TRUTH FUNCTIONS

We learn about truth-functions very early in the study of symbolic logic. Propositional logic is compositional in two related ways. If we begin with atomic well-formed formulas (wffs), represented syntactically by propositional symbols, then we can build up wffs of any desired complexity by linking them together under a set of rules with certain truth-functional propositional connectives. Semantic compositionality extends the concept by implying that the meaning of a complex expression is determined by the meanings and relations holding among the expression's meaningful constituents.

The connectives defined for propositional logic are truth-functions. Propositional connectives standardly include the unary truth-function negation, and the binary truth-functions conjunction, disjunction, conditional and biconditional. These are formally defined by means of truth-tables. As functions, truth-functions transform an input or multiple inputs to a single output. The input to a truth-function consists of propositions bearing specific truth-values, and the output is a resulting complex proposition with a specific truth-value determined by the function. Mappings of all possible input and output truth-evaluated propositions are conveniently displayed in a standard truth-table, but can also be indicated without graphic devices in a metalogical language or language describing the properties of truth-functions in a logical system.

However truth-functions are characterized, they are generally considered to be so immediately surveyable and conceptually straightforward that we may find it hard to imagine that there could be anything mysterious or unanticipated in the elementary formal logic they define.

IDENTITY RELATA

It was Frege who first emphasized the existence of a philosophical connection between the concept of identity and the problems of reference. Frege explains their relationship in his important 1892 essay, “Uber Sinn und Bedeutung” (“On Sense and Reference”). Frege's essay, undoubtedly one of the most influential writings in the history of philosophy, sets the agenda for much of contemporary philosophy of language, as much for logicians and philosophers who take issue with its central teachings as for those who consider themselves modern-day Fregeans.

The concept of identity is at once indispensable and trivial. We need identity in order to explicate basic principles of metaphysics, and to formulate logical and mathematical equations. Identity by reputation is a reflexive, symmetric and transitive relation. Identity is self-identity; a relation that holds between a thing and itself, a = a (and, generally, ∀x[x = x]) (reflexivity); a = b → b = a (∀x,y[x = y → y = x]) (symmetry); [a = b ∧ b = c] a = c (x,y,z[x = y y = z] x = z) (transitivity). We are concerned about identity in cases of establishing uniqueness, as in Russell's analysis of definite descriptions, and in countless other applications. In practical affairs, it is often essential to determine the identity of an heiress to a fortune or the victim or perpetrator of a crime, or the substance in a certain compound.

WHAT IS TRUTH?

We will certainly not be the first to ask the difficult question “What is truth?” Often the query is posed by those who despair of any clear-cut answer or are cynical even about the possibility of understanding or ever arriving at the truth.

When that first postmodern deconstructionist Pontius Pilate cynically asked this of a Nazarene rebel in the Roman province of Judea under his jurisdiction (John 18:38), the problem of the constitutive nature of truth reasserted itself as a philosophical topic in Western consciousness from the time of its ancient Greek origins. The concept of truth is one of the most important, and one of the most elusive, in all of philosophy. By vocation, philosophers are driven by their love of knowledge to try to understand the nature of truth, in recognition of the long-standing analysis whereby knowledge implies truth. Even those who believe that there is no such thing as truth, or that we can never arrive or know with certainty that we have arrived at the truth, are collectively committed to working out a satisfactory understanding of the concept of truth, which they must first explain in order to deny.

TRUTH AND MEANING, MEANING AND TRUTH

We have seen that Davidson believes that philosophical semantics should begin with truth as better understood than the Fregean sense of sentences, and then build upon that understanding to advance toward a general theory of meaning.

CONCEPTS OF LOGIC

If we can explain what logic is, correctly describing in the most general way the nature or concept of logic, then we should have already said enough to understand why logic is important. We may begin, in keeping with customary practice, by saying that logic studies the structural properties of reasoning. Reasoning in turn is an exercise of thought, at least of thoughts considered in the abstract, in which conclusions are drawn inferentially from assumptions.

Assumptions and conclusions, in turn, are propositions. The ontic status of propositions is a frequent subject of philosophical dispute. Propositions can nevertheless be interpreted as sentence tokens or types or the abstract meanings of sentence tokens or types, in which a state of affairs is proposed for consideration, classically as true or false. Logical inference is generally considered to be a syntactical correlation of sentences representing assumptions and conclusions in permissible combinations, or as a semantic relation holding between the possible truth-values of propositions taken respectively as assumptions and conclusions. We may also be able to reason in the sense of drawing inferences from questions and commands, and from direct experiential encounters with the state of the world in the empirical experience of sensation and perception. Logicians have investigated some of these non-propositional formal inferential relations, but the topic has not been widely explored in the philosophical literature.

AN EXPRESSIVE LIMITATION

It would appear that nothing should be easier in standard predicate-quantificational logic than to formalize a relation in which something stands to everything. The surprising fact is that the relation cannot be adequately symbolized for all predicates within the resources of classical logic.

This chapter offers a simple example of the problem and considers the possibilities for symbolizing the relation. The implication is that standard predicate-quantificational logic is inadequate to the task of representing all quantificational propositions. The question, then, if classical logic fails, is which system of non-classical logic to adopt in order to formalize all sentences with the same logical form. The object of the present chapter, accordingly, is not to settle definitively the question of exactly how to proceed in light of this acknowledgment, but merely to appreciate the limitations of classical logic and recognize the need for a non-classical alternative.

Far from proposing an excursion into primate scatology, as our title might suggest, the problem we shall address is the challenge of properly formalizing the sentence “Some monkey devours every raisin”. The inability of classical predicate-quantificational logic to deal adequately with this relatively simple colloquial sentence points toward inherent expressive limitations in standardly interpreted formalisms. As Socrates reminds his interlocutors in Plato's early aporetic dialogues, as true in philosophy of logic as it is in ethics, aporia, an unexpected state of puzzlement about something we thought we understood, the final stage of the Socratic elenchus or interrogative method of enquiry, is the first step on the path toward maieusis, or cognitive (and moral) self-improvement.

AGAINST THE SEMANTIC GRAIN

If in logic we must ultimately have to do with objects and their properties, then it is not surprising to find some logicians and philosophers choosing or being intuitively compelled to begin with objects, making objects logically more basic than properties as the most basic, and others beginning instead with properties, for whatever contrary reasons or impulses, making properties logically more basic than objects. Those who begin with objects, and thus necessarily with entities or existent objects, are extensionalists in logic and semantics; those who begin with properties and thus unavoidably both with instantiated and uninstantiated combinations of properties, or, in any case, with logically possible properties in the abstract, are intensionalists.

Thus, it is no accident that Quine, an arch extensionalist, whose later recently collected writings include the title 2001 essay, “Confessions of a Confirmed Extensionalist”, prescribes in the first chapter of his 1960 book, Word and Object, “Beginning With Ordinary Things”, which is to say, with existent objects, and in its penultimate chapter VI, recommends a “Flight from Intension”. Extensionalists explain true and false predications as a matter of an existent object belonging to the extension of a predicate, including all and only the existent objects possessing the property represented by the predicate. The reason is obvious if we reflect that extensionalists, beginning with objects rather than properties, can only consider existent objects; since nonexistent objects by reputation do not exist, there are no objects other than existent objects for extensionalists to enlist as the objectual foundation of all logic and semantics.

In these chapters, I have tried not only to portray the technical progress of logic as a formal discipline in response to a variety of conceptual problems, but to introduce, albeit implicitly, from a somewhat unorthodox but rigorously argued philosophical perspective, what I consider to be some of the moral lessons that the development of logic and metalogic might be thought to teach.

The moral dimension of modern symbolic logic, as I understand it, recognizes logic as a product of human culture that stands alongside history, science, religion and art. As such, logic is a microcosm and allegory of many aspects of the human condition. When we try to have everything we want in logic or in life, we usually pay an unexpected price for it somewhere later down the line. This seems to have been the fate of the most ambitious formal symbolic logics, of Frege's and Whitehead and Russell's logicism, and of such projects as Hilbert's formalist programme in metamathematics. All of these efforts, despite their first bright promise and intuitive sense of rightness, have turned out to be limited in their pretensions by logic's own symbolic devices as revealed especially in the great variety of paradoxes, involving syntactical logical, semantic and set-theoretical diagonalizations, and formal metatheoretical limitations.

Logic as such offers a moral parable for life. It is not until the full consequences of what we have set in motion by undertaking an action or adopting an axiom in a formal system become clear that the world or logic itself can come crashing down on our most naively cherished aspirations.

What is logic, and to paraphrase American novelist Henry Miller on a rather different topic, how does it get that way? These are questions that we can answer at length only as we proceed to investigate a selection of problems that have shaped the study of logic over the course especially of the last one hundred and fifty years. Miller, who inspires the present book's title, offers the following reflections in his remarkable essay, “Money and How it Gets That Way”:

WHY PARADOXES MATTERA

A paradox nags and gnaws at our peace of mind. Confronted with a particularly diabolical paradox, we may experience an acute intellectual pulling and twisting in opposite directions as we struggle to make sense of a situation that we had previously assumed was unproblematic. When a paradox impinges we realize that we cannot simply have everything we want, or at least not as cheaply and easily as we had supposed in our former innocence. Paradoxes limit concepts in an especially striking way, unless we are obtuse to their impact, requiring us to rethink cherished positions as we try to negotiate passage through an unexpectedly complex conceptual terrain.

Quine, in his essay, “The Ways of Paradox”, distinguishes between paradoxes and antinomies. By this partition of conundrum types, a paradox is merely something surprising and unanticipated, whereas an antinomy is an outright logical contradiction or syntactical inconsistency. The so-called barber's paradox, by Quine's definition, citing his own example, counts only as a paradox rather than an antinomy. We may not have realized, reflecting superficially only on the words, that there cannot possibly be a barber who shaves all and only those persons who do not shave themselves. A para-doxa, tracing the meaning of the term to its Greek roots, is literally something that is beyond belief. It may startle us to learn that there can be no such barber; yet the proposition does not imply a logical contradiction or antinomy of the form p ∧ ┐p.

§1. The Hilbert problems and Hilbert's Program. In 1900 the great mathematician David Hilbert laid down a list of 23 mathematical problems which exercised a great influence on subsequent mathematical research. From the perspective of foundational studies, it is noteworthy that Hilbert's Problems 1 and 2 are squarely in the area of foundations of mathematics, while Problems 10 and 17 turned out to be closely related to mathematical logic.

1. 1. Cantor's Problem of the Cardinal Number of the Continuum.

2. 2. Compatibility of the Arithmetical Axioms.

3. 10. Determination of the Solvability of a Diophantine Equation.

4. 17. Expression of Definite Forms by Squares.

Our starting point here is Problem 2, the consistency (= “compatibility”) of the arithmetical axioms. In a later paper published in 1926, Hilbert further elaborated his ideas on the importance of consistency proofs. Hilbert's Program asks for a finitistic consistency proof for all of mathematics. Although we are not concerned with consistency proofs in Hilbert's sense, we are interested in certain logical structures which grew out of Hilbert's original concerns.

In answer to Hilbert's Problem 2 and Hilbert's Program, Gödel proved the famous Incompleteness Theorems. Let T be any theory in the predicate calculus satisfying certain well-known mild conditions. Then we have the following results:

• T is incomplete (First Incompleteness Theorem, Gödel 1931).

• The statement “T is consistent” is not a theorem of T

• (Second Incompleteness Theorem, Gödel 1931).

• […]