What Is Answer Set Programming?
... (a closed path that passes through each vertex of the graph exactly once). The ASP program below should be combined with definitions of the predicates vertex and edge, as in the previous example. It uses the predicate in to express that an edge belongs to the path; we assume that 0 is one of the ver ...
... (a closed path that passes through each vertex of the graph exactly once). The ASP program below should be combined with definitions of the predicates vertex and edge, as in the previous example. It uses the predicate in to express that an edge belongs to the path; we assume that 0 is one of the ver ...
Proofs as Efficient Programs - Dipartimento di Informatica
... approaches – simple modifications to a general framework allow for the semantical description (and sometimes also for syntactical results, like soundness) of a wide spectrum of formal systems. 2.1 Context Semantics Context semantics [26] is a powerful framework for the analysis of proof and program ...
... approaches – simple modifications to a general framework allow for the semantical description (and sometimes also for syntactical results, like soundness) of a wide spectrum of formal systems. 2.1 Context Semantics Context semantics [26] is a powerful framework for the analysis of proof and program ...
Non-classical metatheory for non-classical logics
... which classical logic is provably sound and complete by its own lights. In order to meet the challenge in a non-classical setting, I propose that we investigate the prospects of a faithful model theory for the non-classical logic. One requirement a faithful model theory must meet is to be able to d ...
... which classical logic is provably sound and complete by its own lights. In order to meet the challenge in a non-classical setting, I propose that we investigate the prospects of a faithful model theory for the non-classical logic. One requirement a faithful model theory must meet is to be able to d ...
Document
... The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. For example, assuming both P and ~P, any arbitrary proposition, Q, can be proved as follows: from P we obtain P Q by the rule of Addition; then from P Q and ~ ...
... The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. For example, assuming both P and ~P, any arbitrary proposition, Q, can be proved as follows: from P we obtain P Q by the rule of Addition; then from P Q and ~ ...
Notes on `the contemporary conception of logic`
... statement forms in ∼ and ⊃ which we’ve previously added to informal mathematical English, but which are now treated as the sole wffs of a special formal syntax. Then we are introduced to some formal ‘proof’ apparatus – i.e. a specification of a class of wffs as axioms, a designated ‘rule of inferenc ...
... statement forms in ∼ and ⊃ which we’ve previously added to informal mathematical English, but which are now treated as the sole wffs of a special formal syntax. Then we are introduced to some formal ‘proof’ apparatus – i.e. a specification of a class of wffs as axioms, a designated ‘rule of inferenc ...
Classical First-Order Logic Introduction
... Free and bound variables The free variables of a formula φ are those variables occurring in φ that are not quantified. FV(φ) denotes the set of free variables occurring in φ. The bound variables of a formula φ are those variables occurring in φ that do have quantifiers. BV(φ) denote the set of boun ...
... Free and bound variables The free variables of a formula φ are those variables occurring in φ that are not quantified. FV(φ) denotes the set of free variables occurring in φ. The bound variables of a formula φ are those variables occurring in φ that do have quantifiers. BV(φ) denote the set of boun ...
Systems of modal logic - Department of Computing
... M, w |= 3A ⇔ R[w] ∩ kAkM 6= ∅ where R[w] =def {t in M : w R t}. R[w] is the set of worlds accessible from w. ...
... M, w |= 3A ⇔ R[w] ∩ kAkM 6= ∅ where R[w] =def {t in M : w R t}. R[w] is the set of worlds accessible from w. ...
Predicate Logic for Software Engineering
... If P and Q are predicate expressions: 1. (xk , P), is the set of all assignments, A, if c is any value in U, A[k c] is in the denotation of P 2. (P) (Q) is the union of P and Q 3. (P) (Q) is the intersection of P and Q, and 4. ¬(P) is the set of all members of Su that are not in P ...
... If P and Q are predicate expressions: 1. (xk , P), is the set of all assignments, A, if c is any value in U, A[k c] is in the denotation of P 2. (P) (Q) is the union of P and Q 3. (P) (Q) is the intersection of P and Q, and 4. ¬(P) is the set of all members of Su that are not in P ...
Logic
... Logic is about how to deduce, on mere form, a valid argument. Valid is a semantic concept. Deduction is a syntactic concept. ...
... Logic is about how to deduce, on mere form, a valid argument. Valid is a semantic concept. Deduction is a syntactic concept. ...
The modal logic of equilibrium models
... for every w, if wRT wT and wRT w0T then wT = w0T ; (heredity) for every w, u, if wRS u then Vu ⊆ Vw ; (fullpast) for every w, for every finite P, Q ⊆ Vw such that P is nonempty, there is u such that: wRS u, Vu ∩ P = ∅ and Q ⊆ Vu ; (mtrans) for every w, u, wT , if wRS u and uRT wT then wRT wT ; (wcon ...
... for every w, if wRT wT and wRT w0T then wT = w0T ; (heredity) for every w, u, if wRS u then Vu ⊆ Vw ; (fullpast) for every w, for every finite P, Q ⊆ Vw such that P is nonempty, there is u such that: wRS u, Vu ∩ P = ∅ and Q ⊆ Vu ; (mtrans) for every w, u, wT , if wRS u and uRT wT then wRT wT ; (wcon ...
x - Stanford University
... A Correct Proof ∀a. ∀b. ∀c. (aR-1b ∧ bR-1c → aR-1c) Theorem: If R is transitive, then R-1 is transitive. Proof: Consider any a, b, and c such that aR-1b and bR-1c. We will prove aR-1c. Since aR-1b and bR-1c, we have that bRa and cRb. Since cRb and bRa, by transitivity we know cRa. Since cRa, we hav ...
... A Correct Proof ∀a. ∀b. ∀c. (aR-1b ∧ bR-1c → aR-1c) Theorem: If R is transitive, then R-1 is transitive. Proof: Consider any a, b, and c such that aR-1b and bR-1c. We will prove aR-1c. Since aR-1b and bR-1c, we have that bRa and cRb. Since cRb and bRa, by transitivity we know cRa. Since cRa, we hav ...
Propositional Logic Syntax of Propositional Logic
... • checking a set of sentences for satisfiability is NP-complete – but there are some circumstances where the proof only involves a small subset of the KB, so can do some of the work in polynomial time – if a KB is monotonic (i.e., even if we add new sentences to a KB, all the sentences entailed by t ...
... • checking a set of sentences for satisfiability is NP-complete – but there are some circumstances where the proof only involves a small subset of the KB, so can do some of the work in polynomial time – if a KB is monotonic (i.e., even if we add new sentences to a KB, all the sentences entailed by t ...
PDF
... D has n ’s, and so by induction, ` D[B/p] ↔ D[C/p], and therefore ` D[B/p] ↔ D[C/p] by 2. This means that ` A[B/p] ↔ A[C/p]. 4. ` A → B implies ` A → B Proof. By assumption, tautology ` (A → B) → (¬B → ¬A), and modus ponens, we get ` ¬B → ¬A. By 1, ` ¬B → ¬A. By another instance of the above ...
... D has n ’s, and so by induction, ` D[B/p] ↔ D[C/p], and therefore ` D[B/p] ↔ D[C/p] by 2. This means that ` A[B/p] ↔ A[C/p]. 4. ` A → B implies ` A → B Proof. By assumption, tautology ` (A → B) → (¬B → ¬A), and modus ponens, we get ` ¬B → ¬A. By 1, ` ¬B → ¬A. By another instance of the above ...
Lecture 3
... all the ambiguities of English. So, we need a separate ‘logical language’ where many of the ambiguities are filtered out. • We start with propositional logic. • There are several mathematical systems (calculi) for reasoning about propositions: – Truth tables (semantic), Equational Logic (syntactic/s ...
... all the ambiguities of English. So, we need a separate ‘logical language’ where many of the ambiguities are filtered out. • We start with propositional logic. • There are several mathematical systems (calculi) for reasoning about propositions: – Truth tables (semantic), Equational Logic (syntactic/s ...
PREPOSITIONAL LOGIS
... • Logical inference creates new sentences that logically follow from a set of sentences (KB) • An inference rule is sound if every sentence X it produces when operating on a KB logically follows from the KB –i.e., inference rule creates no contradictions • An inference rule is complete if it can pro ...
... • Logical inference creates new sentences that logically follow from a set of sentences (KB) • An inference rule is sound if every sentence X it produces when operating on a KB logically follows from the KB –i.e., inference rule creates no contradictions • An inference rule is complete if it can pro ...
Available on-line - Gert
... more adequate account of safety than Anderson’s system did, at least in the particular examples that we have discussed. But we cannot conclude from this that it is generally more adequate. The basic problem is that safety is a complicated and unclear concept with many connotations. The Oxford Englis ...
... more adequate account of safety than Anderson’s system did, at least in the particular examples that we have discussed. But we cannot conclude from this that it is generally more adequate. The basic problem is that safety is a complicated and unclear concept with many connotations. The Oxford Englis ...
Propositional Logic
... We call the literals B and ¬B complementary. The resolution rule deletes a pair of complementary literals from the two clauses and combines the rest of the literals into a new clause. To prove that from a knowledge base KB, a query Q follows, we carry out a proof by contradiction. Following Theorem ...
... We call the literals B and ¬B complementary. The resolution rule deletes a pair of complementary literals from the two clauses and combines the rest of the literals into a new clause. To prove that from a knowledge base KB, a query Q follows, we carry out a proof by contradiction. Following Theorem ...
Discrete Mathematics - Lecture 4: Propositional Logic and Predicate
... USA where every department has at least 20 faculty and at least one noble laureate.” A. There is an university in USA where every department has less than 20 faculty and at least one noble laureate. B. All universities in USA where every department has at least 20 faculty and at least one noble laur ...
... USA where every department has at least 20 faculty and at least one noble laureate.” A. There is an university in USA where every department has less than 20 faculty and at least one noble laureate. B. All universities in USA where every department has at least 20 faculty and at least one noble laur ...
INTRODUCTION TO LOGIC Natural Deduction
... can be derived from the premisses using the specified rules. The notion of proof can be precisely defined. In cases of disagreement, one can always break down an argument into elementary steps that are covered by these rules. The point is that all proofs could in principle be broken down into these ...
... can be derived from the premisses using the specified rules. The notion of proof can be precisely defined. In cases of disagreement, one can always break down an argument into elementary steps that are covered by these rules. The point is that all proofs could in principle be broken down into these ...
A brief introduction to Logic and its applications
... Another reason why one could not prove P ∨ ¬P ? When you prove a statement such as A ∨ B you can extract a proof that answers whether A or B holds. If we were able to prove the excluded middle, we could extract an algorithm that, given some proposition tells us whether it is valid or not (Curry-Howa ...
... Another reason why one could not prove P ∨ ¬P ? When you prove a statement such as A ∨ B you can extract a proof that answers whether A or B holds. If we were able to prove the excluded middle, we could extract an algorithm that, given some proposition tells us whether it is valid or not (Curry-Howa ...
Concept Hierarchies from a Logical Point of View
... Recall the standard definition of an interpretation within the framework of predicate logic: an interpretation of Σ consists of a universe U and a function that takes each monadic predicate p ∈ Σ to a subset of U . Now observe that a formal context hU, Σ, i uniquely corresponds to an interpretation ...
... Recall the standard definition of an interpretation within the framework of predicate logic: an interpretation of Σ consists of a universe U and a function that takes each monadic predicate p ∈ Σ to a subset of U . Now observe that a formal context hU, Σ, i uniquely corresponds to an interpretation ...
Document
... A formal system for describing states of affairs, consisting of syntax (how to make sentences) and semantics (to relate sentences to states of affairs). A proof theory - a set of rules for deducing the entailments of a set of sentences. ...
... A formal system for describing states of affairs, consisting of syntax (how to make sentences) and semantics (to relate sentences to states of affairs). A proof theory - a set of rules for deducing the entailments of a set of sentences. ...
Elements of Modal Logic - University of Victoria
... α1 . . . αn β A set Σ is said to be closed under an inference rule iff β ∈ Σ whenever all of the αi ’s are in Σ. Each system S determines a logic L(S), which is defined as the smallest set containing A that is closed under the rules of R. The logic pc has an associated system. Let Spc = (Apc , Rpc ) ...
... α1 . . . αn β A set Σ is said to be closed under an inference rule iff β ∈ Σ whenever all of the αi ’s are in Σ. Each system S determines a logic L(S), which is defined as the smallest set containing A that is closed under the rules of R. The logic pc has an associated system. Let Spc = (Apc , Rpc ) ...
The Foundations: Logic and Proofs
... In p →q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). ...
... In p →q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). ...
Notes on Propositional and Predicate Logic
... One important way of making proofs is using proof by contradiction. Suppose you have a set of premises Γ and a desired conclusion p. Let Γ0 be obtained by adding (not p) to Γ. If it is possible to prove two propositions q and (not q) from Γ0 , then one has a proof of p from Γ. The argument is that i ...
... One important way of making proofs is using proof by contradiction. Suppose you have a set of premises Γ and a desired conclusion p. Let Γ0 be obtained by adding (not p) to Γ. If it is possible to prove two propositions q and (not q) from Γ0 , then one has a proof of p from Γ. The argument is that i ...