Ontological pluralism is the doctrine that there are different ways or modes of being. In contemporary guise, it is the doctrine that a logically perspicuous description of reality will use multiple quantifiers which cannot be thought of as ranging over a single domain. Although thought defeated for some time, recent defenses have shown a number of arguments against the view unsound. However, another worry looms: that despite looking like an attractive alternative, ontological pluralism is really no different than its counterpart, (...) ontological monism. In this paper, after explaining the worry in detail, I argue that considerations dealing with the nature of the logic ontological pluralists ought to endorse, coupled with an attractive philosophical thesis about the relationship between logic and metaphysics, show this worry to be unfounded. (shrink)

Contemporary logicians continue to address problems associated with the existential import of categorical propositions. One notable problem concerns invalid instances of subalternation in the case of a universal proposition with an empty subject term. To remedy problems, logicians restrict first-order predicate logics to exclude such terms. Examining the historical origins of contemporary discussions reveals that logicians continue to make various category mistakes. We now believe that no proposition per se has existential import as commonly understood and thus it is unnecessary (...) to restrict first-order predicate logics to non-empty classes. After introducing the problem, we trace some nineteenth century treatments of the issue to locate a source of misconstruing propositional import in misconceptions of ‘implies’ and ‘affirms’ and name the process/product fallacy, along with the translation of categorical sentences using quantifiers and accommodating an empty class. Next we treat some metalogical matters to orient our discussion by which we provide a more precise nomenclature about ‘sentence’ and ‘proposition’ to correct previous misconceptions; here we uncover a common category mistake in respect of a proposition’s efficacy. The semantic distinction between agent and force is helpful in this connection. We conclude by showing that logicians have reinserted existence as a predicate, a position previously excised by Kant, and that the Frege-Russell ambiguity thesis applies only to relationships within a categorical sentence between grammatical predicate and subject. (shrink)

Gentzen’s and Jaśkowski’s formulations of natural deduction are logically equivalent in the normal sense of those words. However, Gentzen’s formulation more straightforwardly lends itself both to a normalization theorem and to a theory of “meaning” for connectives . The present paper investigates cases where Jaskowski’s formulation seems better suited. These cases range from the phenomenology and epistemology of proof construction to the ways to incorporate novel logical connectives into the language. We close with a demonstration of this latter aspect by (...) considering a Sheffer function for intuitionistic logic. (shrink)

Ontological commitments and other problems concerning existence arise in connection with various aspects of logical theories. The semantics of quantification theory is usually formulated in such a manner that theorems are all and only those formulae which come out true under all interpretations in all non-empty domains. There are several approaches to include the empty domain. Paradoxically this apparent semantic extension means surrendering several formulae which are valid and intuitively plausible.

Some authors have studied in an ad hoc fashion the inclusive logics, that is the logics which admit or include objects or sets without element. These logics have been recently brought into the limelight because of the use of arbitrary topoi for interpreting languages. (In topoi there are usually many objects without element.)The aim of the paper is to present, for some inclusive logics, an axiomatization as natural and as simple as possible. Because of the intended applications to category theory, (...) the logics studied are many-sorted and intuitionistic. (shrink)

Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that the domain of interpretation is not empty, every name denotes exactly one object in the domain and the quantifiers have existential import. Free logics usually reject the claim that names need to denote in, and of the systems considered in this paper, the positive free logic concedes that (...) some atomic formulas containing non-denoting names are true, while negative free logic rejects even the latter claim. Inclusive logics, which reject, are likewise considered. These logics have complex and varied axiomatizations and semantics, and the goal of this paper is to present an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization, using sequent calculi which possess all the desired structural properties of a good proof system, including admissibility of contraction and cut, while streamlining free logics in a way no other approach has. We then present a simple and unified system of abstract semantics, which allows for a straightforward demonstration of the meta-theoretical properties, and offers insights into the relationship between different logics. The final part of this paper is dedicated to extending the system with modalities by using a labeled sequent calculus, and here we are again able to map out the different approaches and their mutual relations using the same framework. (shrink)

Two subject-predicate calculi with equality,SP = and its extensionUSP =, are presented as systems of natural deduction. Both the calculi are systems of free logic. Their presentation is preceded by an intuitive motivation.It is shown that Aristotle's syllogistics without the laws of identitySaP andSiP is definable withinSP =, and that the first-order predicate logic is definable withinUSP =.

The second printing of Principia Mathematica in 1925 offered Russell an occasion to assess some criticisms of the Principia and make some suggestions for possible improvements. In Appendix A, Russell offered *8 as a new quantification theory to replace *9 of the original text. As Russell explained in the new introduction to the second edition, the system of *8 sets out quantification theory without free variables. Unfortunately, the system has not been well understood. This paper shows that Russell successfully antedates (...) Quine's system of quantification theory without free variables. It is shown as well, that as with Quine's system, a slight modification yields a quantification theory inclusive of the empty domain. (shrink)

For first order languages with no individual constants, empty structures and truth values (for sentences) in them are defined. The first order theories of the empty structures and of all structures (the empty ones included) are axiomatized with modus ponens as the only rule of inference. Compactness is proved and decidability is discussed. Furthermore, some well known theorems of model theory are reconsidered under this new situation. Finally, a word is said on other approaches to the whole problem.

An attractive principle about domains of quantification is the analogue of the Separation Axiom in set theory: restricting a domain by an arbitrary predicate yields a domain. In particular, restricting a domain by a predicate that applies to nothing yields a domain. Thus if there is a nonempty domain, there is an empty domain. But semantics for the empty domain involves some neglected subtleties. Untangling them requires us to revise the usual definition of truth in a model, avoiding the detour (...) through Tarski's notion of satisfaction. (shrink)