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"Tennant, Neil"
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Which ‘Intensional Paradoxes’ are Paradoxes?
We begin with a brief explanation of our proof-theoretic criterion of paradoxicality—its motivation, its methods, and its results so far. It is a proof-theoretic account of paradoxicality that can be given in addition to, or alongside, the more familiar semantic account of Kripke. It is a question for further research whether the two accounts agree in general on what is to count as a paradox. It is also a question for further research whether and, if so, how the so-called Ekman problem bears on the investigations here of the intensional paradoxes. Possible exceptions to the proof-theoretic criterion are Prior’s Theorem and Russell’s Paradox of Propositions—the two best-known ‘intensional’ paradoxes. We have not yet addressed them. We do so here. The results are encouraging. §1 studies Prior’s Theorem. In the literature on the paradoxes of intensionality, it does not enjoy rigorous formal proof of a
Gentzenian
kind—the kind that lends itself to proof-theoretic analysis of recondite features that might escape the attention of logicians using non-Gentzenian systems of logic. We make good that lack, both to render the criterion applicable to the formal proof, and to see whether the criterion gets it right. Prior’s Theorem is a theorem in an
unfree
, classical, quantified propositional logic. But if one were to insist that the logic employed be
free
, then Prior’s Theorem would not be a
theorem
at all. Its proof would have an
undischarged assumption
—the ‘existential presupposition’ that the proposition
∀
p
(
Q
p
→
¬
p
)
exists. Call this proposition
ϑ
. §2 focuses on
ϑ
. We analyse a Priorean
reductio
of
ϑ
along with the possibilitate
◊
∀
q
(
Q
q
↔
(
ϑ
↔
q
)
)
. The attempted
reductio
of this premise-pair, which is constructive, cannot be brought into normal form. The criterion says we have not straightforward inconsistency, but rather genuine paradoxicality. §3 turns to problems engendered by the proposition
∃
p
(
Q
p
∧
¬
p
)
(call it
η
) for the similar possibilitate
◊
∀
q
(
Q
q
↔
(
η
↔
q
)
)
. The attempted disproof of this premise-pair—again, a constructive one—cannot succeed. It cannot be brought into normal form. The criterion says the premise-pair is a genuine paradox. In §4 we show how Russell’s Paradox of Propositions, like the Priorean intensional paradoxes, is to be classified as a genuine paradox by the proof-theoretic criterion of paradoxicality.
Journal Article
One hundred lyrics and a poem, 1979-2016
\"Arranged alphabetically, [this book of lyrics] presents an overview of [Pet Shop Boys member Neil Tennant's] ... achievement as a chronicler of modern life: the romance, the break-ups, the aspirations, the changing attitudes, the history, the politics, the pain\"--Publisher marketing.
Book
Normalizability, cut eliminability and paradox
This is a reply to the considerations advanced by Schroeder-Heister and Tranchini (Ekman’s paradox, Unpublished typescript) as prima facie problematic for the proof-theoretic criterion of paradoxicality, as originally presented in Tennant (Dialectica 36:265–296, 1982) and subsequently amended in Tennant (Analysis 55:199–207, 1995). Countering these considerations lends new importance to the parallelized forms of elimination rules in natural deduction.
Journal Article
WHAT IS A RULE OF INFERENCE?
We explore the problems that confront any attempt to explain or explicate exactly what a primitive logical rule of inference is, or consists in. We arrive at a proposed solution that places a surprisingly heavy load on the prospect of being able to understand and deal with specifications of rules that are essentially self-referring. That is, any rule
$\\rho $
is to be understood via a specification that involves, embedded within it, reference to rule
$\\rho $
itself. Just how we arrive at this position is explained by reference to familiar rules as well as less familiar ones with unusual features. An inquiry of this kind is surprisingly absent from the foundations of inferentialism—the view that meanings of expressions (especially logical ones) are to be characterized by the rules of inference that govern them.
Journal Article
On Tarski's Axiomatization of Mereology
It is shown how Tarski's 1929 axiomatization of mereology secures the reflexivity of the 'part of' relation. This is done with a fusion-abstraction principle that is constructively weaker than that of Tarski; and by means of constructive and relevant reasoning throughout. We place a premium on complete formal rigor of proof. Every step of reasoning is an application of a primitive rule; and the natural deductions themselves can be checked effectively for formal correctness.
Journal Article
A New Unified Account of Truth and Paradox
I propose an anti-realist account of truth and paradox according to which the logico-semantic paradoxes are not genuine inconsistencies. The 'global' proofs of absurdity associated with these paradoxes cannot be brought into normal form. The account combines epistemicism about truth with a proof-theoretic diagnosis of paradoxicality. The aim is to combine a substantive philosophical account of truth with a more rigorous and technical diagnosis of the source of paradox for further consideration by logicians. Core Logic plays a central role in the account on offer. It is shown that the account is not prey to the problem of revenge paradox.
Journal Article
TRANSMISSION OF VERIFICATION
This paper clarifies, revises, and extends the account of the transmission of truthmakers by core proofs that was set out in chap. 9 of Tennant (2017). Brauer provided two kinds of example making clear the need for this. Unlike Brouwer’s counterexamples to excluded middle, the examples of Brauer that we are dealing with here establish the need for appeals to excluded middle when applying, to the problem of truthmaker-transmission, the already classical metalinguistic theory of model-relative evaluations.
Journal Article
THE RELEVANCE OF PREMISES TO CONCLUSIONS OF CORE PROOFS
The rules for Core Logic are stated, and various important results about the system are summarized. We describe its relationship to other systems, such as Classical Logic, Intuitionistic Logic, Minimal Logic, and the Anderson–Belnap relevance logic R. A precise, positive explication is offered of what it is for the premises of a proof to connect relevantly with its conclusion. This characterization exploits the notion of positive and negative occurrences of atoms in sentences. It is shown that all Core proofs are relevant in this precisely defined sense. We survey extant results about variable-sharing in rival systems of relevance logic, and find that the variable-sharing conditions established for them are weaker than the one established here for Core Logic (and for its classical extension). Proponents of other systems of relevance logic (such as R and its subsystems) are challenged to formulate a stronger variable-sharing condition, and to prove that R or any of its subsystems satisfies it, but that Core Logic does not. We give reasons for pessimism about the prospects for meeting this challenge.
Journal Article