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 ...
A short article for the Encyclopedia of Artificial Intelligence: Second
... over functions and predicates. Leibniz’s principle of equality, for example, states that two objects are to be taken as equal if they share the same properties; that is, a = b can be defined as ∀P [P (a) ≡ P (b)]. Of course, first-order logic is very strong and it is possible to encode such a statem ...
... over functions and predicates. Leibniz’s principle of equality, for example, states that two objects are to be taken as equal if they share the same properties; that is, a = b can be defined as ∀P [P (a) ≡ P (b)]. Of course, first-order logic is very strong and it is possible to encode such a statem ...
Logic - UNM Computer Science
... The discipline of logic aims to abstract our thought process and rigorously formalize the rules of inferences. In this course, we study logic to help us form valid arguments and construct correct proofs. However, it would be a mistake to try to convert everything into logic. Instead the correct appr ...
... The discipline of logic aims to abstract our thought process and rigorously formalize the rules of inferences. In this course, we study logic to help us form valid arguments and construct correct proofs. However, it would be a mistake to try to convert everything into logic. Instead the correct appr ...
pdf
... S4 is PSPACE-hard; and is PSPACE-complete for the modal logics K, T, and S4. He also showed that the satisfiability problem for S5 is NP-complete. What causes the gap between NP and PSPACE here? We show that, in a precise sense, it is the negative introspection axiom: ¬Kϕ ⇒ K¬Kϕ. It easily follows f ...
... S4 is PSPACE-hard; and is PSPACE-complete for the modal logics K, T, and S4. He also showed that the satisfiability problem for S5 is NP-complete. What causes the gap between NP and PSPACE here? We show that, in a precise sense, it is the negative introspection axiom: ¬Kϕ ⇒ K¬Kϕ. It easily follows f ...
Chapter 1 Logic and Set Theory
... P ⇔ Q holds if and only if P ⇒ Q and Q ⇒ P are both true. Being able to recognize that two statements are equivalent will become handy. It is sometime possible to demonstrate a result by finding an alternative, equivalent form of the statement that is easier to prove than the original form. A list o ...
... P ⇔ Q holds if and only if P ⇒ Q and Q ⇒ P are both true. Being able to recognize that two statements are equivalent will become handy. It is sometime possible to demonstrate a result by finding an alternative, equivalent form of the statement that is easier to prove than the original form. A list o ...
Propositional Logic: Normal Forms
... assign true to all marked atoms, and false to the others. If φ is not true under ν, it means that there exists a conjunct P1 ∧ . . . ∧ Pki → P 0 of φ that is false. By the semantics, this can only mean that P1 ∧ . . . ∧ Pki is true but P 0 is false. However, by the definition of ν, all Pi s are mark ...
... assign true to all marked atoms, and false to the others. If φ is not true under ν, it means that there exists a conjunct P1 ∧ . . . ∧ Pki → P 0 of φ that is false. By the semantics, this can only mean that P1 ∧ . . . ∧ Pki is true but P 0 is false. However, by the definition of ν, all Pi s are mark ...
Modal Logic
... for basic modal logic is quite general (although it can be further generalized as we will see later) and can be refined to yield the properties appropriate for the intended application. We will concentrate on three different applications: logic of necessity, temporal logic and logic of knowledge. T ...
... for basic modal logic is quite general (although it can be further generalized as we will see later) and can be refined to yield the properties appropriate for the intended application. We will concentrate on three different applications: logic of necessity, temporal logic and logic of knowledge. T ...
Propositional Dynamic Logic of Regular Programs*+
... Lauer [ll]) and the formula semantics is adopted from the relational semantics for modal logic introduced by Kripke [14]. Informally, each program a defines a relation p(u) between program states: (s, t) E p(a) if and only if a executed in state s can terminate in state t. The truth of an assertion ...
... Lauer [ll]) and the formula semantics is adopted from the relational semantics for modal logic introduced by Kripke [14]. Informally, each program a defines a relation p(u) between program states: (s, t) E p(a) if and only if a executed in state s can terminate in state t. The truth of an assertion ...
Module 31
... • If L1 and L2 are are regular languages, then L1*, L1L2, L1 union L2 are regular languages • Use previous constructions to see that these resulting languages are also context-free ...
... • If L1 and L2 are are regular languages, then L1*, L1L2, L1 union L2 are regular languages • Use previous constructions to see that these resulting languages are also context-free ...
this PDF file
... in the ≤k ordering in every model. We feel that this notion of necessary approximation carries some interest given the pivotal role of the approximation (or ‘knowledge’) ordering in the semantics of programming languages. The main purpose of this paper is a simple one. We want to add one more doubli ...
... in the ≤k ordering in every model. We feel that this notion of necessary approximation carries some interest given the pivotal role of the approximation (or ‘knowledge’) ordering in the semantics of programming languages. The main purpose of this paper is a simple one. We want to add one more doubli ...
![[Ch 3, 4] Logic and Proofs (2) 1. Valid and Invalid Arguments (§2.3](http://s1.studyres.com/store/data/014954007_1-d36c768aba23f0b4aa633cb9a2a27ee2-300x300.png)
Chapter 1 Logic and Set Theory
... R ∧ (R → S) and Q = S. Then, the truth of the statement P → Q depends only on the truth of external propositions R and S. The notion of implication can be rigorously defined as follows, P implies Q if the statement P → Q is a tautology. We abbreviate P implies Q by writing P ⇒ Q. It is important to ...
... R ∧ (R → S) and Q = S. Then, the truth of the statement P → Q depends only on the truth of external propositions R and S. The notion of implication can be rigorously defined as follows, P implies Q if the statement P → Q is a tautology. We abbreviate P implies Q by writing P ⇒ Q. It is important to ...
comments on the logic of constructible falsity (strong negation)
... Görnemann’s result suggests the conjecture that a classical model theory for the logic I have described may be obtained by allowing the domain to “grow with time”. This is in fact true. We may define a Nelson model structure as a triple (K, R, D), where K is a non-empty set of “stages of investigat ...
... Görnemann’s result suggests the conjecture that a classical model theory for the logic I have described may be obtained by allowing the domain to “grow with time”. This is in fact true. We may define a Nelson model structure as a triple (K, R, D), where K is a non-empty set of “stages of investigat ...
Document
... denotes “It is not the case that the earth is round,” or more simply “The earth is not round.” ...
... denotes “It is not the case that the earth is round,” or more simply “The earth is not round.” ...
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.