First-Order Theorem Proving and VAMPIRE
... One can see that the proof of Figure 3 is a refutation: the top line explicitly mentions that a refutation is found, the proof derives the false formula $false in inference 203 and inference 6 negates the input conjecture. Finally, the proof output of VAMPIRE contains only a subset of all generated ...
... One can see that the proof of Figure 3 is a refutation: the top line explicitly mentions that a refutation is found, the proof derives the false formula $false in inference 203 and inference 6 negates the input conjecture. Finally, the proof output of VAMPIRE contains only a subset of all generated ...
On modal logics of group belief
... of doxastic mental states, acceptances have only been examined since [57] and since [17]. Some authors (e.g. [16]) claim that acceptance implies belief (at least to some minimal degree as argued in [59]). On the contrary, in [57] acceptance is considered to be stronger than belief. Although belief a ...
... of doxastic mental states, acceptances have only been examined since [57] and since [17]. Some authors (e.g. [16]) claim that acceptance implies belief (at least to some minimal degree as argued in [59]). On the contrary, in [57] acceptance is considered to be stronger than belief. Although belief a ...
Programming in Logic Without Logic Programming
... In KELPS, states are represented by sets of atomic sentences (also called ground atoms, facts or fluents). Events are also represented by atomic sentences. Such sets of atomic sentences can be understood either syntactically as theories or sematically as model-theoretic structures. It is this second ...
... In KELPS, states are represented by sets of atomic sentences (also called ground atoms, facts or fluents). Events are also represented by atomic sentences. Such sets of atomic sentences can be understood either syntactically as theories or sematically as model-theoretic structures. It is this second ...
Acts of Commanding and Changing Obligations
... A word about my choice of monadic deontic operators here may be in order. Monadic deontic logics are known to be inadequate to deal with conditional obligations and R. M. Chisholm’s contrary-to-duty paradox; dyadic deontic logics are better in this respect. But there are still other problems which a ...
... A word about my choice of monadic deontic operators here may be in order. Monadic deontic logics are known to be inadequate to deal with conditional obligations and R. M. Chisholm’s contrary-to-duty paradox; dyadic deontic logics are better in this respect. But there are still other problems which a ...
Curry-Howard Isomorphism - Department of information engineering
... systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λ-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, etc. The ...
... systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed λ-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, etc. The ...
Dialectica Interpretations A Categorical Analysis
... particular, several areas of mathematics, including linear algebra, number theory, probability theory, graph theory, and combinatorics, have been instrumental in the development of computer science. Unlike the natural sciences, however, computer science has also benefited from an extensive and conti ...
... particular, several areas of mathematics, including linear algebra, number theory, probability theory, graph theory, and combinatorics, have been instrumental in the development of computer science. Unlike the natural sciences, however, computer science has also benefited from an extensive and conti ...
Mathematical Logic
... be done without ambiguity. In particular, outermost parentheses can always be omitted, so instead of ((¬ A) ⇒ B) we may write (¬ A) ⇒ B. But we may not write ¬ A ⇒ B, because this would not distinguish the intended formula from ¬ (A ⇒ B). Definition 1.1.8. Let L be a propositional language. A format ...
... be done without ambiguity. In particular, outermost parentheses can always be omitted, so instead of ((¬ A) ⇒ B) we may write (¬ A) ⇒ B. But we may not write ¬ A ⇒ B, because this would not distinguish the intended formula from ¬ (A ⇒ B). Definition 1.1.8. Let L be a propositional language. A format ...
An Institution-Independent Generalization of Tarski`s Elementary
... Institutions are abstract logical frameworks which provide a category of signatures (languages) and signature morphisms (language translations), and, for each signature, a set of sentences, a category of models, and a satisfaction relation. Sentences have translations, and models have reducts, along ...
... Institutions are abstract logical frameworks which provide a category of signatures (languages) and signature morphisms (language translations), and, for each signature, a set of sentences, a category of models, and a satisfaction relation. Sentences have translations, and models have reducts, along ...
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 ...
Algebraic foundations for the semantic treatment of inquisitive content
... constructions that are used to perform the basic algebraic operations on propositions. For instance, it is natural to expect that languages generally have a word that is used (possibly among other things) to construct the join of two propositions, and another word to construct the meet of two propos ...
... constructions that are used to perform the basic algebraic operations on propositions. For instance, it is natural to expect that languages generally have a word that is used (possibly among other things) to construct the join of two propositions, and another word to construct the meet of two propos ...
KURT GÖDEL - National Academy of Sciences
... is just what comes from substituting {Ao, A,, A2, ...} for A in the immediately preceding statement, and noting that, if a contradiction can be deduced from the formulas A(), A,, A2, ..., only a finite number of them can participate in a given deduction of the contradiction. Thus: Either the formula ...
... is just what comes from substituting {Ao, A,, A2, ...} for A in the immediately preceding statement, and noting that, if a contradiction can be deduced from the formulas A(), A,, A2, ..., only a finite number of them can participate in a given deduction of the contradiction. Thus: Either the formula ...
Completeness theorems and lambda
... What happens if one restricts system F to the subsystem F0 where one allows only to form ΠX.T (X) if T (X) is built only with X and →?? Can one use the techniques of proof theory and give a predicative normalisation proof for this fragment? I. Takeuti gave such a proof in 1993, following G. Takeuti, ...
... What happens if one restricts system F to the subsystem F0 where one allows only to form ΠX.T (X) if T (X) is built only with X and →?? Can one use the techniques of proof theory and give a predicative normalisation proof for this fragment? I. Takeuti gave such a proof in 1993, following G. Takeuti, ...
Beginning Logic - University of Notre Dame
... We will define what it means for a statement in a propositional or predicate language to be true in an appropriate formal setting. To show that an argument is not valid, we will look for a “counter-example”, a setting in which the premises are all true and the conclusion is false. IV. Analysis of ar ...
... We will define what it means for a statement in a propositional or predicate language to be true in an appropriate formal setting. To show that an argument is not valid, we will look for a “counter-example”, a setting in which the premises are all true and the conclusion is false. IV. Analysis of ar ...
Modular Construction of Complete Coalgebraic Logics
... structor. The associated language is similar to the standard modal language over the empty set of atomic propositions. However, this language will be interpreted over Id-coalgebras, which provide a trivial model of deterministic systems. (3) If M is a (possibly infinite) set of modal operators with ...
... structor. The associated language is similar to the standard modal language over the empty set of atomic propositions. However, this language will be interpreted over Id-coalgebras, which provide a trivial model of deterministic systems. (3) If M is a (possibly infinite) set of modal operators with ...
Discrete Mathematics: Chapter 2, Predicate Logic
... Once a universe of discourse is specified, variables and quantifiers automatically obtain a definite context and range. Universal statements are understood to be statements about all the objects belonging to that universe of discourse; existential statements claim the existence of an object in the ...
... Once a universe of discourse is specified, variables and quantifiers automatically obtain a definite context and range. Universal statements are understood to be statements about all the objects belonging to that universe of discourse; existential statements claim the existence of an object in the ...
CATEGORICAL MODELS OF FIRST
... In classical categories this operation is idempotent; Φ ∗ Φ = Φ. In the Boolean categories of [45], there are given examples where this is also the case (which are a special case of classical categories) and also models where this equality does not hold; these non-idempotent models are intruiguing ...
... In classical categories this operation is idempotent; Φ ∗ Φ = Φ. In the Boolean categories of [45], there are given examples where this is also the case (which are a special case of classical categories) and also models where this equality does not hold; these non-idempotent models are intruiguing ...
X - UOW
... In a similar way, Logic deals with statements or sentences by defining symbols and establishing ‘rules’. Roughly speaking, in arithmetic an operation is a rule for producing new numbers from a pair of given numbers, like addition (+) or multiplication (× ). In logic, we form new statements by combi ...
... In a similar way, Logic deals with statements or sentences by defining symbols and establishing ‘rules’. Roughly speaking, in arithmetic an operation is a rule for producing new numbers from a pair of given numbers, like addition (+) or multiplication (× ). In logic, we form new statements by combi ...
Logic Part II: Intuitionistic Logic and Natural Deduction
... 2. This proof contains of a proof of a. 3. It also contains a proof of b . 4. So if we take the proof of b and put it together with the proof of a, we obtain a proof of b ...
... 2. This proof contains of a proof of a. 3. It also contains a proof of b . 4. So if we take the proof of b and put it together with the proof of a, we obtain a proof of b ...
The Foundations
... statements built from simpler statements using so-called Boolean connectives. Some applications in computer science: Design of digital electronic circuits. Expressing conditions in programs. George Boole Queries to databases & search engines. (1815-1864) ...
... statements built from simpler statements using so-called Boolean connectives. Some applications in computer science: Design of digital electronic circuits. Expressing conditions in programs. George Boole Queries to databases & search engines. (1815-1864) ...
Refinement Modal Logic
... Definition 1 (Bisimulation, simulation, refinement) Let two models M = (S, R, V ) and M ′ = (S ′ , R′ , V ′ ) be given. A non-empty relation R ⊆ S × S ′ is a bisimulation if for all (s, s′ ) ∈ R and a ∈ A: atoms s ∈ V (p) iff s′ ∈ V ′ (p) for all p ∈ P ; forth-a for all t ∈ S, if Ra (s, t), then the ...
... Definition 1 (Bisimulation, simulation, refinement) Let two models M = (S, R, V ) and M ′ = (S ′ , R′ , V ′ ) be given. A non-empty relation R ⊆ S × S ′ is a bisimulation if for all (s, s′ ) ∈ R and a ∈ A: atoms s ∈ V (p) iff s′ ∈ V ′ (p) for all p ∈ P ; forth-a for all t ∈ S, if Ra (s, t), then the ...
Barwise: Infinitary Logic and Admissible Sets
... relation and operation symbols with finitely many argument places. As usual, by an L-structure M, we mean a universe set M with an interpretation for each symbol of L. In cases where the vocabulary L is clear, we may just say structure. For a given vocabulary L and infinite cardinals µ ≤ κ, Lκµ is t ...
... relation and operation symbols with finitely many argument places. As usual, by an L-structure M, we mean a universe set M with an interpretation for each symbol of L. In cases where the vocabulary L is clear, we may just say structure. For a given vocabulary L and infinite cardinals µ ≤ κ, Lκµ is t ...
Teach Yourself Logic 2017: A Study Guide
... about to mention, skipping quickly over what you already know. L3. If you have taken an elementary logic course based on a substantial text like the ones mentioned in just a moment, then you should be well prepared. Here then, for those that need them, are two initial suggestions of formal logic boo ...
... about to mention, skipping quickly over what you already know. L3. If you have taken an elementary logic course based on a substantial text like the ones mentioned in just a moment, then you should be well prepared. Here then, for those that need them, are two initial suggestions of formal logic boo ...
Tableau-based decision procedure for the full
... hence the latter test is equivalent to the former. The advantage of working with Hintikka structures lies in the fact that they contain just as much semantic information about θ as is necessary for computing its truth value at a distinguished state. More precisely, while models provide the truth val ...
... hence the latter test is equivalent to the former. The advantage of working with Hintikka structures lies in the fact that they contain just as much semantic information about θ as is necessary for computing its truth value at a distinguished state. More precisely, while models provide the truth val ...
doc
... (a) Explain what it means to say that a sentence in the predicate calculus is semantically true. Also state what it means for a sequent to be semantically valid in predicate logic. A sentence in the predicate calculus is semantically true iff the sentence is true in very interpretation of the predic ...
... (a) Explain what it means to say that a sentence in the predicate calculus is semantically true. Also state what it means for a sequent to be semantically valid in predicate logic. A sentence in the predicate calculus is semantically true iff the sentence is true in very interpretation of the predic ...
? A Unified Semantic Framework for Fully
... (generalized) Kripke valuations for which it is strongly sound and complete. In fact, we provide a uniform method to obtain these sets of Kripke valuations. In many important cases, the usual well-known soundness and completeness theorems for known calculi are simple corollaries of this general meth ...
... (generalized) Kripke valuations for which it is strongly sound and complete. In fact, we provide a uniform method to obtain these sets of Kripke valuations. In many important cases, the usual well-known soundness and completeness theorems for known calculi are simple corollaries of this general meth ...