Multi-Agent Only
... If Alice believes that all that Bob knows is that birds normally fly and that Tweety is a bird, then Alice believes that Bob believes that Tweety flies. But technically things were surprisingly cumbersome! The problem lies in the complexity in what agents consider ...
... If Alice believes that all that Bob knows is that birds normally fly and that Tweety is a bird, then Alice believes that Bob believes that Tweety flies. But technically things were surprisingly cumbersome! The problem lies in the complexity in what agents consider ...
Natural Deduction Calculus for Quantified Propositional Linear
... While the propositional quantification does not add any expressiveness to the classical logic QPTL is more expressive than PLTL presenting the same potential of expressiveness as linear-time µ-calculus (linear-time propositional temporal fixpoint logic) [Kaivola (1997)], ETL (propositional linear-ti ...
... While the propositional quantification does not add any expressiveness to the classical logic QPTL is more expressive than PLTL presenting the same potential of expressiveness as linear-time µ-calculus (linear-time propositional temporal fixpoint logic) [Kaivola (1997)], ETL (propositional linear-ti ...
Notes Predicate Logic
... it is quite common to drop the quantification when we state/write theorems. In this case, the theorem could be written: If x is rational, then x is real. In general, a univeral quantification asserts that if an element is within an understood universe or within a specified domain, then the statement ...
... it is quite common to drop the quantification when we state/write theorems. In this case, the theorem could be written: If x is rational, then x is real. In general, a univeral quantification asserts that if an element is within an understood universe or within a specified domain, then the statement ...
Sub-Birkhoff
... name with its rule is called an axiom. Subequational logics generate subequational theories. Definition 2 For a subequational logic L = hS,Ii its theory L is generated by the following inference rules, where an inference rule (i) only applies if i ∈ I. s, t and r range over terms. `sLs ...
... name with its rule is called an axiom. Subequational logics generate subequational theories. Definition 2 For a subequational logic L = hS,Ii its theory L is generated by the following inference rules, where an inference rule (i) only applies if i ∈ I. s, t and r range over terms. `sLs ...
Normalised and Cut-free Logic of Proofs
... From a Gentzen-style point of view, we can formulate two similar sequent calculi for the two systems Lp and Ilp, respectively (see Artemov [2002]). Although simple and cut-free, these sequent calculi fail to satisfy certain properties that are standardly required from a “good" sequent calculus (in P ...
... From a Gentzen-style point of view, we can formulate two similar sequent calculi for the two systems Lp and Ilp, respectively (see Artemov [2002]). Although simple and cut-free, these sequent calculi fail to satisfy certain properties that are standardly required from a “good" sequent calculus (in P ...
term 1 - Teaching-WIKI
... • As usual for a Logic, FOL is defined in several parts: – Syntax: Telling us how well-formed formulae can be built from basic syntactic parts – A proof theory, consisting of: • Inference rules:Telling us what kind of conclusions are admissable base on a set of propositions ((i.e. Modus ponens. see ...
... • As usual for a Logic, FOL is defined in several parts: – Syntax: Telling us how well-formed formulae can be built from basic syntactic parts – A proof theory, consisting of: • Inference rules:Telling us what kind of conclusions are admissable base on a set of propositions ((i.e. Modus ponens. see ...
Sequent calculus - Wikipedia, the free encyclopedia
... operate on the structure of the sequents, ignoring the exact shape of the formulae. The two exceptions to this general scheme are the axiom of identity (I) and the rule of (Cut). Although stated in a formal way, the above rules allow for a very intuitive reading in terms of classical logic. Consider ...
... operate on the structure of the sequents, ignoring the exact shape of the formulae. The two exceptions to this general scheme are the axiom of identity (I) and the rule of (Cut). Although stated in a formal way, the above rules allow for a very intuitive reading in terms of classical logic. Consider ...
4 slides/page
... • variables: x, y, z, . . . • predicate symbols of each arity: P , Q, R, . . . ◦ A unary predicate symbol takes one argument: P (Alice), Q(z) ◦ A binary predicate symbol takes two arguments: Loves(Bob,Alice), Taller(Alice,Bob). An atomic expression is a predicate symbol together with the appropriate ...
... • variables: x, y, z, . . . • predicate symbols of each arity: P , Q, R, . . . ◦ A unary predicate symbol takes one argument: P (Alice), Q(z) ◦ A binary predicate symbol takes two arguments: Loves(Bob,Alice), Taller(Alice,Bob). An atomic expression is a predicate symbol together with the appropriate ...
2 Lab 2 – October 10th, 2016
... U is the set of real numbers, P is interpreted as the property “to be a square root of −1”. Since no real number has the property I(P ), our sentence is true in hU, [[−]]i. On the other hand, consider the interpretation: U 0 is the set of natural numbers, P is interpreted as the property “to be even ...
... U is the set of real numbers, P is interpreted as the property “to be a square root of −1”. Since no real number has the property I(P ), our sentence is true in hU, [[−]]i. On the other hand, consider the interpretation: U 0 is the set of natural numbers, P is interpreted as the property “to be even ...
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 ...
SECOND-ORDER LOGIC, OR - University of Chicago Math
... need to know how we can put those symbols together. Just as we cannot say in English “Water kill John notorious ponder,” we want to rule out pseudo-formulas like “∀(→ x12 ∨.” We therefore define a well-formed formula by formation rules. I omit the details, which are tedious and can be found in any c ...
... need to know how we can put those symbols together. Just as we cannot say in English “Water kill John notorious ponder,” we want to rule out pseudo-formulas like “∀(→ x12 ∨.” We therefore define a well-formed formula by formation rules. I omit the details, which are tedious and can be found in any c ...
Lecture_ai_3 - WordPress.com
... • Interpretation of implication is T if the previous statement has T value • Interpretation of Biconditionalis T only when symbols on the both sides are either T or F ,otherwise F ...
... • Interpretation of implication is T if the previous statement has T value • Interpretation of Biconditionalis T only when symbols on the both sides are either T or F ,otherwise F ...
Lecturecise 19 Proofs and Resolution Compactness for
... then interpretation makes it true. Moreover, all other formulas in T are propositional variables set to true, so the interpretation makes T true. Thus, we see that the inductively proved statement holds even in this case. What the infinite formula D breaks is the second part, which, from the existen ...
... then interpretation makes it true. Moreover, all other formulas in T are propositional variables set to true, so the interpretation makes T true. Thus, we see that the inductively proved statement holds even in this case. What the infinite formula D breaks is the second part, which, from the existen ...
PDF
... Let FO(Σ) be a first order language over signature Σ. Recall that the axioms for FO(Σ) are (universal) generalizations of wff’s belonging to one of the following six schemas: 1. A → (B → A) 2. (A → (B → C)) → ((A → B) → (A → C)) 3. ¬¬A → A 4. ∀x(A → B) → (∀xA → ∀xB), where x ∈ V 5. A → ∀xA, where x ...
... Let FO(Σ) be a first order language over signature Σ. Recall that the axioms for FO(Σ) are (universal) generalizations of wff’s belonging to one of the following six schemas: 1. A → (B → A) 2. (A → (B → C)) → ((A → B) → (A → C)) 3. ¬¬A → A 4. ∀x(A → B) → (∀xA → ∀xB), where x ∈ V 5. A → ∀xA, where x ...
study guide.
... • There are two main normal forms for the propositional formulas. One is called Conjunctive normal form (CNF) and is an ∧ of ∨ of either variables or their negations (here, by ∧ and ∨ we mean several formulas with ∧ between each pair, as in (¬x ∨ y ∨ z) ∧ (¬u ∨ y) ∧ x. A literal is a variable or its ...
... • There are two main normal forms for the propositional formulas. One is called Conjunctive normal form (CNF) and is an ∧ of ∨ of either variables or their negations (here, by ∧ and ∨ we mean several formulas with ∧ between each pair, as in (¬x ∨ y ∨ z) ∧ (¬u ∨ y) ∧ x. A literal is a variable or its ...
WhichQuantifiersLogical
... and quantifiers is to be established semantically in one way or another prior to their inferential role. Their meanings may be the primitives of our reasoning in general“and”, “or”, “not”, “if…then”, “all”, “some”or they may be understood informally like “most”, “has the same number as”, etc. in a ...
... and quantifiers is to be established semantically in one way or another prior to their inferential role. Their meanings may be the primitives of our reasoning in general“and”, “or”, “not”, “if…then”, “all”, “some”or they may be understood informally like “most”, “has the same number as”, etc. in a ...
4 slides/page
... • probabilistic logic: for reasoning about probability • temporal logic: for reasoning about time (and programs) • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the f ...
... • probabilistic logic: for reasoning about probability • temporal logic: for reasoning about time (and programs) • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the f ...
the common rules of binary connectives are finitely based
... A propositional logic (here a standard consequence relation ` in a given propositional language) is said to be f.b. (finitely based) if all its sequential rules derive from a finite subset. A binary propositional connective is proper if it depends on both arguments. Of the 16 binary connectives, 10 ...
... A propositional logic (here a standard consequence relation ` in a given propositional language) is said to be f.b. (finitely based) if all its sequential rules derive from a finite subset. A binary propositional connective is proper if it depends on both arguments. Of the 16 binary connectives, 10 ...
PROVING UNPROVABILITY IN SOME NORMAL MODAL LOGIC
... normal modal logics, turns out redundant in many cases including all considered here. Also let us note that the rule RS can be specified (as it can be seen from the proofs below) in all considered cases as follows: it is enough to admit only 2-free substitutions, i.e. such that every variable is sub ...
... normal modal logics, turns out redundant in many cases including all considered here. Also let us note that the rule RS can be specified (as it can be seen from the proofs below) in all considered cases as follows: it is enough to admit only 2-free substitutions, i.e. such that every variable is sub ...
Relational Predicate Logic
... When the overlapping quantifiers are of the same types, the order in which they occur is not relevant to the meanings of the sentences. But when an existential and a universal quantifier are both involved, order becomes crucial. ...
... When the overlapping quantifiers are of the same types, the order in which they occur is not relevant to the meanings of the sentences. But when an existential and a universal quantifier are both involved, order becomes crucial. ...
chapter 16
... Each function has an arity, which is the number of symbolic terms required by the function in order for it to form a symbolic term. The functions of our language are: f, g, h, i…. There are two quantifiers. — ∀, the universal quantifier. — ∃, the existential quantifier. The connectives are the same ...
... Each function has an arity, which is the number of symbolic terms required by the function in order for it to form a symbolic term. The functions of our language are: f, g, h, i…. There are two quantifiers. — ∀, the universal quantifier. — ∃, the existential quantifier. The connectives are the same ...
Predicate Calculus pt. 2
... Exercise 1 A set of propositional formulas T is called satisfiable iff there is an assignment of the occuring variables which makes all formulas in T true. The compactness theorem of propositional logic says: T is satisfiable iff every finite subset of T is satisfiable. Proof the compactness theorem ...
... Exercise 1 A set of propositional formulas T is called satisfiable iff there is an assignment of the occuring variables which makes all formulas in T true. The compactness theorem of propositional logic says: T is satisfiable iff every finite subset of T is satisfiable. Proof the compactness theorem ...
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 ...
Lecture 11 Artificial Intelligence Predicate Logic
... from the statement that are already known. • It provides a way of deducing new statements ...
... from the statement that are already known. • It provides a way of deducing new statements ...