Introduction to Linear Logic
... defining functions such that a proof of a sequent Γ ` B gives rise to a function which assigns a proof of the formula B to a list of proofs proving the respective formulae in the context Γ. Note that tertium non datur, A ∨ ¬A, which distinguishes Classical Logic from Intuitionistic Logic, cannot be ...
... defining functions such that a proof of a sequent Γ ` B gives rise to a function which assigns a proof of the formula B to a list of proofs proving the respective formulae in the context Γ. Note that tertium non datur, A ∨ ¬A, which distinguishes Classical Logic from Intuitionistic Logic, cannot be ...
Majorana Fermions - Physics | Oregon State University
... • A real part and an imaginary part, each of which is an MF, respectively • Usually, these two states are spatially localized close to each other, and so cannot be addressed independently (i.e. you don’t see the familiar fermions exhibit this behavior) • In some cases, however, it should be possible ...
... • A real part and an imaginary part, each of which is an MF, respectively • Usually, these two states are spatially localized close to each other, and so cannot be addressed independently (i.e. you don’t see the familiar fermions exhibit this behavior) • In some cases, however, it should be possible ...
Introduction to Discrete Structures Introduction
... • There really aren’t ways to represent infinite sets by a computer since a computer has a finite amount of memory • If we assume that the universal set U is finite, then we can easily and effectively represent sets by bit vectors • Specifically, we force an ordering on the objects, say: U={a1, a2,… ...
... • There really aren’t ways to represent infinite sets by a computer since a computer has a finite amount of memory • If we assume that the universal set U is finite, then we can easily and effectively represent sets by bit vectors • Specifically, we force an ordering on the objects, say: U={a1, a2,… ...
Propositional Logic
... Two important properties for inference Soundness: If KB |- Q then KB |= Q – If Q is derived from a set of sentences KB using a given set of rules of inference, then Q is entailed by KB. – Hence, inference produces only real entailments, or any sentence that follows deductively from the premises is ...
... Two important properties for inference Soundness: If KB |- Q then KB |= Q – If Q is derived from a set of sentences KB using a given set of rules of inference, then Q is entailed by KB. – Hence, inference produces only real entailments, or any sentence that follows deductively from the premises is ...
Clausal Connection-Based Theorem Proving in
... For a combined substitution σ to be admissible (see Definition 7) the reduction ordering induced by σ has to be irreflexive. In classical logic this restriction encodes the Eigenvariable condition in the classical sequent calculus [8]. It is usually integrated into the σ-complementary test by usin ...
... For a combined substitution σ to be admissible (see Definition 7) the reduction ordering induced by σ has to be irreflexive. In classical logic this restriction encodes the Eigenvariable condition in the classical sequent calculus [8]. It is usually integrated into the σ-complementary test by usin ...
pdf
... The final important property of first-order logic that we have to investigate is compactness: Given a set F of first-order formulas, what does the satisfiability of finite subsets tell us about the satisfiability of the whole set. In propositional logic we have shown that a set S is uniformly satisf ...
... The final important property of first-order logic that we have to investigate is compactness: Given a set F of first-order formulas, what does the satisfiability of finite subsets tell us about the satisfiability of the whole set. In propositional logic we have shown that a set S is uniformly satisf ...
Certamen 1 de Representación del Conocimiento
... 12 de Octubre, 2012 (a) [1/2 pto] Define a FOL signature S = {Ω, Π} for which formulas in Σ are well-formed. Solution: Ω = {A/0, B/0} and Π = {R/2, P/2} (b) [1/2 pto] Show that Σ is valid (provide an interpretation for S). Solution: Consider the interpretation I = (U, AI , B I , RI , P I ) where U = ...
... 12 de Octubre, 2012 (a) [1/2 pto] Define a FOL signature S = {Ω, Π} for which formulas in Σ are well-formed. Solution: Ω = {A/0, B/0} and Π = {R/2, P/2} (b) [1/2 pto] Show that Σ is valid (provide an interpretation for S). Solution: Consider the interpretation I = (U, AI , B I , RI , P I ) where U = ...
Open Quantum System Studies of Optical Lattices and Nonlinear
... The advent of laser cooling and trapping of neutral atoms and ions [1] has revolutionized the field of atomic physics in several ways. From the standpoint of spectroscopy, the absorption lines of trapped, virtually stationary ultracold atoms or ions can be resolved without spectral lines undergoing ...
... The advent of laser cooling and trapping of neutral atoms and ions [1] has revolutionized the field of atomic physics in several ways. From the standpoint of spectroscopy, the absorption lines of trapped, virtually stationary ultracold atoms or ions can be resolved without spectral lines undergoing ...
Power Point Presentation
... Probability as judgments We assume that probabilities are not objective facts, but subjective judgments. Such a view is more flexible than theories based on logic or on frequencies. It makes sense, for example, that a doctor predicts for a smoker with an otherwise healthy lifestyle a somewhat smalle ...
... Probability as judgments We assume that probabilities are not objective facts, but subjective judgments. Such a view is more flexible than theories based on logic or on frequencies. It makes sense, for example, that a doctor predicts for a smoker with an otherwise healthy lifestyle a somewhat smalle ...
ICS 353: Design and Analysis of Algorithms
... • Computers represent information using bits • A bit string is a sequence of zero or more bits. • 0 represents F and 1 represents T • How does C represent True and False?????? ...
... • Computers represent information using bits • A bit string is a sequence of zero or more bits. • 0 represents F and 1 represents T • How does C represent True and False?????? ...
A SHORT AND READABLE PROOF OF CUT ELIMINATION FOR
... Theorem 2.6 (cf. [10, 12, 9]). If (Γ ⊢ ∆) [a] is provable with order m and b is some other free variable, then (Γ ⊢ ∆) [b] is provable with order ≤ m. Proof. By induction on the order of derivation, m, of (Γ ⊢ ∆)[a]. For m = 0, (Γ ⊢ ∆)[a] is an axiom. Then so is (Γ ⊢ ∆)[b]. For the induction step we ...
... Theorem 2.6 (cf. [10, 12, 9]). If (Γ ⊢ ∆) [a] is provable with order m and b is some other free variable, then (Γ ⊢ ∆) [b] is provable with order ≤ m. Proof. By induction on the order of derivation, m, of (Γ ⊢ ∆)[a]. For m = 0, (Γ ⊢ ∆)[a] is an axiom. Then so is (Γ ⊢ ∆)[b]. For the induction step we ...
CS 399: Constructive Logic Final Exam (Sample Solution) Name Instructions
... Hyp Hyp Γ, ¬A, A ` Γ, ¬A, A ` Γ, ¬A, A ` ⊥ ...
... Hyp Hyp Γ, ¬A, A ` Γ, ¬A, A ` Γ, ¬A, A ` ⊥ ...
Recall... Venn Diagrams Disjunctive normal form Disjunctive normal
... Proof: [take notes...] Example: Write the following Boolean Expression in CNF: ...
... Proof: [take notes...] Example: Write the following Boolean Expression in CNF: ...
Guarded negation
... as a syntactic fragment of first-order logic, it is also natural to ask for syntactic explanations: what syntactic features of modal formulas (viewed as first-order formulas) are responsible for their good behavior? And can we generalize modal logic, preserving these features, while at the same tim ...
... as a syntactic fragment of first-order logic, it is also natural to ask for syntactic explanations: what syntactic features of modal formulas (viewed as first-order formulas) are responsible for their good behavior? And can we generalize modal logic, preserving these features, while at the same tim ...
FC §1.1, §1.2 - Mypage at Indiana University
... The implication (¬q) → (¬p) is called the contrapositive of p → q. An implication is logically equivalent to its contrapositive. The contrapositive of “If this is Tuesday, then we are in Belgium” is “If we aren’t in Belgium, then this isn’t Tuesday.” These two sentences assert exactly the same thing ...
... The implication (¬q) → (¬p) is called the contrapositive of p → q. An implication is logically equivalent to its contrapositive. The contrapositive of “If this is Tuesday, then we are in Belgium” is “If we aren’t in Belgium, then this isn’t Tuesday.” These two sentences assert exactly the same thing ...
Model theory makes formulas large
... numbers by trees of small height that can be controlled by small first-order formulas. In fact, we show — and use — that full arithmetic on a large initial segment of the positive integers can be simulated by comparably small first-order formulas that operate on the tree encodings of the numbers. It ...
... numbers by trees of small height that can be controlled by small first-order formulas. In fact, we show — and use — that full arithmetic on a large initial segment of the positive integers can be simulated by comparably small first-order formulas that operate on the tree encodings of the numbers. It ...
Section I(e)
... We need to construct the truth table for (not Q) implies (not P), Q P . In the first two left hand columns we list all the possible combinations of P and Q . For the next two columns we write down the truth tables for Q (not Q ) and P (not P ) respectively. In the right hand column we ...
... We need to construct the truth table for (not Q) implies (not P), Q P . In the first two left hand columns we list all the possible combinations of P and Q . For the next two columns we write down the truth tables for Q (not Q ) and P (not P ) respectively. In the right hand column we ...
page 3 A CONVERSE BARCAN FORMULA IN ARISTOTLE`S
... First then take a universal negative with the terms a and b. Now if a belongs to no b, b will not belong to any a; for if it, b, does belong to some a (say to c), it will not be true that a belongs to no b — for c is one of the bs (An pr. I.2, 25a14–17).6 It is the cryptic second sentence that sketc ...
... First then take a universal negative with the terms a and b. Now if a belongs to no b, b will not belong to any a; for if it, b, does belong to some a (say to c), it will not be true that a belongs to no b — for c is one of the bs (An pr. I.2, 25a14–17).6 It is the cryptic second sentence that sketc ...
Strong Completeness and Limited Canonicity for PDL
... i.e. when | ϕ implies that there is a finite ⊆ with | ϕ, hence | → ϕ. This is, for example, the case in propositional and predicate logic, and in many modal logics such as K and S5. Segerberg’s axiomatization of PDL is only weakly complete, since PDL is not compact: we have that {[a ...
... i.e. when | ϕ implies that there is a finite ⊆ with | ϕ, hence | → ϕ. This is, for example, the case in propositional and predicate logic, and in many modal logics such as K and S5. Segerberg’s axiomatization of PDL is only weakly complete, since PDL is not compact: we have that {[a ...
