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Chapter 2

The History (and Future) of Logic (and Ethics)

§2.0 Overview of the Chapter

This chapter will focus on the second subdiscipline, which gets placed at center stage in what I call “analytic philosophy the method”—namely, logic. While I am adopting a general policy of discussing formalisms and technicalities as little as possible, this chapter will be the one that discusses minutiae in more detail than any other. That said, the point of this chapter is to show that there can be significant emancipatory potential in sometimes doing so. In order to motivate sticking with this discussion of the minutiae, I begin with a number of thoughts that most clearly express the motivations of my own study of logic. First, John Mohawk on the founding principles of the Haudenosaunee Confederacy—the rightful stewards of every bit of land I have ever lived on, be it John Mohawk’s own Seneca people in Buffalo, the Oneida people in Vernon, the Onondaga people in Syracuse, or the Mohawk people in Potsdam:

If you do not believe in the rational nature of the human being, you cannot believe that you can negotiate with him. If you do not believe that rational people ultimately desire peace, you cannot negotiate confidently with him toward goals you and he share. If you can’t negotiate with him, you are powerless to create peace. If you can’t organize around those beliefs, the principles cannot move from the minds of men into the actions of society. (Mohawk 1989, 221)

Building on this very same idea of logic as tool for peace in the very same year, yet discussing very different temporal and cultural contexts, John Corcoran says,

many exemplary moralists, including Socrates, Plato, Kant, Mill, Gandhi, and Martin Luther King, showed by their teachings and actions a deep commitment to objectivity, the ethical value that motivates logic and is served by logic. (Corcoran 1989b, 37)

Finally, to be clear that these should not be read as implicating the gendered reading of “men,” but the species reading of “men,” my earliest introduction to feminist philosophy:

“When men fight for their freedom, fight to be allowed to judge for themselves concerning their own happiness, isn’t it inconsistent and unjust to hold women down? I know that you firmly believe you are acting in the manner most likely to promote women’s happiness; but who made man the exclusive judge of that if woman shares with him the gift of reason? (Wollstonecraft 1792, 2)

§2.1 Logic, Ethics, and the Discipline of Philosophy

Alongside the centrality of chapter 1’s focus—philosophy of language—to the development of early analytic philosophy was chapter 2’s focus—logic. Logic holds a unique position as the most worked out analytic subdiscipline with the most uncontroversial progress. For somebody interested in the practical, everyday value of analytic philosophy, logic should not be overlooked. It is hard to imagine some individual manifestation of theory from any academic discipline, which more people would be willing to grant had everyday value than the computer—a piece of technology that develops through a story belonging to the history of logic. Given many communities’ growing distrust and skepticism of the value of philosophy, this is something that should make us think that pitching logic is an important piece of giving a pitch for philosophy.

That said, there is a group of common misconceptions about logic¹—that it is cold, unemotional, contrary to feelings, and separate from matters of ethics and value—which keep people from fully recognizing the importance of logic, analytic philosophy, and philosophy generally. Just as we tried to connect analytic philosophy of language to moral improvement and recipes for living one’s life, by looking at its history in the previous chapter, we will connect logic to such practical matters by looking at the history of logic and how it has been connected to ethical matters.

Returning to public skepticism of philosophy, questions about where the intellectual, institutional, and social structure of philosophy will be in 25, 50, 100, and 500 years have all been asked a number of times for a number of reasons. From my perspective, the most pressing of these have to do with how we will be able to undermine the racist, sexist, homophobic, classist, and otherwise oppressive biases that the discipline has inherited from being a part of an extremely oppressive society. Other important inquiries brought up in this regard include how we will deal with shrinking budgets, increasing temporary employment, anti-intellectualism in the broader culture, etc. Among the most important questions asked for a variety of these reasons are:

[Q1] What justifies our spending public money on philosophy?

[Q2] What does philosophy give back to the world?

[Q3] What will philosophy give back to the world in the future?

[Q4] How can philosophy better itself as a community and institution?

The main goal of this chapter is to discuss why I think partial answers to these questions and the future of analytic philosophy lies (or should lay) in logic and ethics coming together to a much greater extent than they do now. In order to achieve this goal, I will enlist help from the following sources:

(1) a certain picture of the development of the modern formal logical revolution, which has come to make Kant’s words infamous² and which gives good reason to think that there has been a natural, historical progression toward logic and ethics coming together more

(2) which will be discussed through specific works from several prominent actors in this story (n.b. Ruth Barcan Marcus, A. N. Prior, and John Corcoran) that show a great deal of potential for important work to come out of connecting logic and ethics and

(3) several examples that show the untapped potential of dedicated attention to nondeductive logics in understanding prejudice and bigotry.

These will be the topics taken up in the subsequent sections of the chapter, before a final section considering a particularly relevant objection from feminists to whom I hope to be an ally.

§2.2 The History of Logic from 1847 to the Present

When trying to determine the trajectory that any discipline is going to take in the future, it is almost always prudent to investigate the path taken during the period just traversed.³ Given that we are coming out of nothing short of a complete logical revolution, this is certainly true of the discipline of logic. For this reason, I will present a broad-strokes picture of the history of this logical revolution. Given how incomplete I will be in this context, it should be clear that this picture will be largely programmatic and will partly be based on what I see to be important for determining where logic will and should go from here. With this in mind, I take the history of the modern logical revolution to be a five-phased story, which looks something like the following:

[P1] Boole and De Morgan bring mathematical methods to the study of logical problems. This gives 1847 a particularly important place in the history of logic.

[P2] Formal systems of sentential logic are formulated, refined, and mastered. Beginning a trend that will be followed in the next two phases, its syntax is first mastered, then its semantics.

[P3] Formal systems of quantificational logic are formulated, refined, and mastered. Its syntax is relatively well understood by the time of Frege (1879). Its semantics is worked out in impressive detail in the 1930s, with the most important work coming from Tarski and Gödel.

[P4] Formal systems of modal logic are formulated, refined, and mastered. Its syntax is developed by Lewis and Marcus and the semantics is understood well enough to have completeness proofs and applications to natural language by Kripke (1963) and Kripke (1980).

[P5] From the time of Kripke’s 1970 lectures, which would become Naming and Necessity up to today, the story of the development of logic becomes a much more disjointed and partitioned one.

With this outline in place, a bit more about each of these phases is in order.

P1: The Mathematization of Logic

Hans-Johann Glock begins his history of analytic philosophy with a section entitled “First Glimmerings: Mathematics and Logic” (Glock 2008, 26). In this section, Glock looks at the ways in which logico-philosophical problems in the foundations of mathematics surrounding algebra, geometry, analysis, and set theory set the stage for increased connections between mathematics and logic. I too believe this is a suitable place to start and, like Glock, that, with the exception of anticipations by Bolzano that have been unfortunately ignored for some time, the new logic begins in Boole’s mathematization of logic (Glock 2008, 27–28). Boole’s logical contributions begin in 1847 with his Mathematical Analysis of Logic, the same year as Augustus De Morgan’s Formal Logic: Or, The Calculus of Inference. Both of them followed up on these works with further significant contributions in Boole’s The Laws of Thought (1854) and De Morgan’s Syllabus of a Proposed System of Logic (1860). Their individual contributions have been noted many times over by historians and logicians much more competent than myself (Kneale & Kneale 1962, Patton 2018, Terzian 2017). So, I will not rehearse these moves here. We should simply note that what is most important about their work from my general perspective is the bringing of mathematical methods to bear on questions in logic. This is what would ultimately allow logicism to play the role it did at the outset of the explicit movement of analytic philosophy. Though, for example, Aristotle had wanted to explain the knowledge gained in doing mathematics with his system of logic, the two had developed somewhat independently up until the nineteenth century. Boole and De Morgan changed this drastically and forever.⁴

This mathematization of the study of logic has been most significant for the methods and tools of clarity, precision, and objectivity that have been brought along with it. Most notable here are the uses of algebraic methods and notation, algorithmic procedures, axiomatic structures, recursive definitions, and set-theoretic frameworks.⁵ Obviously, nothing even slightly resembling modern formal logic could exist without just the members of this short list. Foreshadowing the rest of our sketch of the history of the modern logical revolution, the use of algebra brought with it the modern logician’s lexicon of variable letters, operator symbols, and function-notation. Concern with, and understanding of, algorithms culminates in the famous theorems of decidability and computability of Turing, Church, and others. The use of the axiomatic method in logic has led to the development of modern proof theories, with their bifurcation of axioms and rules of inference. Recursion theory has allowed us to give finitely expressible specifications of the infinity of grammatical constructions in our logical systems (and a means by which to wrangle the theories of their semantics in a finitely expressible way). And, finally, the set-theoretic universe has provided a domain in which to do respectably rigorous semantic work, something whose possibility was doubted even into the twentieth century (e.g., Wittgenstein 1922, Carnap 1934). Thus, the importance of the bringing together of mathematics and logic can simply not be overstated. Likewise, I hope the next revolution in logic will be brought about by the bringing together of logic and ethics.

P2: Sentential Logic

The next three phases in the history of logic form something of a cohesive unit with a similar pattern repeated at each step. That is to say, each phase is defined by the systematic development of a distinct branch of formal deductive logic, each branch being more complex than the last. In particular,

(1) we start with the mastery of sentential logic—the study of the inferential behavior of sentential connectives

(2) then move to a mastery of quantificational logic—the study of the inferential behavior of quantifier phrases, predicates, and singular terms

(3) and finish with a mastery of modal logic—the study of the inferential behavior of modal sentential operators.

This constitutes a natural progression, since we can think of

(1) sentential logic as arising from the logical relations between simple and complex sentences

(2) quantificational logic as arising from the logical structure within the simple sentences, and

(3) modal logic as arising from saying things about the different kinds of completed sentences from the previous two levels.

Thus, it is unsurprising that we find the chronological order to match this progression of complexity—sentential to quantificational to modal.

It is also interesting to note that, within each of these phases, a similar progression is followed. First, the syntactic side of the deductive system is developed and mastered. That is, we start with explicit, systematic treatments of the lexicon, grammar, axioms, and proof theory of the logic under consideration. From there, the semantics of the system is developed second. We then define truth, consequence, denotation, and the like, which allows us to state and prove the main meta-logical results for each of the systems—soundness and completeness. Some have only partially noticed this pattern and have been perplexed. For instance, Burgess says of this situation that

Given how greatly model theory illuminates the significance of formulas in temporal logic, one would expect a modal model theory parallel to temporal model theory to have been developed early, and to have guided the choice of candidate modal axioms to be considered. But the historical development of a science is seldom rational. (Burgess 2009, 47)

Recognizing that the case of mastery of syntax prior to that of semantics in modal logic is part and parcel of a larger trend can make it seem perfectly rational, though. Just as we developed sentential prior to quantificational (and that prior to modal) logic as a result of the fact that one is successively more difficult than the next, semantics (model theory) developed after syntax (proof theory) at each phase because semantics is just harder than syntax.

In the case of sentential logic, this story played out with the mastery of syntax being achieved by Frege (1879) with respect to axioms and proof theory and Sheffer (1913) in terms of lexicographical and grammatical simplicity. Furthermore, the tools needed to understand the semantics of the sentential calculus were developed primarily by Wittgenstein (1922) and Post (1921). Here we find the introduction of models in terms of truth-table assignments and calculations—procedures that make tautology and consequence decidable questions. Despite having the rudiments necessary for a more-or-less complete understanding of the basic meta-logical questions of sentential logic from these works, significant additions and extensions are made by subsequent investigations into the soundness and completeness of extensions of sentential logic. It is to these extensions that we now turn.

P3: Quantificational Logic

The story of quantificational logic is a bit more complicated, but it follows the same basic pattern. Fundamentals of syntax are mastered, followed by semantic ideas, which then allow for the establishment of basic meta-logical results. On the syntactic side, the proof theory for quantificational logic is given in impressive detail with Frege’s (1879) nine axioms and three inference rules. As is well known, though, Frege’s two-dimensional notation was extremely cumbersome and not particularly reader-friendly. As a result, the notation provided by Peano (1889) was a welcome change to the lexicon of quantificational logic. This notation was made most popular by Russell, who made significant strides in our understanding of the logical grammar of descriptions, scopes, and names. That said, there was an independent tracing of this story through Americans like Charles Peirce, Christine Ladd-Franklin, and Oscar Howard Mitchell as well as Lvov-Warsaw School members like Twardowski, Kokoszyńska-Lutmanowa, Łukasiewicz, and Leśniewski.⁶ Of course, the merging of these traditions was cemented by the significance and influence of the semantic work of Tarski.

Despite being the source of the canonical semantics for the subsequent three-quarters of a century of work on quantificational logic, Tarski’s semantic definition of “truth,” his model-theoretic definition of logical consequence, and his work defining models was predated by several years by the standard meta-logical results. In fact, Gödel had proved the completeness of first-order quantificational logic a year prior to his (1930). This may, at first glance, seem to constitute a counter-example to my picture of the development of logic during the modern logical revolution. That said, it is important to note that Gödel’s work has come to be only remembered for establishing that the completeness theorem is true. Once Tarski developed a far superior semantics for quantificational logic than anyone else had prior in his (1933, 1936, 1944, and so on) it is not surprising that a far superior proof of completeness would have been developed. And this is exactly what we find—the canonical proof of completeness coming in Henkin (1949).⁷ Thus, we still have the same pattern within the history of quantificational logic. The development is marked by mastery of syntax followed by mastery of semantics, allowing for the mastery of the basic meta-logical results.

P4: Modal Logic

Before we get into the modern history of modal logic, it is important to correct a mistaken view of how this modern modal story contrasts with the larger history of logic. Aside from Burgess’ earlier mistake, it is often erroneously suggested that the modern modal story is the whole of the modal story. For instance, in a book which is quite good, Sider says that when looking into the history of modal logic, we find that it “arose from dissatisfaction with the material conditional of standard propositional logic” (Sider 2010, 137). Given the twentieth-century origin of this dissatisfaction, this leads to the view that C. I. Lewis was not just the first significant modern modal logician, but the first modal logician. While it was not as systematically developed as Aristotelian quantificational logic, a sophisticated understanding of the logic of various modal operators was achieved by the time of the Islamic Golden Age. Most notably, Ibn Sina’s logic and the Avicennian tradition of logic, which came to rival Aristotelian logic, at this time included a rigorous system for reasoning with temporal modalities. That said, those working in the Aristotelian tradition did interesting related work as well—including al-Farabi, Abu Bishr Matta, and Yahya ibn Adi, among others (Rescher 1963).⁸

As I alluded to, Sider and others are correct in that the modern mathematical study of modal logic begins with Lewis (1918) and Lewis and Langford (1932). Given what we have seen in the previous two sections, it is not surprising that Lewis’ works are syntactic tour de forces. Here, the standard lexicon (‘◊’, ‘’), grammar, and proof theory for modal propositional logics are all established with great precision. On top of that, several axiomatic systems still studied today are developed. That said, modern modal syntax is not mastered until Ruth Barcan Marcus’ pioneering work in Marcus (1946a, 1946b, 1947). Here, we get the first systems of quantified modal logic, which contain all of the innovation in syntax from the entire period under discussion. In addition, Marcus provides the key axiom, which explains the relative logical behavior of the quantifiers and the modal operators—the Barcan Formula. And, not satisfied with purely formal contributions to the understanding of modal logic, Marcus also went on to give important reasons to believe these formal systems could weigh in on long-held metaphysical debates. Given various criticisms from Quine, which were taken to show that anything like this was impossible, Marcus’ work was enormously important.