Modal Logic for Artificial Intelligence
... is valid, regardless of the sentences we use in the place of A and B. The only items that need to be fixed are ‘or’ and ‘not’ in this case. If we would replace ‘not’ by ‘maybe’, then the argument would not be valid anymore. We call ‘or’ and ‘not’ logical constants. Together with ‘and’, ‘if . . . the ...
... is valid, regardless of the sentences we use in the place of A and B. The only items that need to be fixed are ‘or’ and ‘not’ in this case. If we would replace ‘not’ by ‘maybe’, then the argument would not be valid anymore. We call ‘or’ and ‘not’ logical constants. Together with ‘and’, ‘if . . . the ...
Interpreting and Applying Proof Theories for Modal Logic
... 4 Note that from now on the operator will not be taken as primitive but as defined in the following standard way: A = ¬¬A, not because it couldn’t be primitive, but for compactnesss of presentation. 5 In Belnap’s original work on Display Logic, the modal operators are treated with another family ...
... 4 Note that from now on the operator will not be taken as primitive but as defined in the following standard way: A = ¬¬A, not because it couldn’t be primitive, but for compactnesss of presentation. 5 In Belnap’s original work on Display Logic, the modal operators are treated with another family ...
Proof Theory of Finite-valued Logics
... Some of the material presented here has appeared in different forms elsewhere; The main innovations of this report are: the use of signed formula expressions and partial normal forms to provide a unifying framework in which clause translation calculi (Chapter 2), sequent calculi (Chapter 3), natural ...
... Some of the material presented here has appeared in different forms elsewhere; The main innovations of this report are: the use of signed formula expressions and partial normal forms to provide a unifying framework in which clause translation calculi (Chapter 2), sequent calculi (Chapter 3), natural ...
CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction
... determine whether a given expression A ∈ E is in AX or not. Semantical link : For a given semantics M for L and its extension to E, we usually choose as AX a subset of expressions that are tautologies under the semantics M . Component: Rules of Inference The production of provable formulas is to be ...
... determine whether a given expression A ∈ E is in AX or not. Semantical link : For a given semantics M for L and its extension to E, we usually choose as AX a subset of expressions that are tautologies under the semantics M . Component: Rules of Inference The production of provable formulas is to be ...
How complicated is the set of stable models of a recursive logic
... concept of a perfect model is that of a stable model of a program [Gelfond and Lifschitz, 1988]. Stable models of programs have been shown to be default interpretations, see [Reiter, 1980], in [Bidoit and Froidevaux, 1988] and in [Marek and Truszczyński, 1989]. There exist logic programs which are ...
... concept of a perfect model is that of a stable model of a program [Gelfond and Lifschitz, 1988]. Stable models of programs have been shown to be default interpretations, see [Reiter, 1980], in [Bidoit and Froidevaux, 1988] and in [Marek and Truszczyński, 1989]. There exist logic programs which are ...
article in press - School of Computer Science
... in φ so that at most one closure condition is associated with each relation. It is decidable whether φ is satisfiable in a model satisfying C. Proof. The proof is very similar to the proof given in [14] for non-parametrised closure conditions. In fact, it is slightly simpler, because in the original ...
... in φ so that at most one closure condition is associated with each relation. It is decidable whether φ is satisfiable in a model satisfying C. Proof. The proof is very similar to the proof given in [14] for non-parametrised closure conditions. In fact, it is slightly simpler, because in the original ...
Safety Metric Temporal Logic is Fully Decidable
... satisfies ϕ.) In contrast, the results in the present paper involve considering timed alternating automata over infinite timed words. Our main technical contribution is to show the decidability of languageemptiness over infinite timed words for a class of timed alternating automata rich enough to c ...
... satisfies ϕ.) In contrast, the results in the present paper involve considering timed alternating automata over infinite timed words. Our main technical contribution is to show the decidability of languageemptiness over infinite timed words for a class of timed alternating automata rich enough to c ...
An Interpolating Theorem Prover
... Note, this system is not complete, since it has no rule to deal with negated inequalities. Later, after we introduce the equality operator, we will obtain a complete system for the rationals. ...
... Note, this system is not complete, since it has no rule to deal with negated inequalities. Later, after we introduce the equality operator, we will obtain a complete system for the rationals. ...
Document
... (locally) modular, and Pourmahdian [13] has fused Hrushovski’s example with an elementary abelian 2-group, and obtained an ω-categorical group of SU-rank 1 which is not one-based. Moreover, it is unknown whether a supersimple ω-categorical theory can have infinite rank. Nevertheless, we show that at ...
... (locally) modular, and Pourmahdian [13] has fused Hrushovski’s example with an elementary abelian 2-group, and obtained an ω-categorical group of SU-rank 1 which is not one-based. Moreover, it is unknown whether a supersimple ω-categorical theory can have infinite rank. Nevertheless, we show that at ...
lecture notes in logic - UCLA Department of Mathematics
... Basic examples of fields are the rational numbers Q, the real numbers R and the complex numbers C, with universes Q, R, C respectively and the usual operations on these number sets. Definition 1A.5. The universe of sets is the structure ...
... Basic examples of fields are the rational numbers Q, the real numbers R and the complex numbers C, with universes Q, R, C respectively and the usual operations on these number sets. Definition 1A.5. The universe of sets is the structure ...
Per Lindström FIRST
... whose nature need not be specified; the cardinality of D is all that matters. This concept, a forerunner of the present-day notion of truth in a model, was quite foreign to the Frege-Peano-Russell tradition dominating logic at the time and its introduction and the first really significant theorem, t ...
... whose nature need not be specified; the cardinality of D is all that matters. This concept, a forerunner of the present-day notion of truth in a model, was quite foreign to the Frege-Peano-Russell tradition dominating logic at the time and its introduction and the first really significant theorem, t ...
Ways Things Can`t Be
... Now consider the case where there are three propositions, A, B, and C, which are each pairwise consistent but their conjunction A ∧ B ∧ C is inconsistent. So, there is no possible world w such that w A ∧ B ∧ C. What are we to say about a belief state which includes all three beliefs? We have alrea ...
... Now consider the case where there are three propositions, A, B, and C, which are each pairwise consistent but their conjunction A ∧ B ∧ C is inconsistent. So, there is no possible world w such that w A ∧ B ∧ C. What are we to say about a belief state which includes all three beliefs? We have alrea ...
On the use of fuzzy stable models for inconsistent classical logic
... resembles that of a reduct in the classical case, this is why we introduce the following: Definition 5. The program PI is called the reduct of P wrt the interpretation I. As a result of the definition, note that given two fuzzy Linterpretations I and J, then the reducts PI and PJ have the same rules ...
... resembles that of a reduct in the classical case, this is why we introduce the following: Definition 5. The program PI is called the reduct of P wrt the interpretation I. As a result of the definition, note that given two fuzzy Linterpretations I and J, then the reducts PI and PJ have the same rules ...
Large cardinals and the Continuum Hypothesis
... of the concept of a “model” for a given theory (see above in (**) and (***)), or combinatorially – notice that ω itself is regular and strong limit, i.e. no finite subset of ω is cofinal in ω and ∀n < ω 2n < ω. We generalize10 three other properties of ω:11 (C) ω is compact in the sense of the comp ...
... of the concept of a “model” for a given theory (see above in (**) and (***)), or combinatorially – notice that ω itself is regular and strong limit, i.e. no finite subset of ω is cofinal in ω and ∀n < ω 2n < ω. We generalize10 three other properties of ω:11 (C) ω is compact in the sense of the comp ...
Notes on Writing Proofs
... Theorem 1.10 If I is a prime ideal of a ring R, then R/I is an integral domain. The first of these theorems is one you know and love from your calculus class. The second is the famous theorem of Andrew Wiles (it implies Fermat’s last theorem). The third comes from a field of mathematical logic known ...
... Theorem 1.10 If I is a prime ideal of a ring R, then R/I is an integral domain. The first of these theorems is one you know and love from your calculus class. The second is the famous theorem of Andrew Wiles (it implies Fermat’s last theorem). The third comes from a field of mathematical logic known ...
Taming method in modal logic and mosaic method in temporal logic
... (1) ϕ ∈ l(mi ) iff ¬ϕ ∈ / l(mi ); (2) ϕ ∧ ψ ∈ l(mi ) iff {ϕ, ψ} ∈ l(mi ); (3) Gϕ ∈ l(m0 ) =⇒ Gϕ ∈ l(m1 ); (4) Gϕ ∈ l(m0 ) =⇒ ϕ ∈ l(m1 ); (5) Hϕ ∈ l(m1 ) =⇒ Hϕ ∈ l(m0 ); (6) Hϕ ∈ l(m1 ) =⇒ ϕ ∈ l(m0 ). We can allow mosaics with a singleton basis (where m0 = m1 ) requiring conditions one and two only. ...
... (1) ϕ ∈ l(mi ) iff ¬ϕ ∈ / l(mi ); (2) ϕ ∧ ψ ∈ l(mi ) iff {ϕ, ψ} ∈ l(mi ); (3) Gϕ ∈ l(m0 ) =⇒ Gϕ ∈ l(m1 ); (4) Gϕ ∈ l(m0 ) =⇒ ϕ ∈ l(m1 ); (5) Hϕ ∈ l(m1 ) =⇒ Hϕ ∈ l(m0 ); (6) Hϕ ∈ l(m1 ) =⇒ ϕ ∈ l(m0 ). We can allow mosaics with a singleton basis (where m0 = m1 ) requiring conditions one and two only. ...
Provability as a Modal Operator with the models of PA as the Worlds
... out the case where {ϕ: A Bϕ} ⊂ Th(PA) by the first property of B. From this we conclude that {ϕ: A Bϕ}⊆Th(PA), that is, that there’s some ϕ such that A Bϕ but ϕ Th(PA), however, this would imply that PA; ¬ϕ is consistent, in which case there’s some B ∈ W where hN, Bi R, and hence N is not cen ...
... out the case where {ϕ: A Bϕ} ⊂ Th(PA) by the first property of B. From this we conclude that {ϕ: A Bϕ}⊆Th(PA), that is, that there’s some ϕ such that A Bϕ but ϕ Th(PA), however, this would imply that PA; ¬ϕ is consistent, in which case there’s some B ∈ W where hN, Bi R, and hence N is not cen ...
First-Order Intuitionistic Logic with Decidable Propositional
... an example of such formula holding LJ+. See [Ge] for the proof that R\/¬R does not hold in LJ. Also, there exist formulas provable in LK but not in LJ+. This will be proven later - see section ‘Intuitionistic Features’. ∃x P(x)\/¬ ∃x P(x) is an example of a classically provable formula that does not ...
... an example of such formula holding LJ+. See [Ge] for the proof that R\/¬R does not hold in LJ. Also, there exist formulas provable in LK but not in LJ+. This will be proven later - see section ‘Intuitionistic Features’. ∃x P(x)\/¬ ∃x P(x) is an example of a classically provable formula that does not ...
beliefrevision , epistemicconditionals andtheramseytest
... But even if we restrict our attention to epistemic conditionals, the Ramsey test turns out to be problematic. Gärdenfors (1988) shows that this test, despite its initial attractiveness, is incompatible with certain plausible conditions on belief revision — at least as long as belief states are viewe ...
... But even if we restrict our attention to epistemic conditionals, the Ramsey test turns out to be problematic. Gärdenfors (1988) shows that this test, despite its initial attractiveness, is incompatible with certain plausible conditions on belief revision — at least as long as belief states are viewe ...
Relevant deduction
... has the usual properties mathematicians expect from a logic. But apart from some simple paradoxical implications like ex falso quodlibet, the paradoxes mentioned above are not solved by these relevance logics. As will be shown later, it is just the fact that relevance logic keeps certain logical sta ...
... has the usual properties mathematicians expect from a logic. But apart from some simple paradoxical implications like ex falso quodlibet, the paradoxes mentioned above are not solved by these relevance logics. As will be shown later, it is just the fact that relevance logic keeps certain logical sta ...
Cichon`s diagram, regularity properties and ∆ sets of reals.
... the counterpart to the classical Bartoszyński-Raisonnier-Stern implication “Σ1n (B) ⇒ Σ1n (C)” fails to lift to higher levels, for all n ≥ 3 (on the other hand, the existence of a measurable implies Σ13 (B) ⇒ Σ13 (C)). Other “nonliftings” of implications will follow from our results as well, for ex ...
... the counterpart to the classical Bartoszyński-Raisonnier-Stern implication “Σ1n (B) ⇒ Σ1n (C)” fails to lift to higher levels, for all n ≥ 3 (on the other hand, the existence of a measurable implies Σ13 (B) ⇒ Σ13 (C)). Other “nonliftings” of implications will follow from our results as well, for ex ...
Pebble weighted automata and transitive - LSV
... In this section we set up the notation and we recall some basic results on weighted automata and weighted logics. We refer the reader to [6,7] for details. Throughout the paper, Σ denotes a finite alphabet and Σ + is the free semigroup over Σ, i.e., the set of nonempty words. The length of u ∈ Σ + i ...
... In this section we set up the notation and we recall some basic results on weighted automata and weighted logics. We refer the reader to [6,7] for details. Throughout the paper, Σ denotes a finite alphabet and Σ + is the free semigroup over Σ, i.e., the set of nonempty words. The length of u ∈ Σ + i ...
Notes on Modal Logic - Stanford University
... The modal invariance Lemma (Lemma 3.7) can be used to prove what can and cannot be expressed in the basic modal language. Fact 3.9 Let M = hW, R, V i be a relational structure. The universal operator is a unary operator Aϕ defined as follows: M, w |= Aϕ iff for all v ∈ W , M, v |= ϕ The universal o ...
... The modal invariance Lemma (Lemma 3.7) can be used to prove what can and cannot be expressed in the basic modal language. Fact 3.9 Let M = hW, R, V i be a relational structure. The universal operator is a unary operator Aϕ defined as follows: M, w |= Aϕ iff for all v ∈ W , M, v |= ϕ The universal o ...
Proof analysis beyond geometric theories: from rule systems to
... and in general for results that involve the simultaneous analysis of derivability in different theories. The starting point of the investigation is given by an analytic proof system for pure logic, taken as the basis of the extension procedure. Natural deduction is often the privileged proof system ...
... and in general for results that involve the simultaneous analysis of derivability in different theories. The starting point of the investigation is given by an analytic proof system for pure logic, taken as the basis of the extension procedure. Natural deduction is often the privileged proof system ...
Minimal models and fixpoint semantics for definite logic programs
... If M1 is a model of P and M2 is a model of P then M1 ∩ M2 is a model of P . Theorem Let P be a set of definite clauses. Then: • P has a model. (atoms(P ) is one.) • P has a minimal Herbrand model. • P has a unique minimal Herbrand model, denoted M(P ). • M(P ) = the intersection of all Herbrand mode ...
... If M1 is a model of P and M2 is a model of P then M1 ∩ M2 is a model of P . Theorem Let P be a set of definite clauses. Then: • P has a model. (atoms(P ) is one.) • P has a minimal Herbrand model. • P has a unique minimal Herbrand model, denoted M(P ). • M(P ) = the intersection of all Herbrand mode ...