313 research outputs found
Composition and Relative Counting
According to the so-called strong variant of Composition as Identity (CAI), the Principle of Indiscernibility of Identicals can be extended to composition, by resorting to broadly Fregean relativizations of cardinality ascriptions. In this paper we analyze various ways in which this relativization could be achieved. According to one broad variety of relativization, cardinality ascriptions are about objects, while concepts occupy an additional argument place. It should be possible to paraphrase the cardinality ascriptions in plural logic and, as a consequence, relative counting requires the relativization either of quantifiers, or of identity, or of the is one of relation. However, some of these relativizations do not deliver the expected results, and others rely on problematic assumptions. In another broad variety of relativization, cardinality ascriptions are about concepts or sets. The most promising development of this approach is prima facie connected with a violation of the so-called Coreferentiality Constraint, according to which an identity statement is true only if its terms have the same referent. Moreover - even provided that the problem with coreferentiality can be fixed - the resulting analysis of cardinality ascriptions meets several difficulties
Contingent composition as identity
When the necessity of identity is combined with composition as identity, the contingency of composition is at risk. In the extant literature, either NI is seen as the basis for a refutation of CAI or CAI is associated with a theory of modality, such that: either NI is renounced ; or CC is renounced. In this paper, we investigate the prospects of a new variety of CAI, which aims to preserve both NI and CC. This new variety of CAI is the quite natural product of the attempt to make sense of CAI on the background of a broadly Kripkean view of modality, such that one and the same entity is allowed to exist at more than one possible world. CCAI introduces a world-relative kind of identity, which is different from standard identity, and claims that composition is this kind of world-relative identity. CCAI manages to preserve NI and CC. We compare CCAI with Gibbardâs and Galloisâ doctrines of contingent identity and we show that CCAI can be sensibly interpreted as a form of Weak CAI, that is of the thesis that composition is not standard identity, yet is significantly similar to it
On the Infinite in Mereology with Plural Quantification
In Lewis reconstructs set theory using mereology and plural quantification (MPQ). In his recontruction he assumes from the beginning that there is an infinite plurality of atoms, whose size is equivalent to that of the set theoretical universe. Since this assumption is far beyond the basic axioms of mereology, it might seem that MPQ do not play any role in order to guarantee the existence of a large infinity of objects. However, we intend to demonstrate that mereology and plural quantification are, in some ways, particularly relevant to a certain conception of the infinite. More precisely, though the principles of mereology and plural quantification do not guarantee the existence of an infinite number of objects, nevertheless, once the existence of any infinite object is admitted, they are able to assure the existence of an uncountable infinity of objects. So, ifMPQ were parts of logic, the implausible consequence would follow that, given a countable infinity of individuals, logic would be able to guarantee an uncountable infinity of object
Normality Operators and Classical Collapse
In this paper, we extend the expressive power of the logics K3, LP and FDE with anormality operator, which is able to express whether a for-mula is assigned a classical truth value or not. We then establish classical recapture theorems for the resulting logics. Finally, we compare the approach via normality operator with the classical collapse approach devisedby Jc Beall
Normality operators and Classical Recapture in Extensions of Kleene Logics
In this paper, we approach the problem of classical recapture for LP and K3 by using normality
operators. These generalize the consistency and determinedness operators from Logics of Formal Inconsistency and Underterminedness, by expressing, in any many-valued logic, that a given formula has a classical truth value (0 or 1). In particular, in the rst part of the paper we introduce the logics LPe and Ke3 , which extends LP and K3 with normality operators, and we establish a classical recapture result based on the two logics. In the second part of the paper, we compare the approach in terms of normality operators with an established approach to classical recapture, namely minimal inconsistency. Finally, we discuss technical issues connecting LPe and Ke3 to the tradition of Logics of Formal Inconsistency and Underterminedness
On Dialetheic Entailment
The entailment connective is introduced by Priest (2006b). It aims to capture, in a dialetheically acceptable way, the informal notion of logical consequence. This connective does not âfall foulâ of Curryâs Paradox by invalidating an inference rule called âAbsorptionâ (or âContractionâ) and the classical logical theorem called âAssertionâ. In this paper we show that the semantics of entailment, given by Priest in terms of possible worlds, is inadequate. In particular, we will argue that Priestâs counterexamples to Absorption and Assertion use in the metalanguage a dialetheically unacceptable principle. Furthermore, we show that the rejection of Assertion undermines Priestâs claim that the entailment connective expresses the notion of logical consequence
On an Intuitionistic Logic for Pragmatics
We reconsider the pragmatic interpretation of intuitionistic logic [21]
regarded as a logic of assertions and their justications and its relations with classical
logic. We recall an extension of this approach to a logic dealing with assertions
and obligations, related by a notion of causal implication [14, 45]. We focus on
the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on
polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the
S4 modal translation, we give a denition of a system AHL of bi-intuitionistic logic
that correctly represents the duality between intuitionistic and co-intuitionistic logic,
correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism
as a distributed calculus of coroutines is then used to give an operational
interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear
calculus of co-intuitionistic coroutines is dened and a probabilistic interpretation
of linear co-intuitionism is given as in [9]. Also we remark that by extending the
language of intuitionistic logic we can express the notion of expectation, an assertion
that in all situations the truth of p is possible and that in a logic of expectations
the law of double negation holds. Similarly, extending co-intuitionistic logic, we can
express the notion of conjecture that p, dened as a hypothesis that in some situation
the truth of p is epistemically necessary
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 Fregeâs and Dalla Pozzaâs logical systems and implicit signs with no specific sign for assertion, such as Peirceâs algebraic and graphical logics and the recent modification of the latter termed Assertive Graphs. We identify and discuss the main âpointsâ 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
A Note on Gödel, Priest and Naïve Proof
In the 1951 Gibbs lecture, Gödel asserted his famous dichotomy, where the notion of informal proof is at work. G. Priest developed an argument, grounded on the notion of naĂŻve proof, to the effect that Gödelâs first incompleteness theorem suggests the presence of dialetheias. In this paper, we adopt a plausible ideal notion of naĂŻve proof, in agreement with Gödelâs conception, superseding the criticisms against the usual notion of naĂŻve proof used by real working mathematicians. We explore the connection between Gödelâs theorem and naĂŻve proof so understood, both from a classical and a dialetheic perspective
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