page 135 ADAPTIVE LOGICS FOR QUESTION EVOCATION
... 2. Question Evocation As mentioned in the previous section, Wiśniewski’s concept of question evocation can be applied to any logic of questions that satisfies some minimal requirements. An obvious requirement is that its language L consists of a declarative part (some standard formalized language) ...
... 2. Question Evocation As mentioned in the previous section, Wiśniewski’s concept of question evocation can be applied to any logic of questions that satisfies some minimal requirements. An obvious requirement is that its language L consists of a declarative part (some standard formalized language) ...
Integrating Logical Reasoning and Probabilistic Chain Graphs
... describe a probabilistic first-order language that is more expressive than similar languages developed earlier, in the sense that the probabilistic models that can be specified and reasoned about have Bayesian and Markov networks as special cases. This new probabilistic logic is called chain logic. Th ...
... describe a probabilistic first-order language that is more expressive than similar languages developed earlier, in the sense that the probabilistic models that can be specified and reasoned about have Bayesian and Markov networks as special cases. This new probabilistic logic is called chain logic. Th ...
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... computing. Most of that work has used standard Kripke structures to model knowledge, where an agent knows a fact ϕ if ϕ is true in all the worlds that the agent considers possible. While this approach has proved useful for many applications, it suffers from a serious shortcoming, known as the logica ...
... computing. Most of that work has used standard Kripke structures to model knowledge, where an agent knows a fact ϕ if ϕ is true in all the worlds that the agent considers possible. While this approach has proved useful for many applications, it suffers from a serious shortcoming, known as the logica ...
Teach Yourself Logic 2017: A Study Guide
... • However, if you are hoping for help with very elementary logic (e.g. as typically encountered by philosophers in their first-year courses), then let me say straight away that this Guide isn’t designed for you. The only section that directly pertains to this ‘baby logic’ is §1.5; all the rest is ab ...
... • However, if you are hoping for help with very elementary logic (e.g. as typically encountered by philosophers in their first-year courses), then let me say straight away that this Guide isn’t designed for you. The only section that directly pertains to this ‘baby logic’ is §1.5; all the rest is ab ...
CHAPTER 1. SENTENTIAL LOGIC 1. Introduction In sentential logic
... • If 7 is not odd then 2 is odd • If 7 is odd and 2 is odd then 2 is not prime • (7 is odd and 2 is odd) or 2 is prime We have improved on English in the last example by using parentheses to resolve an ambiguity. And, or, not, if . . . then (or implies) are called (sentential) connectives. Using the ...
... • If 7 is not odd then 2 is odd • If 7 is odd and 2 is odd then 2 is not prime • (7 is odd and 2 is odd) or 2 is prime We have improved on English in the last example by using parentheses to resolve an ambiguity. And, or, not, if . . . then (or implies) are called (sentential) connectives. Using the ...
Weak Kripke Structures and LTL
... LTL formula. In [8], Demri and Schnoebelen strengthen this result by showing that satisfiability is hard even for the fragment of LTL with a single atomic proposition. Theorem 3 (Demri and Schnoebelen (2002)). Satisfiability of LTL with a single atomic proposition is PSPACE-complete. t u Restricting ...
... LTL formula. In [8], Demri and Schnoebelen strengthen this result by showing that satisfiability is hard even for the fragment of LTL with a single atomic proposition. Theorem 3 (Demri and Schnoebelen (2002)). Satisfiability of LTL with a single atomic proposition is PSPACE-complete. t u Restricting ...
Teach Yourself Logic 2016: A Study Guide
... That should enable you to sensibly pick and choose your way through the remaining chapters. • However, if you are hoping for help with very elementary logic (e.g. as typically encountered by philosophers in their first-year courses), then let me say straight away that this Guide isn’t designed for y ...
... That should enable you to sensibly pick and choose your way through the remaining chapters. • However, if you are hoping for help with very elementary logic (e.g. as typically encountered by philosophers in their first-year courses), then let me say straight away that this Guide isn’t designed for y ...
One-dimensional Fragment of First-order Logic
... Two-variable logic and guarded-fragment are examples of decidable fragments of first-order logic that are not based on restricting the quantifier alternation patterns of formulae, unlike the prefix classes studied in the context of the classical decision problem. Surprisingly, not many such framewor ...
... Two-variable logic and guarded-fragment are examples of decidable fragments of first-order logic that are not based on restricting the quantifier alternation patterns of formulae, unlike the prefix classes studied in the context of the classical decision problem. Surprisingly, not many such framewor ...
Internal Inconsistency and the Reform of Naïve Set Comprehension
... Set theory has long been instrumental to our ideas of formal logic and mathematics, yet no totally satisfying responses have been given to the following concerns: (1) What is at the root of the logical antinomies? (2) What set descriptions (or their predicative functions) must be excluded from the r ...
... Set theory has long been instrumental to our ideas of formal logic and mathematics, yet no totally satisfying responses have been given to the following concerns: (1) What is at the root of the logical antinomies? (2) What set descriptions (or their predicative functions) must be excluded from the r ...
Studying Sequent Systems via Non-deterministic Multiple
... is of course (the propositional fragment of) Gentzen’s LK (the usual sequent system for classical logic). However, (infinitely) many more sequent systems belong to this family (a particularly useful one was recently suggested in [7]). The study of canonical systems in [3] (see also [4]) mainly conce ...
... is of course (the propositional fragment of) Gentzen’s LK (the usual sequent system for classical logic). However, (infinitely) many more sequent systems belong to this family (a particularly useful one was recently suggested in [7]). The study of canonical systems in [3] (see also [4]) mainly conce ...
Dependence Logic
... ψ of dependence logic such that M |=X φ implies M 6|=X ψ, for all M and all X. Namely, letting X = ∅ would yield a contradiction. The following test is very important and will be used repeatedly in the sequel. Closure downwards is a fundamental property of types in dependence logic. Proposition 8 (C ...
... ψ of dependence logic such that M |=X φ implies M 6|=X ψ, for all M and all X. Namely, letting X = ∅ would yield a contradiction. The following test is very important and will be used repeatedly in the sequel. Closure downwards is a fundamental property of types in dependence logic. Proposition 8 (C ...
An Introduction to Mathematical Logic
... accordingly, then • ∀x (P (y) → Q(y)) ∈ FS , although x does not occur in (P (y) → Q(y)) • ∀x (P (y) → Q(y)) ∈ FS , although y occurs freely in (P (y) → Q(y)), i.e., it occurs not in the range of (“bound by”) ∀y or ∃y • ∃x(x ≡ x ∧ x ≡ x) ∈ FS , although this is obviously redundant • ∃x∃xP (x) ∈ FS , ...
... accordingly, then • ∀x (P (y) → Q(y)) ∈ FS , although x does not occur in (P (y) → Q(y)) • ∀x (P (y) → Q(y)) ∈ FS , although y occurs freely in (P (y) → Q(y)), i.e., it occurs not in the range of (“bound by”) ∀y or ∃y • ∃x(x ≡ x ∧ x ≡ x) ∈ FS , although this is obviously redundant • ∃x∃xP (x) ∈ FS , ...
page 139 MINIMIZING AMBIGUITY AND
... Hence either C 1 or C 2 does not have the intended meaning. The formal solution in this case exists in saying that either C 1 6=C or C 2 6=C. In natural languages we will replace either C 1 or C 2 by another expression. For instance, if we talk about furniture, and especially about chairs, we better ...
... Hence either C 1 or C 2 does not have the intended meaning. The formal solution in this case exists in saying that either C 1 6=C or C 2 6=C. In natural languages we will replace either C 1 or C 2 by another expression. For instance, if we talk about furniture, and especially about chairs, we better ...
Probability Captures the Logic of Scientific
... the following conditions. Here, and in the remainder of this paper, ‘a’ stands for any individual constant. Q1 a = F a.Ga; Q2 a = F a.∼Ga; Q3 a = ∼F a.Ga; Q4 a = ∼F a.∼Ga. A sample is a finite set of individuals. A sample description is a sentence that says, for each individual in some sample, which ...
... the following conditions. Here, and in the remainder of this paper, ‘a’ stands for any individual constant. Q1 a = F a.Ga; Q2 a = F a.∼Ga; Q3 a = ∼F a.Ga; Q4 a = ∼F a.∼Ga. A sample is a finite set of individuals. A sample description is a sentence that says, for each individual in some sample, which ...
Table of mathematical symbols - Wikipedia, the free
... is a subset of (proper subset) A ⊂ B means A ⊆ B but A ≠ B. ℕ ⊂ ℚ set theory (Some writers use the symbol ⊂ as if it were the ℚ ⊂ ℝ same as ⊆.) A ⊇ B means every element of B is also an element of A. ...
... is a subset of (proper subset) A ⊂ B means A ⊆ B but A ≠ B. ℕ ⊂ ℚ set theory (Some writers use the symbol ⊂ as if it were the ℚ ⊂ ℝ same as ⊆.) A ⊇ B means every element of B is also an element of A. ...
Extracting Proofs from Tabled Proof Search
... II History atoms are not tabled; the table uses theories to infer additional atoms. III History atoms can be tabled; the table only infers its members. IV History atoms can be tabled; the table uses theories to infer additional atoms. The first two strategies yield proof certificates that simply us ...
... II History atoms are not tabled; the table uses theories to infer additional atoms. III History atoms can be tabled; the table only infers its members. IV History atoms can be tabled; the table uses theories to infer additional atoms. The first two strategies yield proof certificates that simply us ...
TOWARD A STABILITY THEORY OF TAME ABSTRACT
... the Galois type of the sequence b̄ over the set A, as computed in N ∈ K. In case K has a monster model C, we write gtp(b̄/A) instead of gtp(b̄/A; C). In this case, gtp(b̄/A) = gtp(c̄/A) if and only if there exists an automorphism f of C fixing A such that f (b̄) = c̄. Observe that the definition of ...
... the Galois type of the sequence b̄ over the set A, as computed in N ∈ K. In case K has a monster model C, we write gtp(b̄/A) instead of gtp(b̄/A; C). In this case, gtp(b̄/A) = gtp(c̄/A) if and only if there exists an automorphism f of C fixing A such that f (b̄) = c̄. Observe that the definition of ...
Reasoning about Action and Change
... A logic L is correct with respect to a class of models M iff all theorems of L are valid in all models in M. A logic L is (weakly) complete with respect to a class of models M iff all formulae which are valid in M are theorems of L. Finally, a logic L is (weakly) determined (characterised) by a clas ...
... A logic L is correct with respect to a class of models M iff all theorems of L are valid in all models in M. A logic L is (weakly) complete with respect to a class of models M iff all formulae which are valid in M are theorems of L. Finally, a logic L is (weakly) determined (characterised) by a clas ...
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... quantifiers are typically taken to range over propositions (intuitively, sets of worlds), but this does not work in our setting because awareness is syntactic; when we write, for example, ∀xAi x, we essentially mean that Ai ϕ holds for all formulas ϕ. However, there is another subtlety. If we define ...
... quantifiers are typically taken to range over propositions (intuitively, sets of worlds), but this does not work in our setting because awareness is syntactic; when we write, for example, ∀xAi x, we essentially mean that Ai ϕ holds for all formulas ϕ. However, there is another subtlety. If we define ...
INDEX SETS FOR n-DECIDABLE STRUCTURES CATEGORICAL
... they showed that the index set of computable categorical structures is Π11 -complete. Combining the methods from [2] and from [10], Bazhenov, Goncharov and Marchuk showed that also the index set of computable structures of algorithmic dimension n > 1 is Π11 complete [18]. On the other hand, the inde ...
... they showed that the index set of computable categorical structures is Π11 -complete. Combining the methods from [2] and from [10], Bazhenov, Goncharov and Marchuk showed that also the index set of computable structures of algorithmic dimension n > 1 is Π11 complete [18]. On the other hand, the inde ...
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... true, false, p, ¬ϕ, ϕ ∨ ψ, ϕ, 3ϕ, eϕ, or ϕU ψ, where p ∈ P , and ϕ and ψ are LTL formulas. The temporal operators (“always”), 3 (“eventually”), e(“next”), and U (“until”) enable convenient description of time-dependent events. For example, the LTL formula (request → 3grant) states that every req ...
... true, false, p, ¬ϕ, ϕ ∨ ψ, ϕ, 3ϕ, eϕ, or ϕU ψ, where p ∈ P , and ϕ and ψ are LTL formulas. The temporal operators (“always”), 3 (“eventually”), e(“next”), and U (“until”) enable convenient description of time-dependent events. For example, the LTL formula (request → 3grant) states that every req ...
Proof theory of witnessed G¨odel logic: a
... (First-order) Gödel logic is a prominent example of both a many-valued and a superintuitionistic logic. The importance of Gödel logic is emphasized by the fact that it turns up naturally in a number of different contexts; among them relevance logics, fuzzy logic, and logic programming. Witnessed G ...
... (First-order) Gödel logic is a prominent example of both a many-valued and a superintuitionistic logic. The importance of Gödel logic is emphasized by the fact that it turns up naturally in a number of different contexts; among them relevance logics, fuzzy logic, and logic programming. Witnessed G ...
Judgment and consequence relations
... Here we may simply assume that a circumstance is a valuation. Write “$β ” for the truth definition of such a circumstance. Then we have $β ϕ iff β P ϕ iff βpϕq 1. Alternatively, we can define this using (4) and the base clause stating that $β p iff βp pq 1. It is important to realise that valua ...
... Here we may simply assume that a circumstance is a valuation. Write “$β ” for the truth definition of such a circumstance. Then we have $β ϕ iff β P ϕ iff βpϕq 1. Alternatively, we can define this using (4) and the base clause stating that $β p iff βp pq 1. It is important to realise that valua ...
PLATONISM IN MODERN MATHEMATICS A University Thesis
... Thus, the meaning of mathematical symbols are removed, and a math proof is analyzed by deriving such a proof from a series of formal axioms and rules of inference performed in a series of logical steps. The intuitonist school of mathematics simply assumes that mathematics is a man-made activity stem ...
... Thus, the meaning of mathematical symbols are removed, and a math proof is analyzed by deriving such a proof from a series of formal axioms and rules of inference performed in a series of logical steps. The intuitonist school of mathematics simply assumes that mathematics is a man-made activity stem ...
pdf - at www.arxiv.org.
... logic programs, i.e. if a formula is computable at infinity, it is also in the greatest Herbrand model of the program. Importantly for us, the notion of infinite formula computed at infinity captures the modern-day notion of producing an infinite data structure. We will use the term global productiv ...
... logic programs, i.e. if a formula is computable at infinity, it is also in the greatest Herbrand model of the program. Importantly for us, the notion of infinite formula computed at infinity captures the modern-day notion of producing an infinite data structure. We will use the term global productiv ...