Take the language of Peano Arithmetic and add a new unary predicate T, where the intended meaning of T(x) is “x is (the Godel number of) a true sentence in the augmented language”. VF is one of several established axiomatic theories which can be formulated in this language to express some of our basic intuitions about truth. In our talk we will look at IVF, the inner logic of VF, which by definition is the set of all sentences A such that VF proves T(‘A’). IVF is a demonstrably strong theory, but it is a challenge to provide a useful internal characterization of IVF in a way that reveals the source of this strength. We will discuss why this is an interesting question and report on progress to date.
We deal with an old problem of Skolem (1957), concerning the consistency of the unrestricted comprehension principle in Lukasiewicz infinite valued predicate logic. We are not able to solve the full question, but we survey partial positive results and we consider variants thereof, applying semantical techniques as well as proof-theoretic methods.
We will discuss two desirable properties of deflationary truth theories: conservativeness and maximality. Joining them together, we obtain a notion of a maximal conservative truth theory - a theory which is conservative over its base, but can't be enlarged any further without losing its conservative character. There are indeed such theories; we show however that none of them is axiomatizable, and moreover, that there will be in fact continuum many theories of this sort. It turns out in effect that the deflationist still needs some additional principles, which would permit him to construct his preferred theory of truth. This will be the first part of the talk; in the second part we will discuss some open problems, which in our opinion are of crucial importance to the conservativeness debate.
I consider how to give a theory of truth according to which languages
can express semantic generalizations about themselves. I focus on theories
that result from modifying the supervaluationist scheme.
We will deal with informal provability, i.e., provability as it is understood in mathematical practice (rather than in proof theory). In particular, we will argue that - informal provability differs conceptually from formal provability in an axiomatic system,
- the notion of informal provability has semantic and intuitive components that notions of formal provability lack (and it is possible to make sense of Gödel's view on these kinds of components),
- the logic of informal provability differs from the logic of formal provability, and
- the existence of true but informally unprovable statements follows
from
formalisations of Church's thesis (this last part is discussed in much more detail in a joint paper with Leon Horsten).
From the formalization of Church's Thesis in Epistemic Arithmetic plus some fairly reasonable background assumptions, we deduce that there are absolutely ndecidable propositions, i.e., mathematical sentences that are neither provable nor refutable (in the absolute, or informal, sense of `proof').
At least two prominent positions in the philosophy of mathematics call for absolutely general higher-order reformulations of applied mathematical theories. The reasons are varied, but in each case, I will argue, the appeal to absolute generality gives rise to a difficult epistemological puzzle. The solution suggests itself to discriminate between acceptable and unacceptable absolutely general theories on the basis of a certain model-theoretic constraint. However, I will argue that the constraint is out of line with mathematical practice and provably unattainable in crucial cases. Finally, I will suggest what I take to be the moral of the puzzle in each case.
We give a game theoretic characterisation for Hannes Leitgeb's notion of dependency. This is an epistemic variation on its characterisation over the standard model of arithmetic N as being Pi^1_1(N). Such games have been given before (D.A.Martin?) for constructing the least fixed point for a truth predicate for Strong Kleene Logic. We provide an open game characterisation first for supervaluation over N, and thereby for Leitgeb's dependency set.
last change: 13 May, 2006 e-mail address (please replace "0" by the usual "@" symbol): volker.halbach at philosophy.oxford.ac.uk |