Knowledge Representation: Logic
... the formula may be read: For every x, if x is a car, then there exists s, where s is a set of count 4 and for every w, if w is a member of s, then w is a wheel and w is a part of x. Wojciech Jaworski (MIM UW) ...
... the formula may be read: For every x, if x is a car, then there exists s, where s is a set of count 4 and for every w, if w is a member of s, then w is a wheel and w is a part of x. Wojciech Jaworski (MIM UW) ...
Lectures on Proof Theory - Create and Use Your home.uchicago
... operation s 7→ P (s) α times, starting with the null set ∅. Then ordinary set theory is a theory of pure well-founded sets and its intended models are structures of the form hR(κ), ∈i, where the numbers κ will depend upon the particular axioms included in the theory. There is no appeal here to the e ...
... operation s 7→ P (s) α times, starting with the null set ∅. Then ordinary set theory is a theory of pure well-founded sets and its intended models are structures of the form hR(κ), ∈i, where the numbers κ will depend upon the particular axioms included in the theory. There is no appeal here to the e ...
Lecture Notes on Sequent Calculus
... We have already mentioned that antecedents in sequent proofs are persistent: once an assumption is made, it is henceforth usable above the inference that introduces it. Sequent proofs also obey the important subformula property: if we examine the complete or partial proof above a sequent, we observe ...
... We have already mentioned that antecedents in sequent proofs are persistent: once an assumption is made, it is henceforth usable above the inference that introduces it. Sequent proofs also obey the important subformula property: if we examine the complete or partial proof above a sequent, we observe ...
A System of Interaction and Structure
... calculus. The new formalism is more general than the sequent calculus for logics with involutive negation, like classical and linear logic, and allows a much more refined proof theory than possible in the sequent calculus, without sacrificing simplicity. In particular, rules in the calculus of struc ...
... calculus. The new formalism is more general than the sequent calculus for logics with involutive negation, like classical and linear logic, and allows a much more refined proof theory than possible in the sequent calculus, without sacrificing simplicity. In particular, rules in the calculus of struc ...
Answer Sets for Propositional Theories
... to see that for any formula F , (¬F )X , according to the new definition, is > when X 6|= F , and ⊥ otherwise, as with the traditional definition. Indeed, if X |= F then X 6|= F ⊃ ⊥ and consequently (¬F )X = (F ⊃ ⊥)X = ⊥. Otherwise, X |= F ⊃ ⊥, so that (¬F )X = (F ⊃ ⊥)X = F X ⊃ ⊥ = ⊥ ⊃ ⊥ = >. The fo ...
... to see that for any formula F , (¬F )X , according to the new definition, is > when X 6|= F , and ⊥ otherwise, as with the traditional definition. Indeed, if X |= F then X 6|= F ⊃ ⊥ and consequently (¬F )X = (F ⊃ ⊥)X = ⊥. Otherwise, X |= F ⊃ ⊥, so that (¬F )X = (F ⊃ ⊥)X = F X ⊃ ⊥ = ⊥ ⊃ ⊥ = >. The fo ...
Introduction to proposition
... two squares”. Logic is the basis of all mathematical reasoning. It has practical applications to the design of computing machines, to the specification of systems, to artificial intelligence, to computer programming, to programming languages, and to other areas of computer science, as well as too ma ...
... two squares”. Logic is the basis of all mathematical reasoning. It has practical applications to the design of computing machines, to the specification of systems, to artificial intelligence, to computer programming, to programming languages, and to other areas of computer science, as well as too ma ...
Reasoning about Action and Change
... (1993) (see also Kautz, 1982; Morreau, 1992). In the language of firstorder dynamic logic they propose frame assertions of format A ⊃ [α]A (where α is allowed to be a compound action). We extend this idea to the intermediate states of a plan (typically a sequential composition of actions). Loosely s ...
... (1993) (see also Kautz, 1982; Morreau, 1992). In the language of firstorder dynamic logic they propose frame assertions of format A ⊃ [α]A (where α is allowed to be a compound action). We extend this idea to the intermediate states of a plan (typically a sequential composition of actions). Loosely s ...
Easyprove: a tool for teaching precise reasoning
... Familiar setting. Many proof assistants are based on complex formalisms, which can be hard to understand to freshmen. For example, HOL and Coq use higher-order logics, and it is typical to encode sets as predicates within the logic. This is often advantageous for an expert user, but for a beginner i ...
... Familiar setting. Many proof assistants are based on complex formalisms, which can be hard to understand to freshmen. For example, HOL and Coq use higher-order logics, and it is typical to encode sets as predicates within the logic. This is often advantageous for an expert user, but for a beginner i ...
EXTRA CREDIT PROJECTS The following extra credit projects are
... In this project you will prove that the axiom of choice is equivalent to the statement that all surjective functions have right inverses. Since we already did part of this proof in class, it only remains to show that the existence of right inverses for surjective functions implies that the axiom of ...
... In this project you will prove that the axiom of choice is equivalent to the statement that all surjective functions have right inverses. Since we already did part of this proof in class, it only remains to show that the existence of right inverses for surjective functions implies that the axiom of ...
Examples of Natural Deduction
... • But in the logic problems I am using terms that include a negation: – cannot be wearing ...
... • But in the logic problems I am using terms that include a negation: – cannot be wearing ...
Belief closure: A semantics of common knowledge for
... 2. Syntactical definitions and facts The formal language and axiom systems discussed in this paper derive their special features from the fact that there are belief (knowledge) operators B a, one for each individual or 'agent' a, and, even more importantly, a specific operator C to render 'it is com ...
... 2. Syntactical definitions and facts The formal language and axiom systems discussed in this paper derive their special features from the fact that there are belief (knowledge) operators B a, one for each individual or 'agent' a, and, even more importantly, a specific operator C to render 'it is com ...
Inquiry
An inquiry is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim.