Dependence Logic
... We can replace “all M” by “some M with more than one element in the universe” in the above corollary. Note that in particular we do not have for all X 6= ∅: M |=X ¬ =(x0, x1) ⇐⇒ M 6|=X =(x0, x1). Example 10 Every team X, the domain of which contains xi and xj is of type xi = xj ∨ ¬xi = xj , as we ca ...
... We can replace “all M” by “some M with more than one element in the universe” in the above corollary. Note that in particular we do not have for all X 6= ∅: M |=X ¬ =(x0, x1) ⇐⇒ M 6|=X =(x0, x1). Example 10 Every team X, the domain of which contains xi and xj is of type xi = xj ∨ ¬xi = xj , as we ca ...
Relevant Logic A Philosophical Examination of Inference Stephen Read February 21, 2012
... many readers, particularly those not already well-versed in the subject, will find that work an uncompromising and difficult one. By far the best introduction so far to the subject is Michael Dunn’s survey article, ‘Relevant Logic and Entailment’.3 The present book does not attempt to compete with D ...
... many readers, particularly those not already well-versed in the subject, will find that work an uncompromising and difficult one. By far the best introduction so far to the subject is Michael Dunn’s survey article, ‘Relevant Logic and Entailment’.3 The present book does not attempt to compete with D ...
Intuitionistic and Modal Logic
... with logicism, Frege’s idea that mathematics is no more than logic, since mathematics can be reduced to it, a view supported by Russell (not a Platonist) at the time. • Formalism. Most famous modern representative: Hilbert. View that there are no mathematical objects, no mathematical truths, just fo ...
... with logicism, Frege’s idea that mathematics is no more than logic, since mathematics can be reduced to it, a view supported by Russell (not a Platonist) at the time. • Formalism. Most famous modern representative: Hilbert. View that there are no mathematical objects, no mathematical truths, just fo ...
Lecture - 04 (Logic Knowledge Base)
... Logic and Rule-based Systems • Logic: an important area of mathematics – lots of attention in 19th and early 20th centuries, in an attempt to find a mathematical language for discussing the world – Still many specialists in the area, especially in computer science: logic as computation ...
... Logic and Rule-based Systems • Logic: an important area of mathematics – lots of attention in 19th and early 20th centuries, in an attempt to find a mathematical language for discussing the world – Still many specialists in the area, especially in computer science: logic as computation ...
On the Question of Absolute Undecidability
... discrete topology on ω. As a topological space ω ω is homeomorphic to the standard space of irrationals. In addition to this space we will also be interested in the n-dimensional product spaces (ω ω )n . Given a subset A of (ω ω )n+1 the complement of A is just the set of elements not in A and the p ...
... discrete topology on ω. As a topological space ω ω is homeomorphic to the standard space of irrationals. In addition to this space we will also be interested in the n-dimensional product spaces (ω ω )n . Given a subset A of (ω ω )n+1 the complement of A is just the set of elements not in A and the p ...
Higher Order Logic - Indiana University
... Higher order logics, long considered by many to be an esoteric subject, are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of comp ...
... Higher order logics, long considered by many to be an esoteric subject, are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of comp ...
Higher Order Logic - Theory and Logic Group
... Higher order logics, long considered by many to be an esoteric subject, are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of comp ...
... Higher order logics, long considered by many to be an esoteric subject, are increasingly recognized for their foundational importance and practical usefulness, notably in Theoretical Computer Science. In this chapter we try to present a survey of some issues and results, without any pretense of comp ...
Chapter 6: The Deductive Characterization of Logic
... reflexivity rule for identity (given “nothing”, one is entitled to write down ‘τ = τ’ for any singular term). The existence of zero-place rules is critical if we are to have a non-trivial notion of proof, as defined in the previous section. In particular, a proof must have a first line; since such a ...
... reflexivity rule for identity (given “nothing”, one is entitled to write down ‘τ = τ’ for any singular term). The existence of zero-place rules is critical if we are to have a non-trivial notion of proof, as defined in the previous section. In particular, a proof must have a first line; since such a ...
Peano`s Arithmetic
... this. Other mathematicians, such as Hermann Grassmann and Gottlob Frege, had already been working to systemize arithmetic and its axioms. Grassmann, in fact, was the first to begin this process in his book Lehrbuch der arithmetik (1861). Frege made more progress on this and focused more on the logic ...
... this. Other mathematicians, such as Hermann Grassmann and Gottlob Frege, had already been working to systemize arithmetic and its axioms. Grassmann, in fact, was the first to begin this process in his book Lehrbuch der arithmetik (1861). Frege made more progress on this and focused more on the logic ...
Multiverse Set Theory and Absolutely Undecidable Propositions
... hence neither in sets defined in terms of them. This is particularly 11 The iterative concept of set was first suggested by Mirimano↵ [19] and made explicit by von Neumann [32]. For a thorough discussion of this concept of set see [4] and [20]. 12 Quickly found unnecessary. 13 Apparently this explan ...
... hence neither in sets defined in terms of them. This is particularly 11 The iterative concept of set was first suggested by Mirimano↵ [19] and made explicit by von Neumann [32]. For a thorough discussion of this concept of set see [4] and [20]. 12 Quickly found unnecessary. 13 Apparently this explan ...
full text (.pdf)
... example, Aiken et al. 2] prove that mixed positive and negative set constraints are strictly more expressive than positive constraints alone. The proof of this result is based on a compactness lemma which states that a system of positive constraints is satisable if and only if all nite subsets ar ...
... example, Aiken et al. 2] prove that mixed positive and negative set constraints are strictly more expressive than positive constraints alone. The proof of this result is based on a compactness lemma which states that a system of positive constraints is satisable if and only if all nite subsets ar ...
John Nolt – Logics, chp 11-12
... present, which is in a sense uniquely real (for the past exists no longer, the future not yet), so there is a specific possible world, the actual world, which (for us at least) is uniquely real. A second point of analogy is that in nonpresent moments objects have different properties from those they ...
... present, which is in a sense uniquely real (for the past exists no longer, the future not yet), so there is a specific possible world, the actual world, which (for us at least) is uniquely real. A second point of analogy is that in nonpresent moments objects have different properties from those they ...
abdullah_thesis_slides.pdf
... Given d,t ∈ N, we can define the concept of type signatures of radius d with threshold t such that the values (#Type1 ,...,#Typen ) are counted only upto a threshold t and anything ≥ t is considered ∞. Two structures A and B, are said to be d-equivalent with threshold t if their type signatures with ...
... Given d,t ∈ N, we can define the concept of type signatures of radius d with threshold t such that the values (#Type1 ,...,#Typen ) are counted only upto a threshold t and anything ≥ t is considered ∞. Two structures A and B, are said to be d-equivalent with threshold t if their type signatures with ...
Reading 2 - UConn Logic Group
... fragments of Int, classical logic, etc. ([21], [88], [89]). Abstract computational and functional semantics for Int which did not address the issue of the original BHK semantics for Int were also studied in [71], [94] and many other papers (cf. [18], [22], [106]). Kuznetsov-Muravitsky-Goldblatt sema ...
... fragments of Int, classical logic, etc. ([21], [88], [89]). Abstract computational and functional semantics for Int which did not address the issue of the original BHK semantics for Int were also studied in [71], [94] and many other papers (cf. [18], [22], [106]). Kuznetsov-Muravitsky-Goldblatt sema ...
Implicit Hitting Set Algorithms for Reasoning Beyond NP
... Implicit Hitting Set Algorithms for Reasoning Beyond NP Paul Saikko and Johannes P. Wallner and Matti Järvisalo HIIT, Department of Computer Science, University of Helsinki, Finland ...
... Implicit Hitting Set Algorithms for Reasoning Beyond NP Paul Saikko and Johannes P. Wallner and Matti Järvisalo HIIT, Department of Computer Science, University of Helsinki, Finland ...
PPT - UBC Department of CPSC Undergraduates
... on our way to a full computer: sequential circuits. The pre-class goals are to be able to: – Trace the operation of a deterministic finite-state automaton (represented as a diagram) on an input, including indicating whether the DFA accepts or rejects the input. – Deduce the language accepted by a si ...
... on our way to a full computer: sequential circuits. The pre-class goals are to be able to: – Trace the operation of a deterministic finite-state automaton (represented as a diagram) on an input, including indicating whether the DFA accepts or rejects the input. – Deduce the language accepted by a si ...
A Qualitative Theory of Dynamic Interactive Belief Revision
... [6, 31, 22, 4, 23, 39, 5, 15, 16] etc. In its standard incarnation, as presented e.g., in the recent textbook [25], the DEL approach is particularly well fit to deal with complex multi-agent learning actions by which groups of interactive agents update their beliefs (including higher-level beliefs a ...
... [6, 31, 22, 4, 23, 39, 5, 15, 16] etc. In its standard incarnation, as presented e.g., in the recent textbook [25], the DEL approach is particularly well fit to deal with complex multi-agent learning actions by which groups of interactive agents update their beliefs (including higher-level beliefs a ...
Equality in the Presence of Apartness: An Application of Structural
... This is the “Euclidean” formulation of transitivity of equality. Symmetry follows by setting c identical to a. Brouwer introduced a whole range of notions relating to the notion of apartness, often with very idiosyncratic notations or terminologies. For example, in his (1927, 1950) a strict linear o ...
... This is the “Euclidean” formulation of transitivity of equality. Symmetry follows by setting c identical to a. Brouwer introduced a whole range of notions relating to the notion of apartness, often with very idiosyncratic notations or terminologies. For example, in his (1927, 1950) a strict linear o ...
LOGICAL CONSEQUENCE AS TRUTH-PRESERVATION STEPHEN READ Abstract
... which is truth-preserving, that is, for which it is impossible that the premises be true and the conclusion false. Moreover, suppose that we are persuaded that the premises are true. It follows that it is impossible for the conclusion to be false. Persuaded, for the present, by Restall, Moore, Schur ...
... which is truth-preserving, that is, for which it is impossible that the premises be true and the conclusion false. Moreover, suppose that we are persuaded that the premises are true. It follows that it is impossible for the conclusion to be false. Persuaded, for the present, by Restall, Moore, Schur ...
Logic
... Sometimes we need to say that one or another statement is true. Consider P: Adam’s eyes are blue. Q: Eve’s hair is brown. The statement P ∨Q is read “P or Q” and means that either P is true, Q is true, or both are true. This statement is still true if Adam’s eyes are blue and Eve’s hair is brown. Th ...
... Sometimes we need to say that one or another statement is true. Consider P: Adam’s eyes are blue. Q: Eve’s hair is brown. The statement P ∨Q is read “P or Q” and means that either P is true, Q is true, or both are true. This statement is still true if Adam’s eyes are blue and Eve’s hair is brown. Th ...
Frege, Boolos, and Logical Objects
... the existence of abstract objects in classical logic. The most well-known of these is: Hume’s Principle: The number of F s is identical to the number of Gs iff there is a one-to-one correspondence between the F s and the Gs. #F = #G ↔ F ≈ G When added to classical second-order logic (but not free sec ...
... the existence of abstract objects in classical logic. The most well-known of these is: Hume’s Principle: The number of F s is identical to the number of Gs iff there is a one-to-one correspondence between the F s and the Gs. #F = #G ↔ F ≈ G When added to classical second-order logic (but not free sec ...
Using linear logic to reason about sequent systems
... Cut and initial provide dual information The initial formula bBc ...
... Cut and initial provide dual information The initial formula bBc ...
Ans - Logic Matters
... (d) Since terms cannot contain either ‘=’ or ‘<’, we can simply ask of an expression e: is it of the form e1 = e2 or e1 < e2 for some e1 and e2 ? It is a mechanical business to test if so. If ‘no’, then e isn’t an atomic wff. If ‘yes’, then e is an atomic wff iff both e1 and e2 are terms, and that’s ...
... (d) Since terms cannot contain either ‘=’ or ‘<’, we can simply ask of an expression e: is it of the form e1 = e2 or e1 < e2 for some e1 and e2 ? It is a mechanical business to test if so. If ‘no’, then e isn’t an atomic wff. If ‘yes’, then e is an atomic wff iff both e1 and e2 are terms, and that’s ...
Formale Methoden der Softwaretechnik Formal methods of software
... Till Mossakowski, Christoph Lüth ...
... Till Mossakowski, Christoph Lüth ...
A Logical Expression of Reasoning
... and serious concern from the tradition. That is why for so long, at least until Hume [12], but even after him (deductive) logic and correct reasoning, or even “rationality”, has been taken as synonymous. However, real life reasoning, meaning a wide range of forms of inference, covering from common s ...
... and serious concern from the tradition. That is why for so long, at least until Hume [12], but even after him (deductive) logic and correct reasoning, or even “rationality”, has been taken as synonymous. However, real life reasoning, meaning a wide range of forms of inference, covering from common s ...
Jesús Mosterín
Jesús Mosterín (born 1941) is a leading Spanish philosopher and a thinker of broad spectrum, often at the frontier between science and philosophy.