Race, Gender, and the History of Early Analytic Philosophy - Matt LaVine

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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

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