (.pdf)
... This axiomatization can be thought of as a deductive system for refuting unsatisable systems of mixed positive and negative constraints. Deriving the sequent ` is tantamount to refuting the mixed system fs 6= t j s = t 2 g. Systems of the restricted form ` ? correspond to systems of positive s ...
... This axiomatization can be thought of as a deductive system for refuting unsatisable systems of mixed positive and negative constraints. Deriving the sequent ` is tantamount to refuting the mixed system fs 6= t j s = t 2 g. Systems of the restricted form ` ? correspond to systems of positive s ...
Sets, Logic, Computation
... between and constructions using these. It will be good to have shorthand symbols for these, and think through the general properties of sets, relations, and functions in order to do that. If you are not used to thinking mathematically and to formulating mathematical proofs, then think of the first p ...
... between and constructions using these. It will be good to have shorthand symbols for these, and think through the general properties of sets, relations, and functions in order to do that. If you are not used to thinking mathematically and to formulating mathematical proofs, then think of the first p ...
A Generalization of St˚almarck`s Method
... A tool for validity checking of propositional-logic formulas (also known as a tautology checker) determines whether a given formula ϕ over the propositional variables {pi } is true for all assignments of truth values to {pi }. Validity checking is dual to satisfiability checking; validity of ϕ can b ...
... A tool for validity checking of propositional-logic formulas (also known as a tautology checker) determines whether a given formula ϕ over the propositional variables {pi } is true for all assignments of truth values to {pi }. Validity checking is dual to satisfiability checking; validity of ϕ can b ...
Chapter 8: The Logic of Conditionals
... not attempt to prove that our system (even the part of it we’ve developed so far) has these properties. Rather, we’re just interested in understanding what these properties are, and with getting a rough idea of what would be involved in proving that our system has them. To study more metatheory, you ...
... not attempt to prove that our system (even the part of it we’ve developed so far) has these properties. Rather, we’re just interested in understanding what these properties are, and with getting a rough idea of what would be involved in proving that our system has them. To study more metatheory, you ...
YABLO WITHOUT GODEL
... ∀z > y ¬ Satz (x, x, z) is chosen as ϕ(x, y). It shows that ser and trans together with vs are inconsistent. In contrast to ts, the schema vs by itself is consistent. As long as > is well-founded, that is, for any set of objects there is always a <-maximal element, models of vs can be defined by ind ...
... ∀z > y ¬ Satz (x, x, z) is chosen as ϕ(x, y). It shows that ser and trans together with vs are inconsistent. In contrast to ts, the schema vs by itself is consistent. As long as > is well-founded, that is, for any set of objects there is always a <-maximal element, models of vs can be defined by ind ...
Sets, Logic, Computation
... between and constructions using these. It will be good to have shorthand symbols for these, and think through the general properties of sets, relations, and functions in order to do that. If you are not used to thinking mathematically and to formulating mathematical proofs, then think of the first p ...
... between and constructions using these. It will be good to have shorthand symbols for these, and think through the general properties of sets, relations, and functions in order to do that. If you are not used to thinking mathematically and to formulating mathematical proofs, then think of the first p ...
Godel`s Proof
... be to see that because computers at base manipulate numbers, and because numbers are a universal medium for the embedding of patterns of any sort, computers can deal with arbitrary patterns, whether they are logical or illogical, consistent or inconsistent. In short, when one steps back far enough f ...
... be to see that because computers at base manipulate numbers, and because numbers are a universal medium for the embedding of patterns of any sort, computers can deal with arbitrary patterns, whether they are logical or illogical, consistent or inconsistent. In short, when one steps back far enough f ...