A new system of first-order predicate logic based on Wittgenstein\u27s single operation N

What is a language? One way in which we could approach this question is to ask the related question, “What do we need in order to understand a language?” In his Tractatus Logico-Philosophicus [17], Ludwig Wittgenstein gives an answer to his interpretation of this question, in which he seeks to minimize the amount of information we need to understand the literal meaning of declarative sentences. His solution was to present a non-standard way to express propositions in a logical notation. Through this notation Wittgenstein concludes that, in order to understand the literal meaning of declarative sentences, we need only understand the sense of elementary propositions, as well as how to perform a single operation, which he calls the N operator. For Wittgenstein, every proposition must be capable of being analyzed so that it can be expressed in this novel notation – that is, every proposition shows whether it is a tautology, contradiction, or contingent proposition. To impose this requirement is to “demand” [fordern] an adequate notation (6.1223)1 . This demand is easy to meet in quantifier-free logic, if we simply express each proposition with a truth-table. For Wittgentstein, however, quantified propositions must also meet this demand. The N operator, then, is his attempt to do this for quantified propositions. Wittgenstein therefore demands that all logical propositions be constructed by the successive application of the single operator N. He goes on, in the Tractatus, to give us some idea of what this N operator looks like; it can most easily be understood as a joint-negator, which takes a class of propositions and outputs a proposition that is true if and only if all of them are false. In addition, Wittgenstein also insists on an identity-free system – meaning he bans use of the symbol “=,” instead insisting that different terms always refer to different objects. In this way, we never need to make any statements about two objects being the same. Wittgenstein gives us a few examples in the Tractatus of how this identity-free system works, as well as of the various settings in which the N operator can be used, and of how to express various states of affairs using the operator. However, his account is by no means complete; his development of the N logic system leaves many aspects of the system unclear, and in some cases even impossible to realize if one sticks to the letter of the Tractatus. In other words, Wittgenstein clearly envisioned the rough outlines of an ideal logical notation, based on only one operation; however, he failed to produce a realization of this vision without creating a somewhat flawed, or at least insufficiently explained, system. Hence, in subsequent years, various scholars have attempted to provide their own interpretations of Wittgenstein’s N operator. The goal of such efforts is, in essence, to come up with a logical system that expresses as many propositions as possible that are formulable in standard predicate logic with identity, while also realizing as many as possible of Wittgenstein’s claims about the N operator in the Tractatus. It is our belief that such efforts by commentators have failed to account for some of Wittgenstein’s most basic, important demands regarding the N operator. Most importantly, we insist on Wittgenstein’s clearest assertion: that all propositions may be constructed by finitely many successive applications of the single operator N. We will argue, as does Peter Fogelin (see the Fogelin/Geach/Soames debate in third section), that in previous interpretations this is impossible with respect to universal quantification. We will also argue that different uses of the N operator in other interpretations have actually revealed distinct operations at work. We then argue that the use of distinct letters (x, y, etc.) for variables in quantification stands in unnecessary conflict with what we take to be Wittgenstein’s view that all well-formed formulas are sentences. Finally, many previous interpretors have unnecessarily insisted on the restriction of N to finite universes, and have also insisted on the impossibility of the N system’s being decidable. In this paper, we will present our own interpretation of the N system, and argue that our system much more accurately captures the essence of Wittgenstein’s demands than its predecessors. Our system, which we will call NameBased N-Logic (NBNL), will clearly consist of a single operator that generates all propositions after finitely many applications; we will use only proper names as terms, so that every formula we construct is in fact a sentence. NBNL will have a single syntax and semantics that apply equally effectively to finite and infinite domains. Finally, we maintain that, pace numerous interpreters, the decidability or undecidability of (our version of) the N notation is still an open question. Our goal is not to introduce a system that Wittgenstein had in mind. Instead, it is to give a syntax and semantics of a notation that is as consistent as possible with regard to pseudo-propositions that he regards as nonsensical, and that also employs a heretofore ignored means of quantification that Wittgenstein suggests in the Tractatus (5.501). In presenting our system, we feel we are able to introduce an original interpretation of the N operator, which surpasses its predecessors in matching the demands given by Wittgenstein in the Tractatus. In particular, we are able to provide a system which minimizes the amount of information we need in order to understand a logical language