Knowledge Representation: Logic
... Logics can differ along the following dimensions Syntax: Influences readability, doesn’t change the expressive power Subset: Possible operators and combinations of operators, e.g. the logic ∃∧ or propositional calculus Proof theory: Restrictions on the permissible proofs (e.g. intuitionistic logic, ...
... Logics can differ along the following dimensions Syntax: Influences readability, doesn’t change the expressive power Subset: Possible operators and combinations of operators, e.g. the logic ∃∧ or propositional calculus Proof theory: Restrictions on the permissible proofs (e.g. intuitionistic logic, ...
Classical BI - UCL Computer Science
... Accordingly, the contexts Γ on the left-hand side of the sequents in the rules above are not sets or sequences, as in standard sequent calculi, but rather bunches: trees whose leaves are formulas and whose internal nodes are either ‘;’ or ‘,’ denoting respectively additive and multiplicative combina ...
... Accordingly, the contexts Γ on the left-hand side of the sequents in the rules above are not sets or sequences, as in standard sequent calculi, but rather bunches: trees whose leaves are formulas and whose internal nodes are either ‘;’ or ‘,’ denoting respectively additive and multiplicative combina ...
An Abridged Report - Association for the Advancement of Artificial
... part, we include in addition an infinite stock of predicate symbols of every arity, an infinite collection of (individual) variables, ...
... part, we include in addition an infinite stock of predicate symbols of every arity, an infinite collection of (individual) variables, ...
Justification logic with approximate conditional probabilities
... Fitting [16] provides a possible world semantics for justification logics. Based on this epistemic semantics, a much more general interpretation of t:α is possible, namely t is a justification for the agent’s belief (or knowledge) in α. This interpretation of justification logic has many application ...
... Fitting [16] provides a possible world semantics for justification logics. Based on this epistemic semantics, a much more general interpretation of t:α is possible, namely t is a justification for the agent’s belief (or knowledge) in α. This interpretation of justification logic has many application ...
Automated Deduction
... of assertions expressed in the universal language. Instead of arguing about a statement two persons in disagreement could then calculate who is right using this framework. This idea is embodied in the slogan calculemus!, let us calculate! Another aspect of automated deduction, the automation, was al ...
... of assertions expressed in the universal language. Instead of arguing about a statement two persons in disagreement could then calculate who is right using this framework. This idea is embodied in the slogan calculemus!, let us calculate! Another aspect of automated deduction, the automation, was al ...
HOARE`S LOGIC AND PEANO`S ARITHMETIC
... defined over the structure of the program, and it will be obvious that SP(p, S) can be effectively calculated from p and S. The fact that SP(p, S) does indeed define the strongest postcondition spN(p, S) on the standard model of arithmetic N will be a straightforward exercise whose interest or tedio ...
... defined over the structure of the program, and it will be obvious that SP(p, S) can be effectively calculated from p and S. The fact that SP(p, S) does indeed define the strongest postcondition spN(p, S) on the standard model of arithmetic N will be a straightforward exercise whose interest or tedio ...
Propositional Logic What is logic? Propositions Negation
... – If p is the proposition “ISE students love logic”, and q is the proposition “ISE students are crazy”, then – p ∧ q is the proposition “ISE students love logic and are crazy” – p ∨ q is the proposition “ISE students either love logic, or are crazy, or both” Note the syntax is different to that used ...
... – If p is the proposition “ISE students love logic”, and q is the proposition “ISE students are crazy”, then – p ∧ q is the proposition “ISE students love logic and are crazy” – p ∨ q is the proposition “ISE students either love logic, or are crazy, or both” Note the syntax is different to that used ...
Logic and Computation Lecture notes Jeremy Avigad Assistant Professor, Philosophy
... This document comprises a set of lecture notes that I prepared in the fall of 1998, while teaching a course called Logic and Computation in the Philosophy Department at Carnegie Mellon University. I distributed these notes to the class and then followed them almost word for word, in the hopes that d ...
... This document comprises a set of lecture notes that I prepared in the fall of 1998, while teaching a course called Logic and Computation in the Philosophy Department at Carnegie Mellon University. I distributed these notes to the class and then followed them almost word for word, in the hopes that d ...
PREDICATE LOGIC
... we want to indicate that A is true for all possible values of x, we write ∀ x A. Here ∀ x is called the universal quantifier, and A is called the scope of the quantifier. The variable x is said to be bound by the quantifier. The symbol ∀ is pronounced "for all". The quantifier and the bounded variab ...
... we want to indicate that A is true for all possible values of x, we write ∀ x A. Here ∀ x is called the universal quantifier, and A is called the scope of the quantifier. The variable x is said to be bound by the quantifier. The symbol ∀ is pronounced "for all". The quantifier and the bounded variab ...
(A B) |– A
... Or, In the third case C1 = A, and we are to prove A A (see example 1). b) Induction step: we prove that on the assumption of A Cn being proved for n = 1, 2, ..., i-1 the formula A Cn can be proved also for n = i. For Ci there are four cases: 1. Ci is an assumption of Ai, 2. Ci is an axiom, 3. ...
... Or, In the third case C1 = A, and we are to prove A A (see example 1). b) Induction step: we prove that on the assumption of A Cn being proved for n = 1, 2, ..., i-1 the formula A Cn can be proved also for n = i. For Ci there are four cases: 1. Ci is an assumption of Ai, 2. Ci is an axiom, 3. ...
classden
... interest is in stating and proving metatheorems on completeness, decidability etc. Such aims are best served with logics with a limited expressivity, as increase in expressivity generally leads to loss of metalogical properties. On the other hand, the goal of providing a general logical framework f ...
... interest is in stating and proving metatheorems on completeness, decidability etc. Such aims are best served with logics with a limited expressivity, as increase in expressivity generally leads to loss of metalogical properties. On the other hand, the goal of providing a general logical framework f ...
Fuzzy logic and probability Institute of Computer Science (ICS
... In our opinion any serious discussion on the relation between fuzzy logic and probability must start by mak ing clear the basic differences. Admitting some simpli fication, we cotL'>ider that fuzzy logic is a logic of vague, imprecise notions and propositions, propositions that may be more or less ...
... In our opinion any serious discussion on the relation between fuzzy logic and probability must start by mak ing clear the basic differences. Admitting some simpli fication, we cotL'>ider that fuzzy logic is a logic of vague, imprecise notions and propositions, propositions that may be more or less ...
Lesson 12
... The relation of provability (A1,...,An |– A) and the relation of logical entailment (A1,...,An |= A) are distinct relations. Similarly, the set of theorems |– A (of a calculus) is generally not identical to the set of logically valid formulas |= A. The former is a syntactic issue and defined within ...
... The relation of provability (A1,...,An |– A) and the relation of logical entailment (A1,...,An |= A) are distinct relations. Similarly, the set of theorems |– A (of a calculus) is generally not identical to the set of logically valid formulas |= A. The former is a syntactic issue and defined within ...
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 ...
Failures of Categoricity and Compositionality for
... meanings for the connectives in terms of this interpretation. For example, if we think of an assignments of T and F as representing stages of verification as in the typical presentation of intuitionistic semantics in terms of Kripke models, we can use Garson’s results to derive standard intuitionist ...
... meanings for the connectives in terms of this interpretation. For example, if we think of an assignments of T and F as representing stages of verification as in the typical presentation of intuitionistic semantics in terms of Kripke models, we can use Garson’s results to derive standard intuitionist ...
3.6 First-Order Tableau
... interpretation A0 that is identical to A, except cA = a for the fresh constant c of the δ(c) descendant. The constant c does not occur in (φ1 , . . . , ψ, . . . , φn ), so A0 |= φ1 ∧ . . . ∧ ψ ∧ . . . ∧ φn and by construction A0 |= ψ 0 , for the δ(c) descendant ψ 0 . The case ψ = ¬∀xS .χ can be show ...
... interpretation A0 that is identical to A, except cA = a for the fresh constant c of the δ(c) descendant. The constant c does not occur in (φ1 , . . . , ψ, . . . , φn ), so A0 |= φ1 ∧ . . . ∧ ψ ∧ . . . ∧ φn and by construction A0 |= ψ 0 , for the δ(c) descendant ψ 0 . The case ψ = ¬∀xS .χ can be show ...
Specification Predicates with Explicit Dependency Information
... The notation introduced above provides a concise way to characterise sets of locations. Now we extend the names of non-rigid (predicate and function) symbols by qualifications in the form of location descriptors. The idea is that the value of thus qualified symbols depends at most on the values of t ...
... The notation introduced above provides a concise way to characterise sets of locations. Now we extend the names of non-rigid (predicate and function) symbols by qualifications in the form of location descriptors. The idea is that the value of thus qualified symbols depends at most on the values of t ...
Introduction to Artificial Intelligence
... In propositional logic there are two truth values: t for “true” and f for “false”. Is a formula, such as A ∧ B true? The answer depends on whether the variables A and B are true. Example: If A stands for “It is raining today” and B for “It is cold today” and these are both true, then A ∧ B is true. ...
... In propositional logic there are two truth values: t for “true” and f for “false”. Is a formula, such as A ∧ B true? The answer depends on whether the variables A and B are true. Example: If A stands for “It is raining today” and B for “It is cold today” and these are both true, then A ∧ B is true. ...
On Provability Logic
... The non-existence of “the modal logic” can be explained by the fact that nested modalities are rare in natural language. We seldom say that it is necessary that something is possible and thus there is no agreement whether for instance the modal propositional formula 3p → 23p should be accepted as a ...
... The non-existence of “the modal logic” can be explained by the fact that nested modalities are rare in natural language. We seldom say that it is necessary that something is possible and thus there is no agreement whether for instance the modal propositional formula 3p → 23p should be accepted as a ...
On Provability Logic
... (which is provable in PA) can be read the number three is a prime. The term S(S(S(0))) is denoted 3. More generally, the n-th numeral is defined as the term S(S . . (0) . .) with n occurrence of the symbol S. As an exercise we suggest the reader to formulate the fact that there are infinitely many p ...
... (which is provable in PA) can be read the number three is a prime. The term S(S(S(0))) is denoted 3. More generally, the n-th numeral is defined as the term S(S . . (0) . .) with n occurrence of the symbol S. As an exercise we suggest the reader to formulate the fact that there are infinitely many p ...
(A B) |– A
... The relation of provability (A1,...,An |– A) and the relation of logical entailment (A1,...,An |= A) are distinct relations. Similarly, the set of theorems |– A (of a calculus) is generally not identical to the set of logically valid formulas |= A. The former is a syntactic issue and defined within ...
... The relation of provability (A1,...,An |– A) and the relation of logical entailment (A1,...,An |= A) are distinct relations. Similarly, the set of theorems |– A (of a calculus) is generally not identical to the set of logically valid formulas |= A. The former is a syntactic issue and defined within ...
duality of quantifiers ¬8xA(x) 9x¬A(x) ¬9xA(x) 8x¬A(x)
... Wolves, foxes, birds, caterpillars, and snails are animals, and there are some of each of them. Also there are some grains, and grains are plants. Every animal either likes to eat all plants or all animals much smaller than itself that like to eat some plants. Caterpillars and snails are much smalle ...
... Wolves, foxes, birds, caterpillars, and snails are animals, and there are some of each of them. Also there are some grains, and grains are plants. Every animal either likes to eat all plants or all animals much smaller than itself that like to eat some plants. Caterpillars and snails are much smalle ...
Finite-variable fragments of first
... A terminological note: in books on model theory, the word “type” is standardly used to refer to a maximal consistent set of formulas (over some signature) featuring a fixed collection of variables—including formulas involving quantifiers. What we are calling types here are known, in that nomenclatur ...
... A terminological note: in books on model theory, the word “type” is standardly used to refer to a maximal consistent set of formulas (over some signature) featuring a fixed collection of variables—including formulas involving quantifiers. What we are calling types here are known, in that nomenclatur ...
De Jongh`s characterization of intuitionistic propositional calculus
... Moreover, it can be shown that for every point w in T (n), there exists a point v ∈ U (n) such that wRv. In other words, universal models are “upper parts” of Henkin models. As we saw in Corollary 2.2, the n-universal model of IPC carries all the information about the formulas in n-variables. Unfort ...
... Moreover, it can be shown that for every point w in T (n), there exists a point v ∈ U (n) such that wRv. In other words, universal models are “upper parts” of Henkin models. As we saw in Corollary 2.2, the n-universal model of IPC carries all the information about the formulas in n-variables. Unfort ...
Elementary Logic
... How to read the logical connectives. ¬ (negation): not ∧ (conjunction): and ∨ (disjunction): or → (implication): implies (or if . . . , then . . . ) ↔ (equivalence): is equivalent to (or if and only if) ⊥ (false or bottom): false (or bottom) ...
... How to read the logical connectives. ¬ (negation): not ∧ (conjunction): and ∨ (disjunction): or → (implication): implies (or if . . . , then . . . ) ↔ (equivalence): is equivalent to (or if and only if) ⊥ (false or bottom): false (or bottom) ...