@article {3148353, title = {Numbers and propositions versus nominalists: yellow cards for {S}almon \& {S}oames}, journal = {ERKENNTNIS}, volume = {77}, number = {3}, year = {2012}, pages = {381{\textendash}397}, abstract = {

Salmon and Soames argue against nominalism about numbers and sentence types. They employ (respectively) higher-order and first-order logic to model certain natural language inferences and claim that the natural language conclusions carry commitment to abstract objects, partially because their renderings in those formal systems seem to do that. I argue that this strategy fails because the nominalist can accept those natural language consequences, provide them with plausible and non-committing truth conditions and account for the inferences made without committing themselves to abstract objects. I sketch a modal account of higher-order quantification, on which instead of ranging over sets, higher order quantifiers are used to make (logical) possibility claims about which predicate tokens can be introduced. This approach provides an alternative account of truth conditions for natural language sentences which seem to employ higher-order quantification, thus allowing the nominalist to evade Salmon{\textquoteright}s argument. I also show how the nominalist can account for the occurrence of apparently singular abstract terms in certain true statements. I argue that the nominalist can achieve this by, first, dividing singular terms into real singular terms (referring to concrete objects) and only apparent singular terms (called onomatoids), introduced for the sake of brevity and simplicity, and then providing an account of nominalistically acceptable truth conditions of sentences containing onomatoids. I develop such an account in terms of modally interpreted abstraction principles and argue that applying this strategy to Soames{\textquoteright}s argument allows the nominalists to defend themselves. One would hope and perhaps conjecture that the whole general set theory, however beautiful it is, will in the future disappear. With the higher types Platonism begins. The tendencies of Chwistek and others ({\textquoteright}Nominalism{\textquoteright}) of speaking only of what can be named are healthy. [Alfred Tarski](1)

}, issn = {0165-0106}, doi = {http://dx.doi.org/10.1007/s10670-012-9402-7}, author = {Urbaniak, Rafal} } @article {Urbaniak2010, title = {Neologicist nominalism}, journal = {Studia Logica}, volume = {96}, number = {2}, year = {2010}, pages = {149-173}, abstract = {

The goal is to sketch a nominalist approach to mathematics which just like neologicism employs abstraction principles, but unlike neologicism is not committed to the idea that mathematical objects exist and does not insist that abstraction principles establish the reference of abstract terms. It is well-known that neologicism runs into certain philosophical problems and faces the technical difficulty of finding appropriate acceptability criteria for abstraction principles. I will argue that a modal and iterative nominalist approach to abstraction principles circumvents those difficulties while still being able to put abstraction principles to a foundational use.

}, doi = {10.1007/s11225-010-9279-x}, author = {Urbaniak, Rafal} } @article {urbaniak2009note, title = {A note on identity and higher-order quantification.}, journal = {Australasian Journal of Logic}, volume = {7}, year = {2009}, pages = {48{\textendash}55}, abstract = {

It is a commonplace remark that the identity relation, even though not expressible in a first-order language without identity with classical set-theoretic semantics, can be de ned in a language without identity, as soon as we admit second-order, set-theoretically interpreted quanti ers binding predicate variables that range over all subsets of the domain. However, there are fairly simple and intuitive higher-order languages with set-theoretic semantics (where the variables range over all subsets of the domain) in which the identity relation is not de nable. The point is that the de nability of identity in higher-order languages not only depends on what variables range over, but also is sensitive to how predication is construed.

}, author = {Urbaniak, Rafal} }