article - British Academy
... That is, Lucas only claims to have an argument that he, Lucas, is not machine M, given a relevant, adequate specification of M. David Lewis (1979b) replies as follows. In insisting on the dialectical character of the argument, Lucas is insisting that his output depends on his input: what sentence he ...
... That is, Lucas only claims to have an argument that he, Lucas, is not machine M, given a relevant, adequate specification of M. David Lewis (1979b) replies as follows. In insisting on the dialectical character of the argument, Lucas is insisting that his output depends on his input: what sentence he ...
An Introduction to Proof Theory - UCSD Mathematics
... L-formulas. A language L is complete if and only if every Boolean function can be defined by an L-formula. Propositional logic can be formulated with any complete (usually finite) language L — for the time being, we shall use the language ¬, ∧, ∨ and ⊃. A propositional formula A is said to be a taut ...
... L-formulas. A language L is complete if and only if every Boolean function can be defined by an L-formula. Propositional logic can be formulated with any complete (usually finite) language L — for the time being, we shall use the language ¬, ∧, ∨ and ⊃. A propositional formula A is said to be a taut ...
Problems on Discrete Mathematics1 (Part I)
... stead, we will study some naive concepts of sets; most of them are intuitively understandable from our daily-life experiences. For example, all students of Syracuse University is a set; all students in the United State of America is a superset of the set of students at Syracuse University. Since Den ...
... stead, we will study some naive concepts of sets; most of them are intuitively understandable from our daily-life experiences. For example, all students of Syracuse University is a set; all students in the United State of America is a superset of the set of students at Syracuse University. Since Den ...
The substitutional theory of logical consequence
... ‘intended interpretation’ is not one of these models and cannot easily be identified with one of these models. Models have set-sized domains, while the intended interpretation, if it could be conceived as a model, cannot be limited by any cardinality. Similarly, logical truth defined as truth in all ...
... ‘intended interpretation’ is not one of these models and cannot easily be identified with one of these models. Models have set-sized domains, while the intended interpretation, if it could be conceived as a model, cannot be limited by any cardinality. Similarly, logical truth defined as truth in all ...
Characterizations of stable model semantics for logic programs with
... 2003; Elkabani et al. 2004; Faber et al. 2004; Marek and Remmel 2004; Marek and Truszczynski 2004; Pelov 2004; Pelov and Truszczynski 2004; Calimeri et al . 2005; Elkabani et al. 2005; Ferraris 2005; Liu and Truszczynski 2005; Liu and Truszczynski 2006; Son et al. 2006; Liu et al. 2007; Pelov et al. ...
... 2003; Elkabani et al. 2004; Faber et al. 2004; Marek and Remmel 2004; Marek and Truszczynski 2004; Pelov 2004; Pelov and Truszczynski 2004; Calimeri et al . 2005; Elkabani et al. 2005; Ferraris 2005; Liu and Truszczynski 2005; Liu and Truszczynski 2006; Son et al. 2006; Liu et al. 2007; Pelov et al. ...
Logic and Proof
... Aristotle observed that the correctness of this inference has nothing to do with the truth or falsity of the individual statements, but, rather, the general pattern: Every A is B. Every B is C. Therefore every A is C. We can substitute various properties for A, B, and C; try substituting the propert ...
... Aristotle observed that the correctness of this inference has nothing to do with the truth or falsity of the individual statements, but, rather, the general pattern: Every A is B. Every B is C. Therefore every A is C. We can substitute various properties for A, B, and C; try substituting the propert ...
Chapter 9: Initial Theorems about Axiom System AS1
... Deduction Theorem. [In later chapters, we prove more important theorems, but they are not purely axiomatic, since they concern the relation between deductive entailment and semantic entailment.] In proving the deduction theorem, we employ mathematical induction, which is discussed in Chapter 3, and ...
... Deduction Theorem. [In later chapters, we prove more important theorems, but they are not purely axiomatic, since they concern the relation between deductive entailment and semantic entailment.] In proving the deduction theorem, we employ mathematical induction, which is discussed in Chapter 3, and ...
The Emergence of First
... he took a copy of this article with him and delivered it to De Morgan (Fisch 1984, xxxiii). Unfortunately, De Morgan was already in the decline that led to his death the following March. Peirce did not find a better reception when he gave a copy of the article to Stanley Jevons, who had elaborated B ...
... he took a copy of this article with him and delivered it to De Morgan (Fisch 1984, xxxiii). Unfortunately, De Morgan was already in the decline that led to his death the following March. Peirce did not find a better reception when he gave a copy of the article to Stanley Jevons, who had elaborated B ...
Deductive Databases with Universally Quantified Conditions
... a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. ...
... a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. a n-ary predicate symbol, t1, ..., tn are terms exactly one of are either constant or variable symbols. ...
Default reasoning using classical logic
... In the sequel to this section we will formally justify the translations illustrated above, present the general algorithms, and give more examples. The rest of the paper is organized as follows: After introducing some preliminary de nitions in Section 2, we provide in Section 3 the concept of a mode ...
... In the sequel to this section we will formally justify the translations illustrated above, present the general algorithms, and give more examples. The rest of the paper is organized as follows: After introducing some preliminary de nitions in Section 2, we provide in Section 3 the concept of a mode ...
Inference in FOL Last Lecture
... Nintendo says it is Criminal for a programmer to provide emulators to people. My friends don’t have a Nintendo 64, but they use software that runs N64 games on their PC, which is written by Reality Man, who is a programmer. ...
... Nintendo says it is Criminal for a programmer to provide emulators to people. My friends don’t have a Nintendo 64, but they use software that runs N64 games on their PC, which is written by Reality Man, who is a programmer. ...
article in press - School of Computer Science
... InclPP (z1 , z2 ) = P (z1 , z2 ) ∨ P (z1 , z2 ) Theorem 12. Let φ ∈ GF 2mon and C be an acyclic set of mso closure conditions on relations 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 ...
... InclPP (z1 , z2 ) = P (z1 , z2 ) ∨ P (z1 , z2 ) Theorem 12. Let φ ∈ GF 2mon and C be an acyclic set of mso closure conditions on relations 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 ...
Counterfactuals
... sketched analysis of counterfactuals should give the same result. Unfortunately, it does precisely the opposite. In order for the first counterfactual to be true, φ → ψ must be true in every world sufficiently similar to the actual world; similarly, in order for the second to hold (φ ∧ φ0 ) → ¬ψ mus ...
... sketched analysis of counterfactuals should give the same result. Unfortunately, it does precisely the opposite. In order for the first counterfactual to be true, φ → ψ must be true in every world sufficiently similar to the actual world; similarly, in order for the second to hold (φ ∧ φ0 ) → ¬ψ mus ...
The Computer Modelling of Mathematical Reasoning Alan Bundy
... This book started as notes for a postgraduate course in Mathematical Reasoning given in the Department of Artificial Intelligence at Edinburgh from 1979 onwards. Students on the course are drawn from a wide range of backgrounds: Psychology, Computer Science, Mathematics, Education, etc. The first dr ...
... This book started as notes for a postgraduate course in Mathematical Reasoning given in the Department of Artificial Intelligence at Edinburgh from 1979 onwards. Students on the course are drawn from a wide range of backgrounds: Psychology, Computer Science, Mathematics, Education, etc. The first dr ...
Propositional Logic
... of a theorem of Church, there is no procedure that will halt for every input formula and decide whether or not a formula is valid. There are many ways of proving the completeness of a proof system. Oddly, most proofs establishing completeness only show that if a formula A is valid, then there exists ...
... of a theorem of Church, there is no procedure that will halt for every input formula and decide whether or not a formula is valid. There are many ways of proving the completeness of a proof system. Oddly, most proofs establishing completeness only show that if a formula A is valid, then there exists ...
The Expressive Power of Modal Dependence Logic
... Using this observation Ebbing et al. [3] showed that in terms of expressiveness, EMDL is contained in ML(>). However, it was left open, whether the containment is strict, or whether EMDL and ML(>) are actually equivalent with respect to expressive power. Team semantics is also meaningful in the cont ...
... Using this observation Ebbing et al. [3] showed that in terms of expressiveness, EMDL is contained in ML(>). However, it was left open, whether the containment is strict, or whether EMDL and ML(>) are actually equivalent with respect to expressive power. Team semantics is also meaningful in the cont ...
Inquiry
An inquiry is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim.