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He kept up with current developments, and in particular was informed about the logical work of Whitehead and Russell, largely through his student Heinrich Behmann. When we consider the matter more closely, we soon recognize that the question of the consistency for integers and for sets is not one that stands alone, but that it belongs to a vast domain of difficult epistemological questions which have a specifically mathematical tint: for example to characterize this domain of questions briefly , the problem of the solvability in principle of every mathematical question, the problem of the subsequent checkability of the results of a mathematical investigation, the question of a criterion of simplicity for mathematical proofs, the question of the relationship between content and formalism in mathematics and logic, and finally the problem of the decidability of a mathematical question in a finite number of operations.

Hilbert — Although Bernays had little previous experience in foundations, this turned out to be a shrewd choice, and the beginning of a close and fruitful research partnership. Hilbert for the first time clearly distinguishes metalanguage from object language, and step-by-step presents a sequence of formal logical calculi of gradually increasing strength.

Each calculus is carefully studied in its turn; its strengths and its weaknesses are identified and balanced, and the analysis of the weaknesses is used to prepare the transition to the next calculus. The function calculus is a system of many-sorted first-order logic, with variables for sentences as well as for relations. It is here, for the first time, that we encounter a precise, modern formulation of first-order logic, clearly differentiated from the other calculi, given an axiomatic foundation, and with metalogical questions explicitly formulated.

Hilbert concludes his discussion of first-order logic with the remark:. The basic discussion of the logical calculus could cease here if we had no other end in view for this calculus than the formalization of logical inference. But we cannot be satisfied with this application of symbolic logic. Not only do we want to be able to develop individual theories from their principles in a purely formal way, but we also want to investigate the foundations of the mathematical theories themselves and examine how they are related to logic and how far they can be built up from purely logical operations and concept formations; and for this purpose the logical calculus is to serve us as a tool.

The lecture protocol ends with the sentence:. Thus it is clear that the introduction of the Axiom of Reducibility is the appropriate means to turn the calculus of levels into a system out of which the foundations for higher mathematics can be developed. The following summer, Bernays produced a Habilitation thesis in which he developed, with full rigor, a Hilbert-style, axiomatic analysis of propositional logic. He then proceeds to investigate questions of decidability, consistency, and the mutual independence of various combinations of axioms. The Hilbert lectures and the Bernays Habilitation are a milestone in the development of first-order logic.

In the lectures, for the first time, first-order logic is presented in its own right as an axiomatic logical system, suitable for study using the new metalogical techniques. It was those metalogical techniques that represented the crucial advance over Peirce and Frege and Russell, and that were in time to bring first-order logic into focus. But that did not happen at once, and a great deal of work still lay ahead. It was characteristic of Hilbert to break complex mathematical phenomena into their elements: the sequence of calculi can be viewed as a decomposition of higher-order logic into its simpler component parts, revealing to his students precisely the steps that went into the building of the full system.

Although he discusses the functional calculus, he does not single it out for special attention. In other words and as with Peirce three decades earlier first-order logic is introduced primarily as an expository device: its importance was not yet clear. His proof of the completeness of propositional calculus is a mere sketch, and relegated to a footnote; the parallel problem for first-order logic is not even raised as a conjecture.

Even more strikingly, when Bernays eventually in published his Habilitation , he omitted his proof of the completeness theorem because as he later ruefully said the result seemed at the time straightforward and unimportant. For discussion of this point, see Hilbert [LFL]: For readily available general discussions, see Sieg , Zach , and the essays collected in Sieg ; for the original documents and detailed analysis, see Hilbert [LFL. The Hilbert school throughout the s regarded first-order logic as a fragment of type theory, and made no argument for it as a uniquely favored system.

He did not at the time publish on these topics because, as he later said:. I believed that it was so clear that the axiomatization of set theory would not be satisfactory as an ultimate foundation for mathematics that, by and large, mathematicians would not bother themselves with it very much. To my astonishment I have seen recently that many mathematicians regard these axioms for set theory as the ideal foundation for mathematics. For this reason it seemed to me that the time had come to publish a critique.

Skolem appendix. In the second he provided a new proof of that result. These technical results were of great importance for the subsequent debate over first-order logic. But it is important not to read into Skolem a later understanding of the issues. Skolem at this point did not possess a distinction between the object language and the metalanguage. And although in retrospect his axiomatization of set theory can be interpreted to be first-order, he nowhere emphasizes that fact. Indeed, Eklund presents a compelling argument that Skolem did not yet clearly appreciate the significance of the distinction between first-order and second-order logic, and that the reformulation of the axiom of separation is not in fact as unambiguously first-order as it is often taken to be.

There are two broad tendencies within logic during these years, and they pull in opposite directions. One tendency is towards pruning down logical and mathematical systems so as to accommodate the criticisms of Brouwer and his followers. Set theory was in dispute, and Skolem explicitly presented his results as a critique of set theoretical foundations. To put the matter slightly differently: the very point of axiomatizing set theory was to state its philosophically problematic assumptions in such a way that one could clearly see what they came to.

One possibility was to restrict oneself to first-order logic; another, to adopt some sort of predicative higher-order system.

## Logic: A History of its Central Concepts

Similar broadly constructivist tendencies were also very much in evidence in the proof theoretical work of Hilbert and Bernays and their followers in the s. Hilbert, like C. The epsilon-substitution method was the principal device Hilbert introduced in order to attempt to attain this result. But despite these constructive tendencies, many logicians of the s including Hilbert continued to regard higher-order type theory, and not its first-order fragment, as the appropriate logic for investigations in the foundations of mathematics. The ultimate hope was to provide a consistency proof for the whole of classical mathematics including set theory.

But, in the meanwhile, researchers still were somewhat unclear about certain basic distinctions. So matters remained unclear throughout the s. But the constructivist ambitions of the Hilbert school, the focus on the analysis of the quantifiers, and the explicit posing of metalogical questions had made the emergence of first-order logic as a system worthy of study in its own right all but inevitable.

With these results and others that soon followed it finally became clear that there were important metalogical differences between first-order logic and higher-order logics. Perhaps most significantly, first-order logic is complete, and can be fully formalized in the sense that a sentence is derivable from the axioms just in case it holds in all models.

Second-order logic does not. By the middle of the s these distinctions were beginning to be widely understood, as was the fact that categoricity can in general only be obtained in higher-order systems. But the technical results alone did not settle the matter in favor of first-order logic. In other words, even after the metalogical results there was a choice to be made, and the choice in favor of first-order logic was not inevitable.

After all, the metalogical results can be taken to show a severe limitation of first-order logic: that it is not capable of specifying a unique model even for the natural numbers. At this point in the s, however, several other strands of thinking about logic now coalesced. The intellectual situation was highly complex. A search for secure foundations, and in particular for an avoidance of the set-theoretical paradoxes, was something they shared, and that helped to tip the balance in favor of first-order logic.

As a practical matter, these first-order set theories sufficed to formulate all existing mathematical practice; so for the codification of mathematical proofs, there was no need to resort to higher-order logic. This confirmed an observation that Hilbert had already made as early as , though without himself fully developing the point. Thirdly, there was an increased tendency to distinguish between logic and set theory, and to view set theory as a branch of mathematics.

By the end of the decade, a consensus had been reached that, for purposes of research in the foundations of mathematics, mathematical theories ought to be formulated in first-order terms. Let us now try to draw some lessons, and in particular ask whether the emergence of first-order logic was inevitable. I begin with an observation.

Each stage of this complex history is conditioned by two sorts of shifting background consideration. One is broadly mathematical: the theorems that had been established. The other is broadly philosophical: the assumptions that were made explicitly or tacitly about logic and about the foundations of mathematics. These two things interacted. Each thinker in the sequence starts with some more or less intuitive ideas about logic.

Those ideas prompt mathematical questions: distinctions are drawn: theorems are proved: consequences are noted, and the philosophical understanding is sharpened. Let us now consider the question: When was first-order logic discovered? That question is too general. It needs to be broken down into three subsidiary questions:. Equipped with these distinctions, let us now ask: Why was first-order logic not discovered earlier?

It is striking that Peirce, already in , had clearly differentiated between propositional logic, first-order logic, and second-order logic. He was aware that propositional logic is significantly weaker than quantificational logic, and, in particular, is inadequate to an analysis of the foundations of arithmetic.

### 2. Charles S. Peirce

He could then have gone on to observe that second-order logic is in certain respects philosophically problematic, and that, in general, our grasp on quantification over objects is firmer than our grasp on quantification over properties. The problem arises even if the universe of discourse is finite. We have, for example, a reasonable grasp on what it means to speak in first-order terms of all the planets , or to say that there exists a planet with a particular property.

But what does it mean to talk in second-order terms of all properties of the planets? What is the criterion of individuation for such properties? Is the property of being the outermost planet the same as the property of being the smallest planet? One of the attractions of the Handbook's several volumes is the emphasis they give to the enduring relevance of developments in logic throughout the ages, including some of the earliest manifestations of the subject. The Handbook of the History of Logic will be necessary reading for researchers, and graduate and senior undergraduate students in logic in all its forms, argumentation theory, AI and computer science, cognitive psychology and neuroscience, linguistics, forensics, philosophy and the philosophy and the history of philosophy, and the history of ideas.

Handbook of the History of Logic brings to the development of logic the best in modern techniques of historical and interpretative scholarship. Computational logic was born in the twentieth century and evolved in close symbiosis with the advent of the first electronic computers and the growing importance of computer science, informatics and artificial intelligence. With more than ten thousand people working in research and development of logic and logic-related methods, with several dozen international conferences and several times as many workshops addressing the growing richness and diversity of the field, and with the foundational role and importance these methods now assume in mathematics, computer science, artificial intelligence, cognitive science, linguistics, law and many engineering fields where logic-related techniques are used inter alia to state and settle correctness issues, the field has diversified in ways that even the pure logicians working in the early decades of the twentieth century could have hardly anticipated.

The Dartmouth Conference in — generally considered as the birthplace of artificial intelligence — raised explicitly the hopes for the new possibilities that the advent of electronic computing machinery offered: logical statements could now be executed on a machine with all the far-reaching consequences that ultimately led to logic programming, deduction systems for mathematics and engineering, logical design and verification of computer software and hardware, deductive databases and software synthesis as well as logical techniques for analysis in the field of mechanical engineering.

There are or were philosophers who take mathematics to be no more than a meaningless game played with symbols chapter 8 in this volume , but everyone else holds that mathematics has some sort of meaning. What is this meaning, and how does it relate to the meaning of ordinary nonmathematical discourse? Another group of issues consists of attempts to articulate and interpret particular mathematical theories and concepts.

One example is the foundational work in arithmetic and analysis. Sometimes, this sort of activity has ramifications for mathematics itself, and thus challenges and blurs the boundary between mathematics and its philosophy. Interesting and powerful research techniques are often suggested by foundational work that forges connections between mathematical fields. In addition to mathematical logic, consider the embedding of the natural numbers in the complex plane, via analytic number theory.

Foundational activity has spawned whole branches of mathematics. Sometimes developments within mathematics lead to unclarity concerning what a certain concept is. The example developed in Lakatos [] is a case in point. For another example, work leading to the foundations of analysis led mathematicians to focus on just what a function is, ultimately yielding the modern notion of function as arbitrary correspondence.

The questions are at least partly ontological. This group of issues underscores the interpretive nature of philosophy of mathematics. We need to figure out what a given mathematical concept is , and what a stretch of mathematical discourse says. It is not clear a priori how this blatantly dynamic discourse is to be understood. What is the logical form of the discourse and what is its logic? What is its ontology? The history of analysis shows a long and tortuous task of showing just what expressions like this mean.

Of course, mathematics can often go on quite well without this interpretive work, and sometimes the interpretive work is premature and is a distraction at best. In the present context, the question is whether the mathematician must stop mathematics until he has a semantics for his discourse fully worked out. Surely not. Moreover, we are never certain that the interpretive project is accurate and complete, and that other problems are not lurking ahead. I now present sketches of some main positions in the philosophy of mathematics.

The list is not exhaustive, nor does the coverage do justice to the subtle and deep work of proponents of each view. Nevertheless, I hope it serves as a useful p. Of course, the reader should not hold the advocates of the views to the particular articulation that I give here, especially if the articulation sounds too implausible to be advocated by any sane thinker. According to Alberto Coffa [] , a major item on the agenda of Western philosophy throughout the nineteenth century was to account for the at least apparent necessity and a priori nature of mathematics and logic, and to account for the applications of mathematics, without invoking anything like Kantian intuition.

The main theme—or insight, if you will—was to locate the source of necessity and a priori knowledge in the use of language. Philosophers thus turned their attention to linguistic matters concerning the pursuit of mathematics. What do mathematical assertions mean? What is their logical form? What is the best semantics for mathematical language? The members of the semantic tradition developed and honed many of the tools and concepts still in use today in mathematical logic, and in Western philosophy generally. Michael Dummett calls this trend in the history of philosophy the linguistic turn.

An important program of the semantic tradition was to show that at least some basic principles of mathematics are analytic , in the sense that the propositions are true in virtue of meaning. If the program could be carried out, it would show that mathematical truth is necessary—to the extent that analytic truth, so construed, is necessary. Given what the words mean, mathematical propositions have to be true, independent of any contingencies in the material world.

And mathematical knowledge is a priori—to the extent that knowledge of meanings is a priori. Presumably, speakers of the language know the meanings of words a priori, and thus we know mathematical propositions a priori. The most articulate version of this program is logicism , the view that at least some mathematical propositions are true in virtue of their logical forms chapter 5 in this volume.

According to the logicist, arithmetic truth, for example, is a species of logical truth. The most detailed developments are those of Frege [ , ] and Alfred North Whitehead and Bertrand Russell [].

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Unlike Russell, Frege was a realist in ontology, in that he took the natural numbers to be objects. For any concepts F , G , the number of F 's is identical to the number of G 's if and only if F and G are equinumerous. Frege showed how to define equinumerosity without invoking natural numbers. This became known as the Caesar problem. It is not that anyone would confuse a natural number with the Roman general Julius Caesar, but the underlying idea is that we have not succeeded in characterizing the natural numbers as objects unless and until we can determine how and why any given natural number is the same as or different from any object whatsoever.

The distinctness of numbers and human beings should be a consequence of the theory, and not just a matter of intuition. The number 2, for example, is the extension or collection of all concepts that hold of exactly two elements. The inconsistency in Frege's theory of extensions, as shown by Russell's paradox, marked a tragic end to Frege's logicist program. Russell and Whitehead [] traced the inconsistency in Frege's system to the impredicativity in his theory of extensions and, for that matter, in Hume's principle.

They sought to develop mathematics on a safer, predicative foundation. Their system proved to be too weak, and ad hoc adjustments were made, greatly reducing the attraction of the program. There is a thriving research program under way to see how much mathematics can be recovered on a predicative basis chapter 19 in this volume. Variations of Frege's original approach are vigorously pursued today in the work of Crispin Wright, beginning with [] , and others like Bob Hale [] and Neil Tennant [ , ] chapter 6 in this volume.

The idea is to bypass the treatment of extensions and to work with fully impredicative Hume's principle, or something like it, directly. But what is the philosophical point? On the neologicist approach, Hume's principle is taken to p. Hume's principle is akin to an implicit definition. Indeed, the only essential use that Frege made of extensions was to derive Hume's principle—everything else concerning numbers follows from that.

Neologicism is a reconstructive program showing how arithmetic propositions can become known. Without this feature, the derivation of the Peano axioms from Hume's principle would fail. This impredicativity is consonant with the ontological realism adopted by Frege and his neologicist followers. The neologicist project, as developed thus far, only applies basic arithmetic and the natural numbers. An important item on the agenda is to extend the treatment to cover other areas of mathematics, such as real analysis, functional analysis, geometry, and set theory.

The program involves the search for abstraction principles rich enough to characterize more powerful mathematical theories see, e. Coffa [] provides a brief historical sketch of the semantic tradition, outlining its aims and accomplishments. Many philosophers no longer pay serious attention to notions of meaning, analyticity, and a priori knowledge.

To be precise, such notions are not given a primary role in the epistemology of mathematics, or anything else for that matter, by many contemporary philosophers. Quine e. Quine's view is that the linguistic and factual components of a given sentence cannot be sharply distinguished, and thus there is no determinate notion of a sentence being true solely in virtue of language analytic , as opposed to a sentence whose truth depends on the way the world is synthetic. Then how is mathematics known? Quine is a thoroughgoing empiricist, in the tradition of John Stuart Mill chapter 3 in this volume.

His positive view is that all of our beliefs constitute a seamless web answerable to, and only to, sensory stimulation. Moreover, no part of the web is knowable a priori. This picture gives rise to a now common argument for realism. Their argument begins with the observation that virtually all of science is formulated in mathematical terms. Because mathematics is indispensable for science, and science is well confirmed and approximately true, mathematics is well confirmed and true as well. This is sometimes called the indispensability argument.

Quine, at least, is also a realist in ontology. According to Quine and Putnam, all of the items in our ontology—apples, baseballs, electrons, and numbers—are theoretical posits. We accept the existence of all and only those items that occur in our best accounts of the material universe.

On such views, mathematical knowledge cannot be dependent on anything as blatantly p. The noble science of mathematics is independent of all of that. From the opposing Quinean perspective, mathematics and logic do not enjoy the necessity traditionally believed to hold of them; and mathematics and logic are not knowable a priori.

Indeed, for Quine, nothing is knowable a priori. From this perspective, mathematics is of a piece with highly confirmed scientific theories, such as the fundamental laws of gravitation. No belief is incorrigible. No knowledge is a priori, all knowledge is ultimately based on experience see Colyvan [] , and chapter 12 in this volume.

The idea is to see philosophy as continuous with the sciences, not prior to them in any epistemological or foundational sense. If anything, the naturalist holds that science is prior to philosophy. Naturalized epistemology is the application of this theme to the study of knowledge.

The philosopher sees the human knower as a thoroughly natural being within the physical universe. Any faculty that the philosopher invokes to explain knowledge must involve only natural processes amenable to ordinary scientific scrutiny. Naturalized epistemology exacerbates the standard epistemic problems with realism in ontology. The challenge is to show how a physical being in a physical universe can come to know about abstracta like mathematical objects see Field [, essay 7].

Since abstract objects are causally inert, we do not observe them but, nevertheless, we still seem to know something about them. The Quinean meets this challenge with claims about the role of mathematics in science. Articulations of the Quinean picture thus should, but usually do not, provide a careful explanation of the application of mathematics to science, rather than just noting the existence of this applicability chapter 20 in this volume.

How is it that talk of numbers and functions can shed light on tables, bridge stability, and market stability? Such an analysis would go a long way toward defending the Quinean picture of a web of belief. Once again, it is a central tenet of the naturalistically minded philosopher that there is no first philosophy that stands prior to science, ready to either justify or criticize it. Science guides philosophy, not the other way around.

## Introduction to Logic Diagrams

There is no agreement among naturalists that the same goes for mathematics. Quine himself accepts mathematics as true only to the extent that it is applied in the sciences. In particular, he does not accept the basic assertions of higher set theory because they do not, at present, have any empirical applications. Moreover, he advises mathematicians to conform their practice to his version of naturalism by adopting a weaker and less interesting, but better understood, set theory than the one they prefer to work with.

Mathematicians themselves do not follow the epistemology suggested by the Quinean picture. They do not look for confirmation in science before publishing their results in mathematics journals, or before claiming that their theorems are true. Thus, Quine's picture does not account for mathematics as practiced.

Some philosophers, such as Burgess [] and Maddy [ , ] , apply naturalism to mathematics directly, and thereby declare that mathematics is, and ought to be, insulated from much traditional philosophical inquiry, or any other probes that are not to be resolved by mathematicians qua mathematicians. On such views, philosophy of mathematics—naturalist or otherwise—should not be in the business of either justifying or criticizing mathematics chapters 13 and 14 in this volume.

The most popular way to reject realism in ontology is to flat out deny that mathematics has a subject matter. The nominalist argues that there are no numbers, points, functions, sets, and so on. The burden on advocates of such views is to make sense of mathematics and its applications without assuming a mathematical ontology. A variation on this theme that played an important role in the history of our subject is formalism. Mathematics is likened to the play of a game like chess, where characters written on paper play the role of pieces to be moved. All that matters to the pursuit of mathematics is that the rules have been followed correctly.

As far as the philosophical perspective is concerned, the formulas may as well be meaningless. Opponents of game formalism claim that mathematics is inherently informal and perhaps even nonmechanical. Mathematical language has meaning, and it is a gross distortion to attempt to ignore this. At best, formalism focuses on a small p. It deliberately leaves aside what is essential to the enterprise.

A different formalist philosophy of mathematics was presented by Haskell Curry e. The program depends on a historical thesis that as a branch of mathematics develops, it becomes more and more rigorous in its methodology, the end result being the codification of the branch in formal deductive systems. Curry claimed that assertions of a mature mathematical theory are to be construed not so much as the results of moves in a particular formal deductive system as a game formalist might say , but rather as assertions about a formal system.

In effect, mathematics is metamathematics. See chapter 8 in this volume for a more developed account of formalism. On the contemporary scene, one prominent version of nominalism is fictionalism , as developed, for example, by Hartry Field []. Numbers, points, and sets have the same philosophical status as the entities presented in works of fiction. According to the fictionalist, the number 6 is the same kind of thing as Dr. Watson or Miss Marple. According to Field, mathematical language should be understood at face value. Of course, Field does not exhort mathematicians to settle their open questions via this vacuity.

Unlike Quine, Field has no proposals for changing the methodology of mathematics. His view concerns how the results of mathematics should be interpreted, and the role of these results in the scientific enterprise.

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For Field, the goal of mathematics is not to assert the true. The only mathematical knowledge that matters is knowledge of logical consequences see Field []. As we have seen, more traditional philosophers—and most mathematicians—regard indispensability as irrelevant to mathematical knowledge. In contrast, for thinkers like Field, once one has undermined the indispensability argument, there is no longer any serious reason to believe in the existence of mathematical objects. As Quine and Putnam pointed out, most of the theories developed in scientific practice are not nominalistic, and so begins the indispensability argument.

The first aspect of Field's program is to develop nominalistic versions of p. Of course, Field does not do this for every prominent scientific theory. To do so, he would have to understand every prominent scientific theory, a task that no human can accomplish anymore. Field gives one example—Newtonian gravitational theory—in some detail, to illustrate a technique that can supposedly be extended to other scientific work.

The second aspect of Field's program is to show that the nominalistic theories are sufficient for attaining the scientific goal of determining truths about the physical universe i. Thus, if the mathematical theory is conservative over the nominalist one, then any physical consequence we get via the mathematics we could get from the nominalistic physics alone. This would show that mathematics is dispensable in principle, even if it is practically necessary. The sizable philosophical literature generated by Field [] includes arguments that Field's technique does not generalize to more contemporary theories like quantum mechanics Malament [] ; arguments that Field's distinction between abstract and concrete does not stand up, or that it does not play the role needed to sustain Field's fictionalism Resnik [] ; and arguments that Field's nominalistic theories are not conservative in the philosophically relevant sense Shapiro [].

The collection by Field [] contains replies to some of these objections. The philosopher understands mathematical assertions to be about what is possible, or about what would be the case if objects of a certain sort existed. The formal language developed in Chihara [] includes variables that range over open sentences i. With keen attention to detail, p. Mathematics comes out objective, even if it has no ontology. Chihara's program shows initial promise on the epistemic front.

Perhaps it is easier to account for how the mathematician comes to know about what is possible, or about what sentences can be constructed, than it is to account for how the mathematician knows about a Platonic realm of objects. See chapters 15 and 16 in this volume. Unlike fictionalists, traditional intuitionists , such as L.

Brouwer e. Natural numbers and real numbers are mental constructions or are the result of mental constructions. In mathematics, to exist is to be constructed. Some of their writing seems to imply that each person constructs his own mathematical realm. Communication between mathematicians consists in exchanging notes about their individual constructive activities.

This would make mathematics subjective. It is more common, however, for these intuitionists, especially Brouwer, to hold that mathematics concerns the forms of mental construction as such see Posy []. This follows a Kantian theme, reviving the thesis that mathematics is synthetic a priori.