Partial Grounded Fixpoints
... generalise them to points in the bilattice, while still maintaining the elegance and desirable properties of groundedness. For the case of logic programming, this generalisation boils down to extending groundedness to partial (or three-valued) interpretations. There are several reasons why it is imp ...
... generalise them to points in the bilattice, while still maintaining the elegance and desirable properties of groundedness. For the case of logic programming, this generalisation boils down to extending groundedness to partial (or three-valued) interpretations. There are several reasons why it is imp ...
Inference Tasks and Computational Semantics
... • Theorem prover: A tool that, when given a 1storder formula Φ attempts to prove it. • If Φ is in fact provable a (sound and complete) 1st-order prover can (in principle) prove it. • Model builder: a tool that, when given a 1storder formula Φ, attempts to build a model for it. • It cannot (even in p ...
... • Theorem prover: A tool that, when given a 1storder formula Φ attempts to prove it. • If Φ is in fact provable a (sound and complete) 1st-order prover can (in principle) prove it. • Model builder: a tool that, when given a 1storder formula Φ, attempts to build a model for it. • It cannot (even in p ...
Proof Theory of Finite-valued Logics
... Many-valued logic is not much younger than the whole field of symbolic logic. It was introduced in the early twenties of this century by Lukasiewicz [1920] and Post [1921] and has since developed into a very large area of research. Most of the early work done has concentrated on problems of axiomati ...
... Many-valued logic is not much younger than the whole field of symbolic logic. It was introduced in the early twenties of this century by Lukasiewicz [1920] and Post [1921] and has since developed into a very large area of research. Most of the early work done has concentrated on problems of axiomati ...
One-dimensional Fragment of First-order Logic
... logic GNFO. This logic only allows negations of formulae that are guarded in the sense of the guarded fragment. The guarded negation fragment has been shown complete for 2NEXPTIME in [2]. Two-variable logic and guarded-fragment are examples of decidable fragments of first-order logic that are not ba ...
... logic GNFO. This logic only allows negations of formulae that are guarded in the sense of the guarded fragment. The guarded negation fragment has been shown complete for 2NEXPTIME in [2]. Two-variable logic and guarded-fragment are examples of decidable fragments of first-order logic that are not ba ...
Advanced Topics in Propositional Logic
... Go through these, and whenever you encounter Ai such that neither it nor its negation is derivable from , add Ai to . In view of Lemma 5, you will end up with a formally complete set. To see that the same set is also formally consistent, suppose, for a contradiction, that it is not. Consider the s ...
... Go through these, and whenever you encounter Ai such that neither it nor its negation is derivable from , add Ai to . In view of Lemma 5, you will end up with a formally complete set. To see that the same set is also formally consistent, suppose, for a contradiction, that it is not. Consider the s ...
Using linear logic to reason about sequent systems ?
... The Forum presentation of linear logic [Mil96] relies on the connectives ?, , ?, >, &, ?, ), and 8: this set of connectives is complete for linear logic, in the sense that all other linear logic connectives can be de ned from these. Proof search using this collection of connectives can be restricte ...
... The Forum presentation of linear logic [Mil96] relies on the connectives ?, , ?, >, &, ?, ), and 8: this set of connectives is complete for linear logic, in the sense that all other linear logic connectives can be de ned from these. Proof search using this collection of connectives can be restricte ...
On The Expressive Power of Three-Valued and Four
... Bilattices were further investigated by Fitting, who used them for extending some well known logics (like Kleene 3-valued logics) and for logic programming (see, e.g., [Fi90, Fi91, Fi94]). In [AA96] the set D is also generalized to what is called there a bi lter, and bilattices-based logics are intr ...
... Bilattices were further investigated by Fitting, who used them for extending some well known logics (like Kleene 3-valued logics) and for logic programming (see, e.g., [Fi90, Fi91, Fi94]). In [AA96] the set D is also generalized to what is called there a bi lter, and bilattices-based logics are intr ...
Discrete Mathematics - Lyle School of Engineering
... If P(x) is the statement “x has won a race” where the domain of discourse is all runners, then the universal quantification of P(x) is x, P ( x ) , i.e., every runner has won a race. The negation of this statement is “it is not the case that every runner has won a race. Therefore there exists at ...
... If P(x) is the statement “x has won a race” where the domain of discourse is all runners, then the universal quantification of P(x) is x, P ( x ) , i.e., every runner has won a race. The negation of this statement is “it is not the case that every runner has won a race. Therefore there exists at ...
PPT
... A proof of Q from H1, H2, … Hk is finite sequence of propositional forms Q 1, Q 2, … Qn such that Qn is same as Q and every Qj is either one of Hi, (i = 1, 2, … , k) or it follows from the proceedings by the logic rules. Note: In these proofs we will follow the following formats: We begin with by li ...
... A proof of Q from H1, H2, … Hk is finite sequence of propositional forms Q 1, Q 2, … Qn such that Qn is same as Q and every Qj is either one of Hi, (i = 1, 2, … , k) or it follows from the proceedings by the logic rules. Note: In these proofs we will follow the following formats: We begin with by li ...
Easyprove: a tool for teaching precise reasoning
... big burden for a computer science freshman, who is not yet familiar with any programming language or other formal syntax. To remedy this problem, Easyprove presents proofs using a notation close to natural language, and the terms are displayed using Unicode mathematical symbols, which are already fa ...
... big burden for a computer science freshman, who is not yet familiar with any programming language or other formal syntax. To remedy this problem, Easyprove presents proofs using a notation close to natural language, and the terms are displayed using Unicode mathematical symbols, which are already fa ...
Modal Reasoning
... M, x |= φ if and only if N , y |= φ for all modal formulas φ That is, the pointed models M, x and N , y are modally equivalent or M, x ! N , y. The proof of this result is available on pg. 29 of van Bentham’s Modal Logic for Open Minds. It’s now possible to rigorously show that some properties are u ...
... M, x |= φ if and only if N , y |= φ for all modal formulas φ That is, the pointed models M, x and N , y are modally equivalent or M, x ! N , y. The proof of this result is available on pg. 29 of van Bentham’s Modal Logic for Open Minds. It’s now possible to rigorously show that some properties are u ...
A Paedagogic Example of Cut-Elimination
... Γ → Π, where Π and Γ are lists of predicate formulas, ‘→’ is called the sequent arrow. For someone used to the Hilbert-style calculus, where one only works with single formulas, trying to deduce the end formula from a list of axioms by only two rules, namely modus ponens and generalization, this see ...
... Γ → Π, where Π and Γ are lists of predicate formulas, ‘→’ is called the sequent arrow. For someone used to the Hilbert-style calculus, where one only works with single formulas, trying to deduce the end formula from a list of axioms by only two rules, namely modus ponens and generalization, this see ...
Philosophy as Logical Analysis of Science: Carnap, Schlick, Gödel
... conditions gives one information about meaning. For surely, if ‘S’ is true were apriori equivalent to, or made the same statement as, S, then ‘S’ is true iff S would be apriori equivalent to, or make the same statement as S iff S. But then since knowledge that the earth is round iff the earth ...
... conditions gives one information about meaning. For surely, if ‘S’ is true were apriori equivalent to, or made the same statement as, S, then ‘S’ is true iff S would be apriori equivalent to, or make the same statement as S iff S. But then since knowledge that the earth is round iff the earth ...
Document
... A P P B A B - In the propositional case, two literals are the same if they have the same proposition (negated in exactly one of them). - In first order logic, the situation is more complex, due to the existence of variables. We may assume that variables are universally quantified. - In this c ...
... A P P B A B - In the propositional case, two literals are the same if they have the same proposition (negated in exactly one of them). - In first order logic, the situation is more complex, due to the existence of variables. We may assume that variables are universally quantified. - In this c ...
07.1-Reasoning
... • For example in the KB it will have a sentence that if an agent in 1,1 senses a stench then 1,2 or 2,1 has a wumpus in it. • If in 1,1 the agent sense nothing then it will know that 1,2 2,1 and 1,1 all have neither a wumpus nor a pit in them. ...
... • For example in the KB it will have a sentence that if an agent in 1,1 senses a stench then 1,2 or 2,1 has a wumpus in it. • If in 1,1 the agent sense nothing then it will know that 1,2 2,1 and 1,1 all have neither a wumpus nor a pit in them. ...
pptx
... a distribution over masked examples M(D) if Prρ∈M(D)[ψ|ρ=1] ≥ 1-ε Observation: equal to “ψ is a tautology given ρ” • We will aim to accept φ whenever there exists in standard cases where this is tractable, e.g., a (1-ε)-testable formula that completes a CNFs, intersections of halfspaces; remains sim ...
... a distribution over masked examples M(D) if Prρ∈M(D)[ψ|ρ=1] ≥ 1-ε Observation: equal to “ψ is a tautology given ρ” • We will aim to accept φ whenever there exists in standard cases where this is tractable, e.g., a (1-ε)-testable formula that completes a CNFs, intersections of halfspaces; remains sim ...
PDF
... The proofs of these results can be found here. Theorem 1. (Substitution Theorem) Suppose p1 , . . . , pm are all the propositional variables, not necessarily distinct, that occur in order in A, and if B1 , . . . , Bm , C1 , . . . , Cm are wff ’s such that ` Bi ↔ Ci , then ` A[B1 /p1 , . . . , Bm /pm ...
... The proofs of these results can be found here. Theorem 1. (Substitution Theorem) Suppose p1 , . . . , pm are all the propositional variables, not necessarily distinct, that occur in order in A, and if B1 , . . . , Bm , C1 , . . . , Cm are wff ’s such that ` Bi ↔ Ci , then ` A[B1 /p1 , . . . , Bm /pm ...
1 slide/page
... • This is Goldbach’s conjecture: every even number other than 2 is the sum of two primes. ◦ Is it true? We don’t know. Is there a nice (technically: recursive, so that a program can check whether a formula is an axiom) sound and complete axiomatization for arithmetic? • Gödel’s Incompleteness Theor ...
... • This is Goldbach’s conjecture: every even number other than 2 is the sum of two primes. ◦ Is it true? We don’t know. Is there a nice (technically: recursive, so that a program can check whether a formula is an axiom) sound and complete axiomatization for arithmetic? • Gödel’s Incompleteness Theor ...
POSSIBLE WORLDS SEMANTICS AND THE LIAR Reflections on a
... Now, Kaplan’s argument shows that the principle of plenitude is incompatible with assumptions commonly made in possible worlds semantics. Here is how the argument goes: (i) There is a set W of possible worlds and a set P rop of propositions. (ii) There is, for every subset X of W , a corresponding p ...
... Now, Kaplan’s argument shows that the principle of plenitude is incompatible with assumptions commonly made in possible worlds semantics. Here is how the argument goes: (i) There is a set W of possible worlds and a set P rop of propositions. (ii) There is, for every subset X of W , a corresponding p ...
Chapter 2 Propositional Logic
... So far, we have seen two types of statements: (1) a proposition, which is a statement either always true, or always false, and (2) a paradox, which is a statement whose truth value cannot be assigned. Here are two new types of statements: Definition 13. A contradiction is a statement that is always ...
... So far, we have seen two types of statements: (1) a proposition, which is a statement either always true, or always false, and (2) a paradox, which is a statement whose truth value cannot be assigned. Here are two new types of statements: Definition 13. A contradiction is a statement that is always ...