Problems of ontology
- Key People:
- Hilary Putnam
- Gottlob Frege
- Sir Michael A.E. Dummett
- Related Topics:
- logic
Among the ontological problems—problems concerning existence and existential assumptions—arising in logic are those of individuation and existence.
Individuation
Not all interesting interpretational problems are solved by possible-world semantics, as the developments earlier registered are sometimes called. The systematic use of the idea of possible worlds has raised, however, the subject of cross identification; i.e., of the principles according to which a member of one possible world is to be found identical or nonidentical with one of another. Since one can scarcely be said to have a concept of an individual if he cannot locate it in several possible situations, the problem of cross-identification is also one of the most important ingredients of the logical and philosophical problem of individuation. The criticisms that Quine has put forward concerning modal logic and analyticity (see above Limitations of logic) can be deepened into questions concerning methods of cross identification. Although some such methods undoubtedly belong to everyone’s normal unarticulated conceptual repertoire, it is not clear that they are defined or even definable widely enough to enable philosophers to make satisfactory sense of a quantified logic of logical necessity and logical possibility. The precise principles used in ordinary discourse—or even in the language of science—pose a subtle philosophical problem. The extent to which special “essential properties” are relied on in individuation and the role of spatio-temporal frameworks are moot points here. It has also been suggested that essentially different methods of cross identification are actually used together, some of them depending on impersonal descriptive principles and others on the perspective of a person.
Existence and ontology
Because one of the basic concepts of first-order logic is that of existence, as codified by the existential quantifier “(∃x),” one might suppose that there is little room left for any separate philosophical problem of existence. Yet existence, in fact, does seem to pose a problem, as witnessed by the bulk of the relevant literature. Some issues are relatively easy to clarify. In the usual formulations of first-order logic, for instance, there are “existential presuppositions” present to the effect that none of the singular terms employed is without a bearer (as “Pegasus” is). It is a straightforward matter, however, to dispense with these presuppositions. Though this seems to involve the procedure, branded as inadmissible by many philosophers, of treating existence as a predicate, this can nonetheless be easily done on the formal level. Given certain assumptions, it may even be shown that this “predicate” will have to be “(∃x) (x = a)” (for “a exists”—literally, “There exists an x such that x is a”) or something equivalent. Furthermore, the logical peculiarities of this predicate seem to explain amply philosophers’ apparent denial of its reality.
The interest in the notion of existence is connected with the question of what entities a theory commits its holder to or what its “ontology” is. The “predicate of existence” just mentioned recalls Quine’s criterion of ontological commitment: “To be is to be a value of a bound variable”—i.e., of the x in (∀x) or in (∃x). According to Quine, a theory is committed to those and only those entities that in the last analysis serve as the values of its bound variables. Thus ordinary first-order theory commits one to an ontology only of individuals (particulars), whereas higher order logic commits one to the existence of sets—i.e., of collections of definite and distinct entities (or, alternatively, of properties and relations). Likewise, if bound first-order variables are assumed to range over sets (as they do in set theory), a commitment to the existence of these sets is incurred.
The doctrine that an ontology of individuals is all that is needed is known as (the modern version of) nominalism. The opposite view is known as (logical) realism. Even those philosophers who profess sympathy with nominalism find it hard, however, to maintain that mathematics could be built on a consistently nominalistic foundation.
The precise import of Quine’s criterion of ontological commitment, however, is not completely clear. Nor is it clear in what other sense one is perhaps committed by a theory to those entities that are named or otherwise referred to in it but not quantified over in it. Questions can also be raised concerning the very distinction between what in modern logic are usually called individuals (“particulars” would be a more traditional designation) and such universals as their properties and relations; and these questions can be combined with others concerning the “tie” that binds particulars and universals together in predication.
An interesting approach to these problems is the distinction made by Gottlob Frege, a pioneer of mathematical logic in the late 19th century, between individuals—he called them objects—and what he called functions (which in his view include concepts) and his doctrine of the unsaturated character of the latter, according to which a function (as it were) contains a gap, which can be filled by an object. Another approach is the “picture theory of language” of Wittgenstein’s Tractatus Logico-Philosophicus, according to which a simple sentence presents a person with an isomorphic representation (a “picture”) of reality as it would be if the sentence were true. According to this view (which was later given up by Wittgenstein), “a sentence [or proposition, Satz] is a model of reality such as we think of it as being.”
Alternative logics
The natures of most of the so-called nonclassical logics can be understood against the background of what has here been said. Some of them are simply extensions of the “classical” first-order logic—e.g., modal logics and many versions of intensional logic. The so-called free logics are simply first-order (or modal) logics without existential presuppositions.
One of the most important nonclassical logics is intuitionistic logic, first formalized by the Dutch mathematician Arend Heyting in 1930. It has been shown that this logic can be interpreted in terms of the same kind of modal logic serving as a system of epistemic logic. In the light of its purpose to consider only the known, this isomorphism is suggestive. The avowed purpose of the intuitionist is to consider only what can actually be established constructively in logic and in mathematics—i.e., what can actually be known. Thus, he refuses to consider, for example, “Either A or not-A” as a logical truth, for it does not actually help one in knowing whether A or not-A is the case. This does not close, however, the philosophical problem about intuitionism. Special problems arise from intuitionists’ rejection (in effect) of the nonepistemic aspects of logic, as illustrated by the fact that only a part of epistemic logic is needed in this translation of intuitionistic logic into epistemic logic.
Other new logics are obtained by modifying the rules of those games that are involved in the game-theoretical interpretation of first-order logic mentioned above. The logician may reject, for instance, the assumption that he possesses perfect information, an assumption that characterizes classical first-order logic. One may also try to restrict the strategy sets of the players—to recursive strategies, for example.
Among the oldest kinds of alternative logics are many-valued logics. In them, more truth values than the usual true and false are assumed. The idea seems very natural when considered in abstraction from the actual use of logic. But a philosophically satisfactory interpretation of many-valued logics is not equally straightforward. The interest in finite-valued logics and the applicability of them are sometimes exaggerated. The idea, however, of using the elements of an arbitrary Boolean algebra—a generalized calculus of classes—as abstract truth-values has provided a powerful tool for systematic logical theory.
Logic and other disciplines
Technical disciplines
The relations of logic to mathematics, to computer technology, and to the empirical sciences are here considered.
Mathematics
It is usually said that all of mathematics can, in principle, be formulated in a sufficiently theorem-rich system of axiomatic set theory. What the axioms of a set theory that could accomplish this might be, however, and whether they are at all natural is not obvious in every case. (The recent development in abstract algebra known as category theory offers the most conspicuous examples of these problems.) The axioms of set theory may be presumed to hold in virtue of the meanings of the terms set, member of, and so on. Thus, in some loose sense all of pure mathematics falls within the scope of logic in the wider sense. This assertion is not very informative, however, as long as the logician has no ways of analyzing these meanings so as to be able to tell what assumptions (axioms of set theory) should be adopted. The definitions of basic mathematical concepts (such as “number”) in logical terms proposed by Gottlob Frege (in 1884), by Bertrand Russell (in 1903), and by their successors do not help in this enterprise. It is not clear that more recent insights in logic help very much, either, in the search for strong set-theoretical assumptions. The relationship of mathematics to logic on this level therefore remains ambiguous.
Notwithstanding these deep problems, virtually all normal mathematical argumentation is carried out in logical terms—mostly in first-order terms, but with a generous sprinkling of second-order reasoning and various principles of set theory. Historically speaking, most specific early examples of nontrivial logical reasoning were taken from mathematics.
Often these examples were set in contrast to logical arguments understood in a narrow traditional sense—in a sense narrower still than the idea of logic as being exhausted by quantification theory. According to this traditional view, logic is equated with syllogistic; i.e., with a part of that part of first-order logic that deals with properties and not with relations. Much of what earlier philosophers said of mathematical reasoning must, thus, be understood as applying to relational (first-order) reasoning. The present-day philosophy of logic is therefore as much an heir to traditional philosophy of mathematics as to traditional philosophy of logic.
Specific logical results are applicable in several parts of mathematics, especially in algebra, and various concepts and techniques used by logicians have often been borrowed from mathematics. (Thus one can even speak of “the mathematics of metamathematics.”)
Computers
It has already been indicated that recursive function theory is, in effect, the study of certain idealized automata (computers). It is, in fact, a matter of indifference whether this theory belongs to logic or to computer science. The idealized assumption of a potentially infinite computer tape, however, is not a trivial one: Turing machines typically need plenty of tape in their calculations. Hence the step from Turing machines to finite automata (which are not assumed to have access to an infinite tape) is an important one.
This limitation does not dissociate computer science from logic, however, for other parts of logic are also relevant to computer science and are constantly employed there. Propositional logic may be thought of as the “logic” of certain simple types of switching circuits. There are also close connections between automata theory and the logical and algebraic study of formal languages. An interesting topic on the borderline of logic and computer science is mechanical theorem proving, which derives some of its interest from being a clear-cut instance of the problems of artificial intelligence, especially of the problems of realizing various heuristic modes of thinking on computers. In theoretical discussions in this area, it is nevertheless not always understood how much textbook logic is basically trivial and where the distinctively nontrivial truths of logic (including first-order logic) lie.
Methodology of the empirical sciences
The quest for theoretical self-awareness in the empirical sciences has led to interest in methodological and foundational problems as well as to attempts to axiomatize different empirical theories. Moreover, general methodological problems, such as the nature of scientific explanations, have been discussed intensively among philosophers of science. In all of these endeavours, logic plays an important role.
By and large, there are here three different lines of thought. (1) Often, only the simplest parts of logic—e.g., propositional logic—are appealed to (over and above the mere use of logical notation). Sometimes, claims regarding the usefulness of logic in the methodology of the empirical sciences are, in effect, restricted to such rudimentary applications. This restriction is misleading, however, for most of the interesting and promising connections between methodology and logic lie on a higher level, especially in the area of model theory. In econometrics, for instance, a special case of the logicians’ problems of definability plays an important role under the title “identification problem.” On a more general level, logicians have been able to clarify the concept of a model as it is used in the empirical sciences.
In addition to those employing simple logic, two other contrasting types of theorists can be distinguished: (2) philosophers of science, who rely mostly on first-order formulations, and (3) methodologists (e.g., Patrick Suppes, a U.S. philosopher and behavioral scientist), who want to use the full power of set theory and of the mathematics based on it. Both approaches have advantages. Usually realistic axiomatizations and other reconstructions of actual scientific theories are possible only in terms of set theoretical and other strong mathematical conceptualizations (theories conceived of as “set-theoretical predicates”). In spite of the oversimplification that first-order formulations often entail, however, they can yield theoretical insights because first-order logic (including its model theory) is mastered by logicians much more thoroughly than is set theory.
Many empirical sciences, especially the social sciences, use mathematical tools borrowed from probability theory and statistics, together with such outgrowths of these as decision theory, game theory, utility theory, and operations research. A modest but not uninteresting beginning in the study of their foundations has been made in modern inductive logic.
