On the Numbers of Minimal Tautologies and Properties of Their Proofs in Classical and Nonclassical Logic

It is proved in this paper that the number of minimal tautologies for any given logic tautology of size п can be an exponential function in п, and it is also proved that for every tautology of the given logic there is some minimal tautology such that the number of its sequential form proof steps is equal to minimal steps in the proof of sequential form for the given tautology in cut-free sequent systems for classical, intuitionistic, Joganssons and monotone logics.

On Williamson's new Quinean argument against nonclassical logic

In "Semantic paradoxes and abductive methodology", Williamson presents a new Quinean argument based on central ingredients of common pragmatism about theory choice (including logical theory, as is common). What makes it new is that, in addition to avoiding Quine's unfortunate charge of mere terminological squabble, Williamson's argument explicitly rejects at least for purposes of the argument Quine's key conservatism premise. In this paper I do two things. First, I argue that Williamson's new Quinean argument implicitly relies on Quine's conservatism principle. Second, by way of answering his charges against nonclassical logic I directly defend a particular subclassical account of logical consequence.

The Silliness of Magical Realism

Professors Allen and Pardo champion "relative plausibility" to explain the standards of proof. But even after countless explanatory articles, it remains an underdeveloped model bereft of underlying theory. Multivalent logic, a fully developed and accepted system of logic, comes to the same endpoint as relative plausibility. Multivalent logic would thus provide the missing theory, while it would resolve all the old problems of using traditional probability theory to explain the standards of proof as well as the new problems raised by the relative plausibility model. For example, multivalent logic resolves the infamous "conjunction paradox" that traditional probability creates for itself, and which relative plausibility tries to sweep under the rug.Yet Allen & Pardo dismiss multivalent logic as magical realism when applied to legal factfinding. They reject this ring buoy because they misunderstand nonclassical logic, as this response explains.

The silliness of magical realism

Relative plausibility, even after countless explanatory articles, remains an underdeveloped model bereft of underlying theory. Multivalent logic, a fully developed and accepted system of logic, comes to the same endpoint as relative plausibility. Multivalent logic would thus provide the missing theory, while it would resolve all the old problems of using traditional probability theory to explain the standards of proof as well as the new problems raised by the relative plausibility model. For example, multivalent logic resolves the infamous ‘conjunction paradox’ that traditional probability creates for itself, and which relative plausibility tries to sweep under the rug. Yet Professors Allen and Pardo dismiss multivalent logic as magical realism when applied to legal factfinding. They reject this ring buoy because they misunderstand nonclassical logic, as this response explains.

NAIVE TRUTH AND NAIVE LOGICAL PROPERTIES

A unified answer is offered to two distinct fundamental questions: whether a nonclassical solution to the semantic paradoxes should be extended to other apparently similar paradoxes (in particular, to the paradoxes of logical properties) and whether a nonclassical logic should be expressed in a nonclassical metalanguage. The paper starts by reviewing a budget of paradoxes involving the logical properties of validity, inconsistency, and compatibility. The author’s favored substructural approach to naive truth is then presented and it is explained how that approach can be extended in a very natural way so as to solve a certain paradox of validity. However, three individually decisive reasons are later provided for thinking that no approach adopting a classical metalanguage can adequately account for all the features involved in the paradoxes of logical properties. Consequently, the paper undertakes the task to do better, and, building on the system already developed, introduces a theory in a nonclassical metalanguage that expresses an adequate logic of naive truth and of some naive logical properties.

NAIVE TRUTH AND RESTRICTED QUANTIFICATION: SAVING TRUTH A WHOLE LOT BETTER

Restricted quantification poses a serious and under-appreciated challenge for nonclassical approaches to both vagueness and the semantic paradoxes. It is tempting to explain “All A are B” as “For all x, if x is A then x is B”; but in the nonclassical logics typically used in dealing with vagueness and the semantic paradoxes (even those where ‘if … then’ is a special conditional not definable in terms of negation and disjunction or conjunction), this definition of restricted quantification fails to deliver important principles of restricted quantification that we’d expect. If we’re going to use a nonclassical logic, we need one that handles restricted quantification better.The challenge is especially acute for naive theories of truth—roughly, theories that take True(〈A〉) to be intersubstitutable with A, even when A is a “paradoxical” sentence such as a Liar-sentence. A naive truth theory inevitably involves a somewhat nonclassical logic; the challenge is to get a logic that’s compatible with naive truth and also validates intuitively obvious claims involving restricted quantification (for instance, “If S is a truth stated by Jones, and every truth stated by Jones was also stated by Smith, then S is a truth stated by Smith”). No extant naive truth theory even comes close to meeting this challenge, including the theory I put forth in Saving Truth from Paradox. After reviewing the motivations for naive truth, and elaborating on some of the problems posed by restricted quantification, I will show how to do better. (I take the resulting logic to be appropriate for vagueness too, though that goes beyond the present paper.)In showing that the resulting logic is adequate to naive truth, I will employ a somewhat novel fixed point construction that may prove useful in other contexts.

MODEL THEORY OF MEASURE SPACES AND PROBABILITY LOGIC

We study the model-theoretic aspects of a probability logic suited for talking about measure spaces. This nonclassical logic has a model theory rather different from that of classical predicate logic. In general, not every satisfiable set of sentences has a countable model, but we show that one can always build a model on the unit interval. Also, the probability logic under consideration is not compact. However, using ultraproducts we can prove a compactness theorem for a certain class of weak models.