Logics of Truth - Project Euclid
... to Scott [10] and Aczel [1]. The central notion is AczePs concept of a Frege structure. These structures are models of the Lambda Calculus together with two distinguished subsets —a set of propositions and a subset of this set called truths. In addition, such structures come equipped with the usual ...
... to Scott [10] and Aczel [1]. The central notion is AczePs concept of a Frege structure. These structures are models of the Lambda Calculus together with two distinguished subsets —a set of propositions and a subset of this set called truths. In addition, such structures come equipped with the usual ...
Fuzzy logic and probability Institute of Computer Science (ICS
... b + c - d. Thus P is a probability. {2) Conversely, assume that P is a probability on crisp formulas and put e{f"') = P(cp). We verify that e ass igns 1 to each axiom of F P. Clearly, if cp is an axiom of classical logic then cp is a Boolean tautology and hence e{f"') = P(cp) = 1. This verifies {FP1 ...
... b + c - d. Thus P is a probability. {2) Conversely, assume that P is a probability on crisp formulas and put e{f"') = P(cp). We verify that e ass igns 1 to each axiom of F P. Clearly, if cp is an axiom of classical logic then cp is a Boolean tautology and hence e{f"') = P(cp) = 1. This verifies {FP1 ...
MUltseq: a Generic Prover for Sequents and Equations*
... logics. This means that it takes as input the rules of a many-valued sequent calculus as well as a many-sided sequent and searches – automatically or interactively – for a proof of the latter. For the sake of readability, the output of MUltseq is typeset as a LATEX document. Though the sequent rules ...
... logics. This means that it takes as input the rules of a many-valued sequent calculus as well as a many-sided sequent and searches – automatically or interactively – for a proof of the latter. For the sake of readability, the output of MUltseq is typeset as a LATEX document. Though the sequent rules ...
byd.1 Second-Order logic
... relations with subsets of the domain, and so in particular you can quantify over these sets; for example, one can express induction for the natural numbers with a single axiom ∀R ((R() ∧ ∀x (R(x) → R(x0 ))) → ∀x R(x)). If one takes the language of arithmetic to have symbols , 0, +, × and <, one ca ...
... relations with subsets of the domain, and so in particular you can quantify over these sets; for example, one can express induction for the natural numbers with a single axiom ∀R ((R() ∧ ∀x (R(x) → R(x0 ))) → ∀x R(x)). If one takes the language of arithmetic to have symbols , 0, +, × and <, one ca ...
Lecture 3
... • “If you clean the car then you can go out” • Could we infer either of the following? – “if you don't clean the car then you can't go out” or – “if you were allowed out, then you must have cleaned the car”. ...
... • “If you clean the car then you can go out” • Could we infer either of the following? – “if you don't clean the car then you can't go out” or – “if you were allowed out, then you must have cleaned the car”. ...
If T is a consistent theory in the language of arithmetic, we say a set
... We relegate further examples of this kind to the problems at the end of the chapter. Once we have the basic laws of arithmetic, we can go on to prove various elementary lemmas of number theory such as the facts that a divisor of a divisor of a number is a divisor of that number, that every number ha ...
... We relegate further examples of this kind to the problems at the end of the chapter. Once we have the basic laws of arithmetic, we can go on to prove various elementary lemmas of number theory such as the facts that a divisor of a divisor of a number is a divisor of that number, that every number ha ...
L-spaces and the P
... Definition IX.4.5 of Shelah Proper and Improper Forcing Usually S is costationary. We call a forcing notion P (T, S)-preserving if the following holds: T is an Aronszajn tree, S ⊆ ω1 , and for every λ > (2|P |+ℵ1 )+ and countable N ≺ H(λ, ∈) such that P, T, S ∈ N and δ = N ∩ ω1 6∈ S, and every p ∈ N ...
... Definition IX.4.5 of Shelah Proper and Improper Forcing Usually S is costationary. We call a forcing notion P (T, S)-preserving if the following holds: T is an Aronszajn tree, S ⊆ ω1 , and for every λ > (2|P |+ℵ1 )+ and countable N ≺ H(λ, ∈) such that P, T, S ∈ N and δ = N ∩ ω1 6∈ S, and every p ∈ N ...
Second-order Logic
... Proof. Suppose M |= PA2 Of course, for any n ∈ N, ValM (n) ∈ |M|, so N ⊆ |M|. Let N = {ValM (n) : n ∈ N} and s(X) = N . By assumption, M |= ∀X (X() ∧ ∀x (X(x) → X(x0 ))) → ∀x X(x) and so M, s |= (X() ∧ ∀x (X(x) → X(x0 ))) → ∀x X(x). ValM () ∈ N , and so M, s |= X(). Also, since if x ∈ N then als ...
... Proof. Suppose M |= PA2 Of course, for any n ∈ N, ValM (n) ∈ |M|, so N ⊆ |M|. Let N = {ValM (n) : n ∈ N} and s(X) = N . By assumption, M |= ∀X (X() ∧ ∀x (X(x) → X(x0 ))) → ∀x X(x) and so M, s |= (X() ∧ ∀x (X(x) → X(x0 ))) → ∀x X(x). ValM () ∈ N , and so M, s |= X(). Also, since if x ∈ N then als ...
- Free Documents
... By the completeness of L noninterderivable and give rise to distinct and n . This is in general not so for theories. An example is the theory axiomatized by p on the one hand, and the theory T axiomatized by m p for each m, on the other. The sets p and T are the same, consisting of all nodes that to ...
... By the completeness of L noninterderivable and give rise to distinct and n . This is in general not so for theories. An example is the theory axiomatized by p on the one hand, and the theory T axiomatized by m p for each m, on the other. The sets p and T are the same, consisting of all nodes that to ...
Concept Hierarchies from a Logical Point of View
... logic is commonly seen as the most basic sort of logic – witness any textbook on logic. Conceptually, however, it seems rather awkward to regard attributes as propositions. If attributes are formalized within a logical language at all then the most natural way to do so is to represent them as monadi ...
... logic is commonly seen as the most basic sort of logic – witness any textbook on logic. Conceptually, however, it seems rather awkward to regard attributes as propositions. If attributes are formalized within a logical language at all then the most natural way to do so is to represent them as monadi ...
Aristotle`s particularisation
... denoted in the interpretation by ‘P ∗ (x)’, and the formula ‘[¬(∀x)¬P (x)]’ as the arithmetical proposition denoted by ‘¬(∀x)¬P ∗ (x)’, the formula ‘[(∃x)P (x)]’ cannot be assumed to always interpret as the arithmetical proposition ‘There exists some object a in the domain of the interpretation such ...
... denoted in the interpretation by ‘P ∗ (x)’, and the formula ‘[¬(∀x)¬P (x)]’ as the arithmetical proposition denoted by ‘¬(∀x)¬P ∗ (x)’, the formula ‘[(∃x)P (x)]’ cannot be assumed to always interpret as the arithmetical proposition ‘There exists some object a in the domain of the interpretation such ...
Probabilistic Theorem Proving - The University of Texas at Dallas
... function of a PKB K is given by Z(K) = x i φi i . The conditional probability P (Q|K) is simply a ratio of two partition functions: P (Q|K) = Z(K ∪ {Q, 0})/Z(K), where Z(K ∪ {Q, 0}) is the partition function of K with Q added as a hard formula. The main idea in PTP is to compute the partition functi ...
... function of a PKB K is given by Z(K) = x i φi i . The conditional probability P (Q|K) is simply a ratio of two partition functions: P (Q|K) = Z(K ∪ {Q, 0})/Z(K), where Z(K ∪ {Q, 0}) is the partition function of K with Q added as a hard formula. The main idea in PTP is to compute the partition functi ...
preference based on reasons
... In the fire alarm example, A envisions his home with a new fire alarm, but with the same furniture, cat, and fireplace as before. Home with no fire alarm is the actual situation, hence especially easy to envision. If u 1 measures safety, and p is “A will purchase a fire alarm” then p 1 ¬ p holds in ...
... In the fire alarm example, A envisions his home with a new fire alarm, but with the same furniture, cat, and fireplace as before. Home with no fire alarm is the actual situation, hence especially easy to envision. If u 1 measures safety, and p is “A will purchase a fire alarm” then p 1 ¬ p holds in ...
Paper - Christian Muise
... The lack of disjunction is a restriction that is acceptable for some scenarios, but clearly not all; however extending PEKBs to include disjunction would eliminate the desirable computational properties. We believe the next best opportunity is to include a restricted form of disjunction: ‘knowing wh ...
... The lack of disjunction is a restriction that is acceptable for some scenarios, but clearly not all; however extending PEKBs to include disjunction would eliminate the desirable computational properties. We believe the next best opportunity is to include a restricted form of disjunction: ‘knowing wh ...
PDF
... We briefly review the five standard systems of reverse mathematics. For completeness, we include systems stronger than arithmetical comprehension, but these will play no part in this paper. Details, general background, and results, as well as many examples of reversals, can be found in Simpson [1999 ...
... We briefly review the five standard systems of reverse mathematics. For completeness, we include systems stronger than arithmetical comprehension, but these will play no part in this paper. Details, general background, and results, as well as many examples of reversals, can be found in Simpson [1999 ...
The strong completeness of the tableau method 1 The strong
... In particular, does not generate what they call a ‘canonical derivation’ (pp. 123 and 131132), since in such a tree all the premises in question must be present. Hence the result called ‘Lemma I’ on p. 132 does not follow. And this is one of the bases on which they build up their proof of the comp ...
... In particular, does not generate what they call a ‘canonical derivation’ (pp. 123 and 131132), since in such a tree all the premises in question must be present. Hence the result called ‘Lemma I’ on p. 132 does not follow. And this is one of the bases on which they build up their proof of the comp ...
Cut-Free Sequent Systems for Temporal Logic
... temporal logic with first-order quantifiers. However, this fragment does not include full propositional linear temporal logic. Jäger et al. propose using the small model property to finitise the ω-rule for the logic of common knowledge [8] and for the full modal µ-calculus in [7]. This leads to fin ...
... temporal logic with first-order quantifiers. However, this fragment does not include full propositional linear temporal logic. Jäger et al. propose using the small model property to finitise the ω-rule for the logic of common knowledge [8] and for the full modal µ-calculus in [7]. This leads to fin ...
Transcendental values of the digamma function
... is either zero or transcendental. Let us note that here and later, we interpret log as the principal value of the logarithm with argument in (−π, π]. A creative application of this theorem was made by Baker, Birch and Wirsing [5] to resolve a problem of Chowla [6], which we now describe. In a lectur ...
... is either zero or transcendental. Let us note that here and later, we interpret log as the principal value of the logarithm with argument in (−π, π]. A creative application of this theorem was made by Baker, Birch and Wirsing [5] to resolve a problem of Chowla [6], which we now describe. In a lectur ...
On Equivalent Transformations of Infinitary Formulas under the
... (a) if a formula F is provable in the basic system then H ∪ {F } has the same stable models as H; (b) if F is equivalent to G in the basic system then H ∪ {F } and H ∪ {G} have the same stable models. Lemma 1. For any formula F and interpretation I, if I does not satisfy F then F I ⇒ ⊥ is a theorem ...
... (a) if a formula F is provable in the basic system then H ∪ {F } has the same stable models as H; (b) if F is equivalent to G in the basic system then H ∪ {F } and H ∪ {G} have the same stable models. Lemma 1. For any formula F and interpretation I, if I does not satisfy F then F I ⇒ ⊥ is a theorem ...
CS311H: Discrete Mathematics Cardinality of Infinite Sets and
... Example: Find a number congruent to 7 modulo 4. ...
... Example: Find a number congruent to 7 modulo 4. ...
Variations on a Montagovian Theme
... object. The subject is the person who knows or believes; the object is that which is known or believed. But what kind of object is this? Two answers have been popular in the more systematic branches of epistemology and philosophy of mind. The first identifies objects of attitudes with something like ...
... object. The subject is the person who knows or believes; the object is that which is known or believed. But what kind of object is this? Two answers have been popular in the more systematic branches of epistemology and philosophy of mind. The first identifies objects of attitudes with something like ...
A systematic proof theory for several modal logics
... so to is its subsystem aKS, in the sense that looking at the inferences going either up or down, structure is rearranged, or atoms introduced, abandoned or duplicated, but arbitrarily large substructures are never introduced, abandoned or duplicated. Bruennler also discusses an important advantage c ...
... so to is its subsystem aKS, in the sense that looking at the inferences going either up or down, structure is rearranged, or atoms introduced, abandoned or duplicated, but arbitrarily large substructures are never introduced, abandoned or duplicated. Bruennler also discusses an important advantage c ...
Tautologies Arguments Logical Implication
... • i.e., there is an inference rule A1, . . . , An ` B such that Ai = Cji for some ji < N and B = CN . This is said to be a derivation or proof of CN . A derivation is a syntactic object: it’s just a sequence of formulas that satisfy certain constraints. • Whether a formula is derivable depends on th ...
... • i.e., there is an inference rule A1, . . . , An ` B such that Ai = Cji for some ji < N and B = CN . This is said to be a derivation or proof of CN . A derivation is a syntactic object: it’s just a sequence of formulas that satisfy certain constraints. • Whether a formula is derivable depends on th ...
Bounded Functional Interpretation
... given f , compactness reasons (that can be put in intuitionistic clothes) yield a bound for the n’s. Notwithstanding, it is not continuity that is responsible for the elimination of the FAN theorem by b.f.i.: It is majorizability! This was first observed in [15] in connection with the FAN rule (see ...
... given f , compactness reasons (that can be put in intuitionistic clothes) yield a bound for the n’s. Notwithstanding, it is not continuity that is responsible for the elimination of the FAN theorem by b.f.i.: It is majorizability! This was first observed in [15] in connection with the FAN rule (see ...
Incompleteness - the UNC Department of Computer Science
... In The Emperor's New Mind [Penrose, 1989] and especially in Shadows of the Mind [Penrose, 1994], Roger Penrose argues against the “strong artificial intelligence thesis," contending that human reasoning cannot be captured by an artificial intellect because humans detect nontermination of programs in ...
... In The Emperor's New Mind [Penrose, 1989] and especially in Shadows of the Mind [Penrose, 1994], Roger Penrose argues against the “strong artificial intelligence thesis," contending that human reasoning cannot be captured by an artificial intellect because humans detect nontermination of programs in ...