Bristol 6.8. January 2005This is the first of a series of meetings (supported by the British Academy) of the research group in Logical Methods in Epistemology, Semantics and Philosophy of Mathematics which aims to investigate, and implement, the use of mathematical and technical logical apparatus in philosophical fields, in particular, but not exclusively, philosophical logic, theories of truth, and the philosophy of mathematics. The first of these will take place at the University of Bristol Mathematics Department. Participation is free, but we ask you to inform us at least by 20th December if you plan to attend the meeting: volker.halbach at philosophy.oxford.ac.uk Please replace "at" by the usual symbol. VenueAll talks will be in the School of Mathematics in Seminar Room SM3, which is on the 1st Floor relative to the Main entrance on University Walk. A map of the University precinct and further information on reaching Bristol and on hotel accomodation are also available. There is a direct coach service from Heathrow to Bristol. On the National Express website you make make reservations for tickets. It is much cheaper to buy a return ticket than to buy to single tickets. Bristol also has an airport. From the Train Station you can take the No. 8 or 8A Bus to the Clifton Down Road/Princess Victoria Street stop. From there the Rodney Hotel is about 80ms. From the Train Station to the University take the No. 9 Bus. TimetableThursday 6th January 2005
Friday 7th January 2005
Saturday 7th January 2005
Note for SpeakersThere will be an Overhead Slides Projector, Computer Projector, and White Boards available.
AbstractsBranden Fitelson: A UserFriendly Decision Procedure for the Probability Calculus, with Applications to Bayesian Confirmation TheoryA general mechanical procedure for reasoning about the probability calculus is presented. The procedure involves (1) a translation from probability calculus into the theory of real closed fields (TRCF), and (2) an application of a recent implementation of the CAD procedure for TRCF. The procedure is then used to solve various problems in Bayesian confirmation theory (some of which were open). Some issues of computational complexity and problem size will also be discussed. All necessary technical, historical, and philosophical background will be provided during the talk. Michael Glanzberg: Contexts and Unrestricted QuantificationMuch of the debate over the possibility of ‘absolutely unrestricted quantification’—quantification over a domain of ‘absolutely everything’—has centered on the paradoxes. To some, such as myself, careful consideration of the paradoxes shows absolutely unrestricted quantification to be impossible. But those of us who hold this sort of view face a challenge. For even if we establish on such general grounds that absolutely unrestricted quantification is impossible, we have still to account for the ways that our prima facie unrestricted quantifiers really function. This challenge is made all the more pressing, as Timothy Williamson has recently argued, by the appearance that some applications of quantifiers require them to be absolutely unrestricted. This paper takes up these challenges, by presenting a contextualist approach to unrestricted quantification. It argues that even our widest, syntactically unrestricted quantifiers, are subject to a special kind of contextual domain restriction. However, it also argues that this is not an instance of the ordinary sorts of contextual domain restrictions which apply to our uses of restricted quantifiers. Instead, our widest quantifiers are subject to a distinct sort of context dependence, which I label ‘extraordinary context dependence’. This paper argues that though unusual, extraordinary context dependence is possible, and sketches how it arises. It goes on to show that the rules governing extraordinary context dependence ensure that the domain of any maximal quantifier in a context functions nearly enough as if it were absolutely unrestricted. Because of this, the paper argues, our maximal quantifiers are suitable for their intended applications, in spite of being subject to extraordinary contextual restrictions. Jeff Ketland: SpeedUp, Indispensability and UnfeasibilityA standard argument against nominalism is the indispensability argument, which points out that our scientific understanding of the world is thoroughly mathematicized. Most examples thus far examined in the literature involve very abstract mathematics, usually from theoretical physics. I wish to discuss the application of mathematics to very simple problems of logical validity. The topic of applying mathematics to problems of logical validity has not been widely discussed. There is a paper by Hartry Field, "Is Mathematical Knowledge Just Logical Knowledge?" (1985) and some papers by George Boolos, concerning speedup. Nominalists seem to think that the nominalizability of finite cardinality statements like "There are n Fs" represents a success for their programme. I argue that even this is a failure. Consider the inference PHP(100) (thanks to my wife for the example): There are 101 dalmatians We all know that this is valid, but how? I call such logically valid inferences "quasiarithmetic". The validity of PHP(100) is an example of a quasiarithmetic "validity fact". Simple combinatorial mathematics is indispensable for seeing that PHP(100) is valid. The reason is that a direct logical verification for this inference would fill a 3,000 page book. In fact, we know that PHP(n) is valid for all n, and thus that PHP(100) is valid. But the nominalist rejects these assumptions about numbers, finite set and functions as false. So why does a false theory of numbers, sets, etc., yield true classification of such validityfacts? This is nothing short of a miracle! And if we reject miracles, then we should reject nominalism too. Postulating numbers, sets and functions is indispensable for seeing that certain valid inference are indeed valid. This "speedup argument against nominalism" is briefly sketched in a short paper "Some More Curious Inferences" (Analysis, Jan 2005). A forthcoming paper by John Burgess ("Protocol Sentences for Lite Logicism" on Burgess’s webpage) contains a similar argument. James Ladyman: Mathematical Structuralism and
the Identity of Mathematical Objects
