Neil Tennant

Given the harmony principle for logical operators, compositionality ought to ensure that harmony should obtain at the level of whole contents. That is, the role of a content qua premise ought to be balanced exactly by its role as a conclusion. Frege's contextual definition of propositional content happens to exploit this balance, and one appeals to the Cut rule to show that the definition is adequate. We show here that Frege's definition remains adequate even when one relevantizes logic by abandoning an unrestricted Cut rule. The proof exploits the fact that in the relevantized logic, which abandons the unrestricted rule of Cut, any failure of the transitivity of deduction is offset by the epistemic gain involved in learning that a stronger-than-expected result holds

I present a general theory of abstraction operators which treats them as variable-binding term- forming operators, and provides a reasonably uniform treatment for definite descriptions, set abstracts, natural number abstraction, and real number abstraction. This minimizing, extensional and relational theory reveals a striking similarity between definite descriptions and set abstracts, and provides a clear rationale for the claim that there is a logic of sets (which is ontologically non- committal). The theory also treats both natural and real numbers as answering to a two-fold process of abstraction. The first step, of conceptual abstraction, yields the object occupying a particular position within an ordering of a certain kind. The second step, of objectual abstraction, yields the number sui generis, as the position itself within any ordering of the kind in question.

Peacocke argues for a ‘generalized rationalism’, holding that ‘all entitlement has a fundamentally a priori component.’ (2) But his rationalism ‘differs from those of Frege and Gödel, just as theirs differ from that of Leibniz.’ He requires both substantive theories of intentional content and of understanding, and systematic formal theories of referential semantics and truth. We need an externalist theory of content: ‘Only mental states with externally individuated contents can make judgements about the external, mind-independent world rational.’ (123) Purely evidential conceptions of meaning and content are inadequate. (34-49) They cannot account for the following: a thinker often has to work out what would be evidence for a content; contents cannot depend, for their identity, on all of the infinitely ramifying evidential connections among them; and thinkers conceive, however tacitly, of (at least some) observed properties as categorical. By contrast with an evidential theory, a truthconditional theory of content can account for all these problematic facts. Peacocke states, develops and defends three principles of rationalism which collectively ‘relate entitlement to truth, to the identity of states and their intentional contents, and to the a priori.’ (3-4) He does not thoroughly explain his central notion of entitlement, but this much is clear: any thinker is entitled to various transitions in, or into, thought. An example of a transition into thought would be that from one’s perceptual experience to an observational judgment. An example of a transition in thought would be a logical inference from certain premises to a conclusion. A transition is rational just in case the thinker is entitled to it. (Note that this aims to explain rationality in terms of entitlement, not the other way round.) It is clear from 28 that Peacocke needs an abstract ontology of entitlements (such as proofs, in the case of mathematics). Yet he does not endorse ‘Gödel’s obscure quasi-perceptual and quasi-causal epistemology of mathematics and the abstract sciences.’ (54..

This is Part I of a two-part study of the foundations of mathematics through the lenses of (i) apriority and analyticity, and (ii) the resources supplied by Core Logic. Here we explain what is meant by apriority, as the notion applies to knowledge and possibly also to truths in general. We distinguish grounds for knowledge from grounds of truth, in light of our recent work on truthmakers. We then examine the role of apriority in the realism/anti-realism debate. We raise a hitherto unnoticed problem for any Orthodox Realist who attempts to explain the a priori

We explore the consequences, for logical system-building, of taking seriously the aim of having irredundant rules of inference, and a preference for proofs of stronger results over proofs of weaker ones. This leads one to reconsider the structural rules of REFLEXIVITY, THINNING, and CUT. REFLEXIVITY survives in the minimally necessary form $\varphi:\varphi$. Proofs have to get started. CUT is subject to a CUT-elimination theorem, to the effect that one can always make do without applications of CUT. So CUT is redundant, and should not be a rule of the system. CUT-elimination, however, in the context of the usual forms of logical rules, requires the presence, in the system, of THINNING. But THINNING, it turns out, is not really necessary. Provided only that one liberalizes the statement of certain logical rules in an appropriate way, one can make do without CUT or THINNING. From the methodological point of view of this study, the logical rules ought to be framed in this newly liberalized form. These liberalized logical rules determine the system of core logic. Given any intuitionistic Gentzen proof of $\Delta:\varphi$, one can determine from it a Core proof of some subsequent of $\Delta:\varphi$. Given any classical Gentzen proof of $\Delta:\varphi$, one can determine from it a classical Core proof of some subsequent of $\Delta:\varphi$. In both cases the Core proof is of a result at least as strong as that of the Gentzen proof; and the only structural rule used is $\varphi:\varphi$.

Changing the Theory of Theory Change: Towards a Computational Approach’ (Tennant [1994]; henceforth CTTC) claimed that the AGM postulate of recovery is false, and that AGM contractions of theories can be more than minimally mutilating. It also described an alternative, computational method for contracting theories, called the Staining Algorithm. Makinson [1995] and Hansson and Rott [1995] criticized CTTC's arguments against AGM-theory, and its specific proposals for an alternative, computational approach. This paper replies as comprehensively as space allows.

Christopher Peacocke, in A Study of Concepts, motivates his account of possession conditions for concepts by means of an alleged parallel with the conditions under which numbers are abshacted to give the numerosity of a predicate. There are, however, logical mistakes in Peacocke's treatment of numbers, which undermine his intended analogy. Nevertheless Peacocke's account of possession conditions for concepts is not rendered inadequate simply by virtue of being deprived of the intended analogy and the motivation it was supposed to afford. His account of concepts deserves still to be considered on its own merits, even if it is more idiosyncratic for being isolated from the paradigm case of numerical abstraction. Peacocke's own account of concepts as abstract objects turns out, though, not to have the logical form that he himself was seeking for it. We show how to re-cast it in an equivalent form of the kind he requires. Then we re-formulate it so as to achieve complete generality. This exercise helps to clarify the central theses in Peacocke's account of concepts. It invites the conclusion that his account of content-determination is rather platitudinous and unoriginal — except for a claim of 'doxastic sufficiency' for content-determination, which emerges at the end of the discussion.

I use the Corcoran–Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen–Prawitz system of natural deduction, using the universal and existential quantifiers of standard first-order logic, and the usual formalizations of Aristotle's sentence-forms. I explain how the syllogistic is a fragment of my system of Core Logic. Then I introduce my main innovation: the use of binary quantifiers, governed by introduction and elimination rules. The syllogisms in all four figures are re-proved in the binary system, and are thereby revealed as all on a par with each other. I conclude with some comments and results about grammatical generativity, ecthesis, perfect validity, skeletal validity and Aristotle's chain principle.

Using a quantified propositional logic involving the operators it is known that and it is possible to know that, we formalize various interesting philosophical claims involved in the realism debate. We set out inferential rules for the epistemic modalities, ranging from ones that are obviously analytic, to ones that are epistemologically more substantive or even controversial. Then we investigate various aporias for the realism debate. These are constructively inconsistent triads of claims from our list: a claim expressing some sort of common ground in the debate; a characteristically anti-realist thesis about truth and knowability; and a characteristically realist thesis about determinacy of truth value. Various patterns of reductio proof for these inconsistent triads are generated, so as to display their variety. The reductio proofs use only the inferential rules set out earlier. The philosophical utility of each aporia for the anti-realist is then assessed. This involves consideration of the acceptability of the premiss expressing common ground; the strength and plausibility of the anti-realist premiss; the strength of the realist premiss that is the ultimate target of the reductio; and the analytic status of the inferential rules applied within the proof.

This study continues the anti-realist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the anti-realist’s knowability principle ‘ϕ, therefore 3Kϕ’ should be formulated as a consistency requirement not on the premise ϕ of an application of the rule, but rather on the set of assumptions on which the relevant occurrence of ϕ depends. It is stressed, by reference to illustrative proofs, how important it is to have proofs in normal form before applying the proposed restriction. A similar restriction is proposed for the converse inference, the so-called Rule of Factiveness ‘3Kϕ therefore ϕ’. The proposed restriction appears to block another Fitch-style derivation that uses the KK -thesis in order to get around..

Any consistent and sufficiently strong system of first-order formal arithmetic fails to decide some independent Gödel sentence. We examine consistent first-order extensions of such systems. Our purpose is to discover what is minimally required by way of such extension in order to be able to prove the Gödel sentence in a non-trivial fashion. The extended methods of formal proof must capture the essentials of the so-called 'semantical argument' for the truth of the Gödel sentence. We are concerned to show that the deflationist has at his disposal such extended methods-methods which make no use or mention of a truth-predicate. This consideration leads us to reassess arguments recently advanced-one by Shapiro and another by Ketland-against the deflationist's account of truth. Their main point of agreement is this: they both adduce the Gödel phenomena as motivating a 'thick' notion of truth, rather than the deflationist's 'thin' notion. But the so-called 'semantical argument', which appears to involve a 'thick' notion of truth, does not really have to be semantical at all. It is, rather, a reflective argument. And the reflections upon a system that are contained therein are deflationarily licit, expressible without explicit use or mention of a truth-predicate. Thus it would appear that this anti-deflationist objection fails to establish that there has to be more to truth than mere conformity to the disquotational T-schema.

Written for any readers interested in better harnessing philosophy's real value, this book covers a broad range of fundamental philosophical problems and certain intellectual techniques for addressing those problems. In Introducing Philosophy: God, Mind, World, and Logic, Neil Tennant helps any student in pursuit of a 'big picture' to think independently, question received dogma, and analyse problems incisively. It also connects philosophy to other areas of study at the university, enabling all students to employ the concepts and techniques of this millennia-old discipline throughout their college careers - and beyond. KEY FEATURES AND BENEFITS: -- Investigates the philosophy of various subjects, helping students contextualize philosophy and view it as an interdisciplinary pursuit; also helps students with majors outside of philosophy to see the relationship between philosophy and their own focused academic pursuits -- Author comes from a distinguished background in Logic and Philosophy of Language, which gives the book a level of rigor, balance, and analytic focus sometimes missing from primers to philosophy -- Introduces students to various important philosophical distinctions providing skills that are important for undergraduates to develop in order to inform their study at higher levels. They are essential for further work in philosophy but they are also very beneficial for students pursuing most other disciplines -- Is much more methodologically comprehensive than competing introductions, giving the student the ability to address a wide range of philosophical problems - and not just the ones reviewed in the book -- Offers a companion website with links to apt primary sources, organized chapter-by-chapter, making unnecessary a separate Reader/Anthology of primary sources - thus providing students with all reading material necessary for the course -- Provides five to ten discussion questions for each chapter, helping instructors and students better interact with the ideas and concepts in the text.