Logic seminar
... – “Snow is white and the sky is clear” – “If John is not at home, then Mary is at home” ...
... – “Snow is white and the sky is clear” – “If John is not at home, then Mary is at home” ...
Lecture 14 Notes
... Let ϕ be a mapping from the set of parameters to U. For a formula A define Aϕ to be the result of replacing every parameter ai in A by ϕ(ai ). We say that A is true under ϕ and v if v[Aϕ ] = t. The standard semantics of first-order formulas can be linked to the above as follows. Let E define the set ...
... Let ϕ be a mapping from the set of parameters to U. For a formula A define Aϕ to be the result of replacing every parameter ai in A by ϕ(ai ). We say that A is true under ϕ and v if v[Aϕ ] = t. The standard semantics of first-order formulas can be linked to the above as follows. Let E define the set ...
Propositional Logic
... In propositional logic there are two truth values: t for “true” and f for “false”. We begin with an example and ask ourselves whether the formula A ∧ B is true. The answer is: it depends on whether the variables A and B are true. For example, if A stands for “It is raining today” and B for “It is co ...
... In propositional logic there are two truth values: t for “true” and f for “false”. We begin with an example and ask ourselves whether the formula A ∧ B is true. The answer is: it depends on whether the variables A and B are true. For example, if A stands for “It is raining today” and B for “It is co ...
ARITHMETIC AND GEOMETRIC SEQUENCES
... to calculate any term in a sequence without knowing the previous term. For example, the tenth term in the sequence determined by the formula t n=2 n3 is 2(10) + 3, or 23 It is sometimes more convenient to calculate a term in a sequence from one or more previous terms in the sequence. Formulas that ...
... to calculate any term in a sequence without knowing the previous term. For example, the tenth term in the sequence determined by the formula t n=2 n3 is 2(10) + 3, or 23 It is sometimes more convenient to calculate a term in a sequence from one or more previous terms in the sequence. Formulas that ...
Powerpoint
... • So to find a50 I need to take d, which is 3, and add it to my a1, which is 2, 49 times. That’s a lot of adding. • But if we think back to elementary school, repetitive adding is just multiplication. ...
... • So to find a50 I need to take d, which is 3, and add it to my a1, which is 2, 49 times. That’s a lot of adding. • But if we think back to elementary school, repetitive adding is just multiplication. ...
(A B) |– A
... if T, A |– B and |– A, then T |– B. It is not necessary to state theorems in the assumptions. if A |– B, then T, A |– B. (Monotonicity of proving) if T |– A and T, A |– B, then T |– B. if T |– A and A |– B, then T |– B. if T |– A; T |– B; A, B |– C then T |– C. if T |– A and T |– B, then T |– A B. ...
... if T, A |– B and |– A, then T |– B. It is not necessary to state theorems in the assumptions. if A |– B, then T, A |– B. (Monotonicity of proving) if T |– A and T, A |– B, then T |– B. if T |– A and A |– B, then T |– B. if T |– A; T |– B; A, B |– C then T |– C. if T |– A and T |– B, then T |– A B. ...
CC-20CC-2 - Reeths
... A sequence is a function whose domain is the natural numbers, and whose outputs are the terms of the sequence. You can write a sequence using a recursive formula. A recursive formula is a function rule that relates each term of a sequence after the first to the ones before it. Consider the sequence ...
... A sequence is a function whose domain is the natural numbers, and whose outputs are the terms of the sequence. You can write a sequence using a recursive formula. A recursive formula is a function rule that relates each term of a sequence after the first to the ones before it. Consider the sequence ...
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 ...
Quantification - Rutgers Philosophy
... Those who find it odd that the universal generalization (∀x)(R(x)→B(x)) is not logically stronger than the existential generalization (∃x)(R(x)∧B(x)) will perhaps be mollified by the observation that the true universal correlate of (∃x)(R(x)∧B(x)) is not (∀x)(R(x)→B(x)) but rather the sentence (∀x)( ...
... Those who find it odd that the universal generalization (∀x)(R(x)→B(x)) is not logically stronger than the existential generalization (∃x)(R(x)∧B(x)) will perhaps be mollified by the observation that the true universal correlate of (∃x)(R(x)∧B(x)) is not (∀x)(R(x)→B(x)) but rather the sentence (∀x)( ...
EVERYONE KNOWS THAT SOMEONE KNOWS
... 5. π : P → P(W ) is a function that maps propositional variables into sets of epistemic worlds. In this article, we write u ∼X v if u ∼x v for each x ∈ X. The next definition introduces the update operation on an arbitrary function. This operation changes the value of the function at a single point. ...
... 5. π : P → P(W ) is a function that maps propositional variables into sets of epistemic worlds. In this article, we write u ∼X v if u ∼x v for each x ∈ X. The next definition introduces the update operation on an arbitrary function. This operation changes the value of the function at a single point. ...
Exam 2 Sample
... 6. (6 pts) Consider a relation R on the set of all living people on Earth, where x R y means "x is y's parent." For example, if Phil is Sandy's parent, then Phil relates to Sandy, i.e., "Phil R Sandy" is true. a. Is R a transitive relation? ________ Explain: b. Use plain English to describe the inve ...
... 6. (6 pts) Consider a relation R on the set of all living people on Earth, where x R y means "x is y's parent." For example, if Phil is Sandy's parent, then Phil relates to Sandy, i.e., "Phil R Sandy" is true. a. Is R a transitive relation? ________ Explain: b. Use plain English to describe the inve ...
On Gabbay`s temporal fixed point operator
... all i ≤ min(n, m). Note that in general, equivalence is not transitive; but the definition is no less useful for that. ...
... all i ≤ min(n, m). Note that in general, equivalence is not transitive; but the definition is no less useful for that. ...
2. First Order Logic 2.1. Expressions. Definition 2.1. A language L
... • ∀x∀y∀z x < y ∧ y < z → x < z, • ∀x∀y(x < y → ∃z x < z ∧ z < y), • ∀x∀y x < y → x 6= y, • ∃x∃y x < y. From these, we can easily deduce, for any n, ∃x1 ∃x2 · · · ∃xn (x1 6= x2 ∧ x2 6= x3 ∧ · · · ∧ x1 6= xn ∧ x2 6= x2 ∧ · · · ). In other words, the finite list of axioms above implies that the model i ...
... • ∀x∀y∀z x < y ∧ y < z → x < z, • ∀x∀y(x < y → ∃z x < z ∧ z < y), • ∀x∀y x < y → x 6= y, • ∃x∃y x < y. From these, we can easily deduce, for any n, ∃x1 ∃x2 · · · ∃xn (x1 6= x2 ∧ x2 6= x3 ∧ · · · ∧ x1 6= xn ∧ x2 6= x2 ∧ · · · ). In other words, the finite list of axioms above implies that the model i ...
Modal Logics Definable by Universal Three
... I Theorem 1. There exists a three-variable universal formula Γ0 , without equality, such that the local satisfiability problem for modal logic over KΓ0 is undecidable. Our formula, despite the fact that it uses much smaller number of variables, is also simpler than the formula from [8]. Actually, if ...
... I Theorem 1. There exists a three-variable universal formula Γ0 , without equality, such that the local satisfiability problem for modal logic over KΓ0 is undecidable. Our formula, despite the fact that it uses much smaller number of variables, is also simpler than the formula from [8]. Actually, if ...
A Uniform Proof Procedure for Classical and Non
... complementary connection has been identified all paths containing this connection are deleted. This is similar to Bibel’s connection method for classical logic [5] but without necessity for transforming the formula into any normal form. The algorithm avoids the notational redundancies occurring in s ...
... complementary connection has been identified all paths containing this connection are deleted. This is similar to Bibel’s connection method for classical logic [5] but without necessity for transforming the formula into any normal form. The algorithm avoids the notational redundancies occurring in s ...
Deciding Global Partial-Order Properties
... Partial order specifications are also interesting due to their compatibility with the so-called partial order reductions. The partial-order equivalence among sequences can be exploited to reduce the state-space explosion problem: the cost of generating at least one representative per equivalence cla ...
... Partial order specifications are also interesting due to their compatibility with the so-called partial order reductions. The partial-order equivalence among sequences can be exploited to reduce the state-space explosion problem: the cost of generating at least one representative per equivalence cla ...
slides - Computer and Information Science
... • If there are n different atomic propositions in some formula, then there are different lines in the truth table for that formula. (This is because each proposition can take one 1of 2 values—i.e., true or false.) • Let us write T for truth, and F for falsity. Then the truth table for p q is: ...
... • If there are n different atomic propositions in some formula, then there are different lines in the truth table for that formula. (This is because each proposition can take one 1of 2 values—i.e., true or false.) • Let us write T for truth, and F for falsity. Then the truth table for p q is: ...
Lecture 6e (Ordered Monoids and languages in 1 and 2 )
... reflexive and transitive but not anti-symmetric. As a matter of fact, its symmetric core (i.e) tpx, yq | x ďL y ^ y ďL xu is exactly the ”L relation. As with any pre-order, it induces a partial-order on the equivalence classes of its symmetric core. In what follows we write rxsL to denote the equiva ...
... reflexive and transitive but not anti-symmetric. As a matter of fact, its symmetric core (i.e) tpx, yq | x ďL y ^ y ďL xu is exactly the ”L relation. As with any pre-order, it induces a partial-order on the equivalence classes of its symmetric core. In what follows we write rxsL to denote the equiva ...
characterization of classes of frames in modal language
... Let us consider Kripke1 (relational) semantics hT,
... Let us consider Kripke1 (relational) semantics hT,
Definition - Rogelio Davila
... Definition. A formula is valid when every interpretation of it is a model. This formula is called a tautology. Definition . A formula is contingent when it is not valid, but consistent. ...
... Definition. A formula is valid when every interpretation of it is a model. This formula is called a tautology. Definition . A formula is contingent when it is not valid, but consistent. ...
When is Metric Temporal Logic Expressively Complete?
... Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany ...
... Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany ...
Arithmetic Series - Henry County Schools
... • So to find a50 I need to take d, which is 3, and add it to my a1, which is 2, 49 times. That’s a lot of adding. • But if we think back to elementary school, repetitive adding is just multiplication. ...
... • So to find a50 I need to take d, which is 3, and add it to my a1, which is 2, 49 times. That’s a lot of adding. • But if we think back to elementary school, repetitive adding is just multiplication. ...
1 Introduction 2 Formal logic
... • A semantics that explains the meaning of statements in our formal language in informal terms. • A deductive system that establishes formal rules of reasoning about logical statements which we can apply without having to constantly consider their informal explanation. It is important to remember th ...
... • A semantics that explains the meaning of statements in our formal language in informal terms. • A deductive system that establishes formal rules of reasoning about logical statements which we can apply without having to constantly consider their informal explanation. It is important to remember th ...
The strong completeness of the tableau method 1 The strong
... 2.1. DEFINITION. Let be an arbitrary set of formulas and any formula. We say that is deducible from by the first-order tableau method T (briefly, ├─T ) when there is a finite subset 0 of such that the set 0 { } can be confuted ─i.e., generates a closed tableau─. Notice that, by co ...
... 2.1. DEFINITION. Let be an arbitrary set of formulas and any formula. We say that is deducible from by the first-order tableau method T (briefly, ├─T ) when there is a finite subset 0 of such that the set 0 { } can be confuted ─i.e., generates a closed tableau─. Notice that, by co ...
Arithmetic Series
... • So to find a50 I need to take d, which is 3, and add it to my a1, which is 2, 49 times. That’s a lot of adding. • But if we think back to elementary school, repetitive adding is just multiplication. ...
... • So to find a50 I need to take d, which is 3, and add it to my a1, which is 2, 49 times. That’s a lot of adding. • But if we think back to elementary school, repetitive adding is just multiplication. ...