Putnam, Peano, and the Malin Génie: could we possibly bewrong about elementary number-theory?

This article examines Hilary Putnam's work in the philosophy of mathematics and - more specifically - his arguments against mathematical realism or objectivism. These include a wide range of considerations, from Gödel's incompleteness-theorem and the limits of axiomatic set-theory as formalised in the Löwenheim-Skolem proof to Wittgenstein's sceptical thoughts about rule-following (along with Saul Kripke's ‘scepticalsolution’), Michael Dummett's anti-realist philosophy of mathematics, and certain problems – as Putnam sees them – with the conceptual foundations of Peano arithmetic. He also adopts a thought-experimental approach – a variant of Descartes' dream scenario – in order to establish the in-principle possibility that we might be deceived by the apparent self-evidence of basic arithmetical truths or that it might be ‘rational’ to doubt them under some conceivable (even if imaginary) set of circumstances. Thus Putnam assumes that mathematical realism involves a self-contradictory ‘Platonist’ idea of our somehow having quasi-perceptual epistemic ‘contact’ with truths that in their very nature transcend the utmost reach of human cognitive grasp. On this account, quite simply, ‘nothing works’ in philosophy of mathematics since wecan either cling to that unworkable notion of objective (recognition-transcendent) truth or abandon mathematical realism in favour of a verificationist approach that restricts the range of admissible statements to those for which we happen to possess some means of proof or ascertainment. My essay puts the case, conversely, that these hyperbolic doubts are not forced upon us but result from a false understanding of mathematical realism – a curious mixture of idealist and empiricist themes – which effectively skews the debate toward a preordained sceptical conclusion. I then go on to mount a defence of mathematical realism with reference to recent work in this field and also to indicate some problems – as I seethem – with Putnam's thought-experimental approach as well ashis use of anti-realist arguments from Dummett, Kripke, Wittgenstein, and others.     

Teaching Genesis: A Present-Day Approach Inspired by the Prophet Nathan

The prophets Nathan (2 Samuel 12:1–15) and John the Baptist (Mark 6:16–28) had comparable tasks before them: to convince their respective kings about the wrongs of taking somebody else's wife and marrying her. Nathan succeeded, while John failed and furthermore lost his life. What made the difference? One possible explanation is that Nathan proceeded in two steps: (1) Tell an interesting, nonthreatening story that nevertheless makes the point at issue; (2) transfer that message to the case at hand. In contrast, John used a direct approach, which raised apprehension, even fear (on the part of Herodias, the woman involved), and led to failure. That lesson has wider applications, as illustrated here for teaching the biblical Genesis narration. The other ingredient in this teaching is relational and contextual reasoning (RCR), the use of which is also indicated for other issues besides teaching Genesis.     

Logic Games are Complete for Game Logics

Game logics describe general games through powers of players for forcing outcomes. In particular, they encode an algebra of sequential game operations such as choice, dual and composition. Logic games are special games for specific purposes such as proof or semantical evaluation for first-order or modal languages. We show that the general algebra of game operations coincides with that over just logical evaluation games, whence the latter are quite general after all. The main tool in proving this is a representation of arbitrary games as modal or first-order evaluation games. We probe how far our analysis extends to product operations on games. We also discuss some more general consequences of this new perspective for standard logic.     

Memory of Past Beliefs and Actions

Two notions of memory are studied both syntactically and semantically: memory of past beliefs and memory of past actions. The analysis is carried out in a basic temporal logic framework enriched with beliefs and actions.     

The Undecidability of Grisin's Set Theory

We investigate a contractionless naive set theory, due to Grisin [11]. We prove that the theory is undecidable.     

Monotonicity in Practical Reasoning

Classic deductive logic entails that once a conclusion is sustained by a valid argument, the argument can never be invalidated, no matter how many new premises are added. This derived property of deductive reasoning is known as monotonicity. Monotonicity is thought to conflict with the defeasibility of reasoning in natural language, where the discovery of new information often leads us to reject conclusions that we once accepted. This perceived failure of monotonic reasoning to observe the defeasibility of natural-language arguments has led some philosophers to abandon deduction itself (!), often in favor of new, non-monotonic systems of inference known as `default logics'. But these radical logics (e.g., Ray Reiter's default logic) introduce their desired defeasibility at the expense of other, equally important intuitions about natural-language reasoning. And, as a matter of fact, if we recognize that monotonicity is a property of the form of a deductive argument and not its content (i.e., the claims in the premise(s) and conclusion), we can see how the common-sense notion of defeasibility can actually be captured by a purely deductive system.     

Toward an Alliance Between the Issue-Processing Approach and Pragma-Dialectical Analysis

On the approach to discussions of policy choices that treats such discussions as instances of issue-processing, the joint use of the logic of questions and the logic of rules gives precise formulation to two sorts of issues. To one sort of issue belong issue-circumscribing questions; to another sort, issues-simplicter, which consist of disjunctions of policy proposals – so many proposed social rules – that are answers, in the case of each disjunction, to a given issue-circumscribing question. Work in pragma-dialectics can take over the issue-processing approach; and by doing so add to the pragma-dialectical repertory further dimensions in the analysis of issues and in protocol-narratives of discussion. The analysis and narratives would now include accounts of how issue-circumscribing questions generate initial standpoints and how discussions sometimes end with compromises between standpoints. Further research questions follow about transformations of issues and the comparison of successive rounds of discussion. A narrative of one period of discussion during `the War on Drugs' in the United States illustrates these points.     

Axioms for Deliberative Stit

Based on a notion of companions to stit formulas applied in other papers dealing with astit logics, we introduce choice formulas and nested choice formulas to prove the completeness theorems for dstit logics in a language with the dstit operator as the only non-truth-functional operator. The main logic discussed in this paper is the basic logic of dstit with multiple agents, other logics discussed include the basic logic of dstit with a single agent and some logics of dstit with multiple agents each of which corresponds to a semantic condition concerning the number of possible choices for agents.     

Labeled Calculi and Finite-Valued Logics

A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite-valued logic if the labels are interpreted as sets of truth values (sets-as-signs). Furthermore, it is shown that any finite-valued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic in the number of truth values, and it is shown that this bound is tight.     

Embedding Logics into Product Logic

We construct a faithful interpretation of ukasiewicz's logic in product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable.We prove a completeness theorem for product logic extended by a unary connective of Baaz [1]. We show that Gödel's logic is a sublogic of this extended product logic.We also prove NP-completeness of the set of propositional formulas satisfiable in product logic (resp. in Gödel's logic).