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Logic and Proof - Numeracy Workshop
Logic and Proof - Numeracy Workshop

... The truth or falsity of a converse can not be inferred from the truth or falsity of the original statement. For example, x = 2 ⇒ x2 = 4 is true, but . . . its converse x2 = 4 ⇒ x = 2 is false, because x could be equal to −2. ...
A Pebble Weighted Automata and Weighted Logics
A Pebble Weighted Automata and Weighted Logics

... Hoogeboom 1999; Bojańczyk et al. 2006; Samuelides and Segoufin 2007; Bojańczyk 2008], we actually define weighted generalizations that preserve their natural connections with logic. More precisely, we introduce pebble weighted automata on words and establish expressive equivalence to weighted firs ...
Reasoning about Complex Actions with Incomplete Knowledge: A
Reasoning about Complex Actions with Incomplete Knowledge: A

... focuses on ramification problem but does not provide a formalization of incomplete initial states with an explicit representation of undefined fluents. Such an explicit representation is needed if we want to model an agent which is capable of reasoning and acting on the basis of its (dis)beliefs. In ...
LPF and MPLω — A Logical Comparison of VDM SL and COLD-K
LPF and MPLω — A Logical Comparison of VDM SL and COLD-K

... the third idea. The corresponding notion of logical consequence seems too strong for a logic for the formal specification and verified design of software systems, which is only concerned with drawing true conclusions from true premises. For LPF, the underlying notion of logical consequence is the th ...
Constraint propagation
Constraint propagation

... There exists a reasoning technique, such that for any theory T and formula F, such that T |= F, the reasoning technique proves T |= F. ...
logic, programming and prolog (2ed)
logic, programming and prolog (2ed)

... and also provides a logical justification of the rule by extending the user’s program with additional axioms. Thereafter definite programs are generalized to general programs. The resulting proof-technique of this language is called SLDNF-resolution and is a result of combining SLD-resolution with t ...
Constructing Cut Free Sequent Systems With Context Restrictions
Constructing Cut Free Sequent Systems With Context Restrictions

... easily modified to treat e.g. minimal logic [9] or the {∧, ∨}-fragment of intuitionistic logic as base logics, but since we are not aware of modal logics based on either of these we restrict ourselves to the classical and intuitionistic cases. Related Work: The method of cut elimination by saturatio ...
The History of Categorical Logic
The History of Categorical Logic

... was also the year of the Durham meeting on applications of sheaf theory to logic, algebra and analysis, whose proceedings were published in 1979. We submit that around the end of the nineteen seventies, categorical logic was on firm ground and could be developed in various directions, which is preci ...
CATEGORICAL MODELS OF FIRST
CATEGORICAL MODELS OF FIRST

... as an infinitary conjunction or disjunction over the domain of quantification. While it does not directly address the other notions of model above, it is worth noting that techniques developed in this thesis should adapt successfully to them. The issue of finding a notion of equality on proofs is se ...
Proof Theory for Propositional Logic
Proof Theory for Propositional Logic

... Proof Theory for Propositional Logic ........................................................................................... 20 Miminal Logic .......................................................................................................................... 20 6- Propositional Minimal Log ...
Beginning Logic - University of Notre Dame
Beginning Logic - University of Notre Dame

... we compare the sizes of the set of counting numbers 1, 2, 3, . . . and the set of real numbers between 0 and 1? Cantor developed a way to compare these sets and then gave a surprising construction to show that the set of real numbers is larger. Cantor’s idea, diagonalization, has been used in many o ...
The Pure Calculus of Entailment Author(s): Alan Ross Anderson and
The Pure Calculus of Entailment Author(s): Alan Ross Anderson and

... One might find those who would object that "if ... then -" doesn't seem to be symmetrical, and that the third axiom is objectionable. But our logician has an answer to that. "There is nothing paradoxical about the third axiom; it is just a matter of understanding the formalism properly. 'If A then B ...
The Journal of Functional and Logic Programming The MIT Press
The Journal of Functional and Logic Programming The MIT Press

... constraint containing them is processed. From that point on, they can only be bound to terms of a particular domain. Thus, although in one way or another all CLP systems use more than one constraint domain, they do not freely allow mixed terms or constraints, that is, constraint expressions containi ...
Dealing with imperfect information in Strategy Logic
Dealing with imperfect information in Strategy Logic

... Concerning ATL, many variants have been introduced that deal with imperfect information [12, 14, 22, 13]. Some of these numerous logics deal with strategizing under imperfect information, some with reasoning about knowledge; because it is not natural to reason about the knowledge of agents with impe ...
P,Q
P,Q

...  1. each Ai (i
Extracting Proofs from Tabled Proof Search
Extracting Proofs from Tabled Proof Search

... be valid, i.e., proved elsewhere using (co-)inductive techniques. Actually, Strategy I collapses into Strategy II if the empty set is an allowed theory. Soundness of these two strategies is not difficult to establish and it follows the work presented in [8]. Strategy III is sound only when the table ...
Combinaison des logiques temporelle et déontique pour la
Combinaison des logiques temporelle et déontique pour la

... In order to formally specify a security policy, it is natural to reason about time on the one hand, and obligations, permissions, and prohibitions on the other hand. Indeed, we have to express for instance the permission to access a resource for a certain period, the obligation to release a resource ...
Weyl`s Predicative Classical Mathematics as a Logic
Weyl`s Predicative Classical Mathematics as a Logic

... holds that the use of the axiom of excluded middle is illegitimate, and predicativism, which holds that definitions that involve a certain form of circularity are illegitimate. Proof assistants or proof checkers are tools that help the user to construct formal proofs of theorems in these formal logi ...
A Judgmental Reconstruction of Modal Logic
A Judgmental Reconstruction of Modal Logic

... connective should be locally sound and complete in order to have a satisfactory meaning explanation for the connective. Local soundness and completeness are witnessed by local reductions and expansions of proofs, respectively. Note that there are other ways to define meaning. For example, we frequen ...
Argumentations and logic
Argumentations and logic

... the basis of what we already know or do we need new information? 1. Settling Hypotheses by Argumentations. Argumentation is involved in settling hypotheses on the basis of what we already know. Every argumentation that deduces the hypothesis from premises already known to be true proves the hypothes ...
? A Unified Semantic Framework for Fully
? A Unified Semantic Framework for Fully

... The names given to the rules in this table will be used below. Definition 3.4. A basic system is a set of basic rules in which (cut), (id), and the two weakening rules, (W ⇒) and (⇒ W ), are all included. We denote by ΥG all other rules of a basic system G, and by ΠG the set of context-relations app ...
An Introduction to Proof Theory - UCSD Mathematics
An Introduction to Proof Theory - UCSD Mathematics

... Proof Theory is the area of mathematics which studies the concepts of mathematical proof and mathematical provability. Since the notion of “proof” plays a central role in mathematics as the means by which the truth or falsity of mathematical propositions is established; Proof Theory is, in principle ...
Die Grundlagen der Arithmetik §§82–83
Die Grundlagen der Arithmetik §§82–83

... 6. Every number except 0 immediately follows a number in the natural sequence of numbers. It is clear from §44 of Grundgesetze6 that Frege did not take (6) to imply that 0 does not immediately follow a number, that ¬xP0. This proposition is proved separately in Grundgesetze, as Theorem 108, and will ...
Revisiting Preferences and Argumentation
Revisiting Preferences and Argumentation

... - the ordinary and assumption premises in A are exactly those in {A1 , . . . , An }; - the defeasible rules in A are exactly those in {A1 , . . . , An }; - the strict rules and axiom premises of A are a superset of the strict rules and axiom premises in {A1 , . . . , An }. Notice that if B defeats s ...
tbmk5ictk6
tbmk5ictk6

... two examples of valid arguments; one of them was sound and the other was not. Since both examples were valid, the one with true premises was the one that was sound. We also saw two examples of invalid arguments. Both of those are unsound simply because they are invalid. Sound arguments have to be v ...
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History of logic

The history of logic is the study of the development of the science of valid inference (logic). Formal logic was developed in ancient times in China, India, and Greece. Greek logic, particularly Aristotelian logic, found wide application and acceptance in science and mathematics.Aristotle's logic was further developed by Islamic and Christian philosophers in the Middle Ages, reaching a high point in the mid-fourteenth century. The period between the fourteenth century and the beginning of the nineteenth century was largely one of decline and neglect, and is regarded as barren by at least one historian of logic.Logic was revived in the mid-nineteenth century, at the beginning of a revolutionary period when the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in mathematics. The development of the modern ""symbolic"" or ""mathematical"" logic during this period is the most significant in the two-thousand-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history.Progress in mathematical logic in the first few decades of the twentieth century, particularly arising from the work of Gödel and Tarski, had a significant impact on analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic.
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