Bristol 6.-8. January 2005
This 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.
All 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.
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.
Thursday 6th January 2005
Friday 7th January 2005
Saturday 7th January 2005
Note for Speakers
There will be an Overhead Slides Projector, Computer Projector, and White Boards available.
Branden Fitelson: A User-Friendly Decision Procedure for the Probability Calculus, with Applications to Bayesian Confirmation Theory
A 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.
Much 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.
A 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 speed-up. 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 "quasi-arithmetic". The validity of PHP(100) is an example of a quasi-arithmetic "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 validity-facts? 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 "speed-up 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 Burgesss webpage) contains a similar argument.
Mathematical structuralism comes in various varieties and faces many problems. In this paper I offer a solution to two problems that have been raised for non-eliminative structuralism about mathematical objects of the kind that has been defended by Stewart Shapiro. The first concerns the truth-values of identity statements concerning mathematical objects in different structures, for example, 'the natural number 1 is identical to the real number 1'. The second concerns the violation of the principle of the identity of indiscernibles to which the non-eliminative structuralist seems committed.
Carnap's Aufbau is usually regarded as a famous - perhaps even notorious - failure. I want to show that certain parts of the Aufbau programme can actually be saved and be put to work. However, in order to do so, Carnap's original intentions have to be lowered and various problems that affect the original Aufbau system have to be solved or circumvented. In my talk I will give a sketch of a new constitution system and I will try to outline how it addresses the well-known difficulties concerning (i) the constitutional basis, (ii) quasi-analysis, (iii) dimensionality, (iv) holism and theoretical terms, (v) disposition terms, and (vi) structuralism in an Aufbau-like setting.
In this talk, we discuss the object-/metalanguage distinction and Tarski's theorem on the undefinability of truth in the context of the category of interpretations. This category offers the proper framework for the study the interaction between truth and translation. We explicate the notion of object-/metalanguage pair as an arrow in the category of interpretations. In these terms,Tarski's result means that certain arrows cannot be `restricted'.
We apply the framework to discuss a construction of Orey Sentences of
a theory T, i.e. sentences such that both T+A and T+¬A are interpretable
in T. Moreover, we prove that arithmetical theories extending PA cannot
be bi-interpretable with set theories extending ZF. This shows that the
difference between arithmetical theories and set theories extends below
the surface constituted by an arbitrary choice of language. At the same
time, the result illustrates that mutual interpretability differs from
bi-interpretability, since all set theories extending ZF are mutually
interpretable with an arithmetical theory extending PA.
last change: 13 May, 2006
e-mail address (please replace "0" by the usual "@" symbol): volker.halbach at philosophy.oxford.ac.uk