623 research outputs found
The bearable lightness of being
How are philosophical questions about what kinds of things there are to be understood and how are they to be answered? This paper defends broadly Fregean answers to these questions. Ontological categories-such as object, property, and relation-are explained in terms of a prior logical categorization of expressions, as singular terms, predicates of varying degree and level, etc. Questions about what kinds of object, property, etc., there are are, on this approach, reduce to questions about truth and logical form: for example, the question whether there are numbers is the question whether there are true atomic statements in which expressions function as singular terms which, if they have reference at all, stand for numbers, and the question whether there are properties of a given type is a question about whether there are meaningful predicates of an appropriate degree and level. This approach is defended against the objection that it must be wrong because makes what there depend on us or our language. Some problems confronting the Fregean approach-including Frege's notorious paradox of the concept horse-are addressed. It is argued that the approach results in a modest and sober deflationary understanding of ontological commitments
Predicativity, the Russell-Myhill Paradox, and Church's Intensional Logic
This paper sets out a predicative response to the Russell-Myhill paradox of
propositions within the framework of Church's intensional logic. A predicative
response places restrictions on the full comprehension schema, which asserts
that every formula determines a higher-order entity. In addition to motivating
the restriction on the comprehension schema from intuitions about the stability
of reference, this paper contains a consistency proof for the predicative
response to the Russell-Myhill paradox. The models used to establish this
consistency also model other axioms of Church's intensional logic that have
been criticized by Parsons and Klement: this, it turns out, is due to resources
which also permit an interpretation of a fragment of Gallin's intensional
logic. Finally, the relation between the predicative response to the
Russell-Myhill paradox of propositions and the Russell paradox of sets is
discussed, and it is shown that the predicative conception of set induced by
this predicative intensional logic allows one to respond to the Wehmeier
problem of many non-extensions.Comment: Forthcoming in The Journal of Philosophical Logi
Logicism, Ontology, and the Epistemology of Second-Order Logic
In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second-order domain in terms of definability, can serve a neo-logicist's purposes. The problem, in both cases, is similar: neither Wright nor Hale is sufficiently sensitive to the demands that impredicativity imposes. Finally, I defend my own earlier attempt to finesse this issue, in "A Logic for Frege's Theorem", from Hale's criticisms
Defining Original Presentism
It is surprisingly hard to define presentism. Traditional definitions
of the view, in terms of tensed existence statements, have
turned out not to to be capable of convincingly distinguishing
presentism from eternalism. Picking up on a recent proposal
by Tallant, I suggest that we need to locate the break between
eternalism and presentism on a much more fundamental level.
The problem is that presentists have tried to express their
view within a framework that is inherently eternalist. I call
that framework the Fregean nexus, as it is defined by Frege’s
atemporal understanding of predication. In particular, I show
that the tense-logical understanding of tense which is treated
as common ground in the debate rests on this very same
Fregean nexus, and is thus inadequate for a proper definition
of presentism. I contrast the Fregean nexus with what I call
the original temporal nexus, which is based on an alternative,
inherently temporal form of predication. Finally, I propose
to define presentism in terms of the original temporal nexus,
yielding original presentism. According to original presentism,
temporal propositions are distinguished from atemporal ones not
by aspects of their content, as they are on views based on the
Fregean nexus, but by their form—in particular, by their form of
predication
Some Logical Notations for Pragmatic Assertions
The pragmatic notion of assertion has an important inferential role in logic. There are also many notational forms to express assertions in logical systems. This paper reviews, compares and analyses languages with signs for assertions, including explicit signs such as Frege’s and Dalla Pozza’s logical systems and implicit signs with no specific sign for assertion, such as Peirce’s algebraic and graphical logics and the recent modification of the latter termed Assertive Graphs. We identify and discuss the main ‘points’ of these notations on the logical representation of assertions, and evaluate their systems from the perspective of the philosophy of logical notations. Pragmatic assertions turn out to be useful in providing intended interpretations of a variety of logical systems
Some remarks on semantics and expressiveness of the Sentential Calculus with Identity
Suszko's Sentential Calculus with Identity SCI results from classical
propositional calculus CPC by adding a new connective and axioms for
identity (which we interpret here as `propositional
identity'). We reformulate the original semantics of SCI in terms of Boolean
prealgebras establishing a connection to `hyperintensional semantics'.
Furthermore, we define a general framework of dualities between certain
SCI-theories and Lewis-style modal systems in the vicinity of S3. Suszko's
original approach to two SCI-theories corresponding to S4 and S5 can be
formulated as a special case. All these dualities rely particularly on the fact
that Lewis' `strict equivalence' is axiomatized by the SCI-principles of
`propositional identity'.Comment: 31 page
Identity, many-valuedness and referentiality
In the paper * we discuss a distinctive versatility of the non-Fregean approach to the sentential identity. We present many-valued and referential counterparts of the systems of SCI, the sentential calculus with identity, including Suszko’s logical valuation programme as applied to many-valued logics. The similarity of different constructions: many-valued, referential and mixed, leads us to the conviction of the universality of the non-Fregean paradigm of sentential identity as distinguished from the equivalence, cf. [9]
Logics and operators
Two connectives are of special interest in metalogical investigations — the connective of implication which is important due to its connections to the notion of inference, and the connective of equivalence. The latter connective expresses, in the material sense, the fact that two sentences have the same logical value while in the strict sense it expresses the fact that two sentences are interderivable on the basis of a given logic. The process of identification of equivalent sentences relative to theories of a logic C defines a class of abstract algebras. The members of the class are called Lindenbaum-Tarski algebras of the logic C. One may abstract from the origin of these algebras and examine them by means of purely algebraic methods
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