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Interpretability logic
 Mathematical Logic, Proceedings of the 1988 Heyting Conference
, 1990
"... Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength ..."
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Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove
The many faces of interpretability
"... Abstract. In this paper we discus work in progress on interpretability logics. We show how semantical considerations have allowed us to formulate nontrivial principles about formalized interpretability. In particular we falsify the conjecture about the nature of the interpretability logic of all re ..."
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Abstract. In this paper we discus work in progress on interpretability logics. We show how semantical considerations have allowed us to formulate nontrivial principles about formalized interpretability. In particular we falsify the conjecture about the nature of the interpretability logic of all reasonable arithmetical theories. We consider this an interesting example of how purely semantical considerations give new nontrivial facts about syntactical and arithmetical notions. In addition we give some apparatus that allows us to push ‘global ’ semantical properties into more ‘local ’ syntactical ones. With this apparatus, the rather wild behavior of the different interpretability logics are nicely formulated in a single notion that expresses their differences in a uniform way. This paper consists of three parts. We start of by giving a short introduction to interpretability logics. In the second part we discuss how a careful analysis of the modal semantical behaviour of interpretability logics lets us formulate nontrivial interpretability principles. In the third part we present a semantical bookkeeping tool which pushes ‘global ’ semantical considerations into ‘local’ syntactical ones. The hope is that this machinery will provide a general and uniform treatment of the bewildering field of modal interpretability logics. 1
ROBOT FREE WILL NOTICE
"... Abstract. We introduce the perspex machine which unifies projective geometry and Turing computation and results in a supraTuring machine. We show two ways in which the perspex machine unifies symbolic and nonsymbolic AI. Firstly, we describe concrete geometrical models that map perspexes onto neur ..."
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Abstract. We introduce the perspex machine which unifies projective geometry and Turing computation and results in a supraTuring machine. We show two ways in which the perspex machine unifies symbolic and nonsymbolic AI. Firstly, we describe concrete geometrical models that map perspexes onto neural networks, some of which perform only symbolic operations. Secondly, we describe an abstract continuum of perspex logics that includes both symbolic logics and a new class of continuous logics. We argue that an axiom in symbolic logic can be the conclusion of a perspex theorem. That is, the atoms of symbolic logic can be the conclusions of subatomic theorems. We argue that perspex space can be mapped onto the spacetime of the universe we inhabit. This allows us to discuss how a robot might be conscious, feel, and have free will in a deterministic, or semideterministic, universe. We ground the reality of our universe in existence. On a theistic point, we argue that preordination and free will are compatible. On a theological point, we argue that it is not heretical for us to give robots free will. Finally, we give a pragmatic warning as to the doubleedged risks of creating robots that do, or alternatively do not, have free will. 1
Press. Mathematical Induction and Induction in Mathematics
"... However much we many disparage deduction, it cannot be denied that the laws established by induction are not enough. Frege (1884/1974, p. 23) At the yearly proseminar for firstyear graduate students at Northwestern, we presented some evidence that reasoning draws on separate cognitive systems for a ..."
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However much we many disparage deduction, it cannot be denied that the laws established by induction are not enough. Frege (1884/1974, p. 23) At the yearly proseminar for firstyear graduate students at Northwestern, we presented some evidence that reasoning draws on separate cognitive systems for assessing deductive versus inductive arguments (Rips, 2001a, 2001b). In the experiment we described, separate groups of participants evaluated the same set of arguments for deductive validity or inductive strength. For example, one of the validity groups decided whether the conclusions of these arguments necessarily followed from the premises, while one of the strength groups decided how plausible the premises made the conclusions. The results of the study showed that the percentages of “yes ” responses (“yes ” the argument is deductively valid or “yes ” the argument is inductively strong) were differently ordered for the validity and the strength judgments. In some cases, for example, the validity groups judged Argument A to be valid more often than Argument B, but the strength groups judged B inductively strong more often than A. Reversals of this sort suggest that people do not just see arguments as ranging along a single
Crossing Two lines Zero pics Separated
, 2007
"... The fifth postulate of Euclid, the parallel postulate, was formulated in 300 B.C., and is still not proved as a theorem, although many have tried to do that. The reason for this is not to be found in the postulate per se, but in the definition of parallelism. The most common definition states that p ..."
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The fifth postulate of Euclid, the parallel postulate, was formulated in 300 B.C., and is still not proved as a theorem, although many have tried to do that. The reason for this is not to be found in the postulate per se, but in the definition of parallelism. The most common definition states that parallel lines never meet. It will be demonstrated here why this definition is wrong, and why parallelism is hard to define. One pic
Awareness Month. The Mathematical Structure of Escher’s Print Gallery
"... Month theme “Mathematics and Art”, this article brings together three different pieces about intersections between mathematics and the artwork of M. C. Escher. For more information about Mathematics Awareness Month, visit the website ..."
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Month theme “Mathematics and Art”, this article brings together three different pieces about intersections between mathematics and the artwork of M. C. Escher. For more information about Mathematics Awareness Month, visit the website
School Geometry
, 2008
"... In the following, Geometry refers to plane geometry. In designing the programme in Geometry for schools, it is essential to ensure the logical coherence of the system that is used. This is not something that can be taken for granted, and in fact, things have gone wrong in the past in Ireland[8]. We ..."
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In the following, Geometry refers to plane geometry. In designing the programme in Geometry for schools, it is essential to ensure the logical coherence of the system that is used. This is not something that can be taken for granted, and in fact, things have gone wrong in the past in Ireland[8]. We distinguish three levels: Level 1: The fullyrigorous level, likely to be intelligible only to professional mathematicians and advanced third and fourthlevel students. Level 2: The semiformal level, suitable for digestion by many children from (roughly) the age of 14 and upwards. Level 3: The informal level, suitable for younger children. In schools, one should begin with Level 3, developing the pupils ’ geometrical intuition and teaching them geometrical language, such as the names of various shapes. One should progress to geometrical problemsolving, addressing concrete questions. At an appropriate point (Secondary Year 3 seems to be favoured), one should introduce the semiformal Level 2, involving such things as definitions, theorems, and proofs. At no stage should the material 1 of the fullyrigorous Level 1 be used in school. Its purpose is to ensure that the semiformal level is gounded on a sound system, and to act as a guide to the construction of Level 2.