South American Journal of Logic Vol. 4, n. 1, pp. 29{50, 2018 ISSN: 2446-6719 Term Functor Logic Tableaux J.-Mart´ınCastro-Manzano and Paniel-O. Reyes-C´ardenas Abstract The plus-minus calculus of Term Functor Logic features a peculiar algebra that, as of today, has not been used to produce a full tableaux method: here we offer one. Keywords: Term logic, syllogistic, semantic tree. Introduction Logic is about inference and in order to study it we usually make use of first order languages: first order logic and first order logic with identity are logi- cal systems defined after first order languages. The origin of this habit has a complex and interesting history [10], but it is certainly related to the repre- sentative advantages first order languages offer when compared to traditional or Aristotelian term logic. By 1860, Augustus De Morgan had already pointed out the inability of Aristotelian term logic to deal with relations [9], but it was Russell who, around 1900, made popular the idea that the limits of the traditional logic programme, i.e. syllogistic, were due to a commitment to a ternary syntax, that is, a grammar of triads composed by a subject term and a predicate term joined by a copula [31]. Later, in 1930, Carnap would generalize this judgment to all traditional logic by claiming that its available syntax was predicative only, as in \All (some) Greeks are (not) mortal" or \Socrates is (not) mortal" [3]. It is true that the syntactical shortcomings of term logic cause big troubles when trying to represent propositions other than predicative, say relational (e.g., \Socrates and Plato are friends"), singular (e.g., \Socrates is mortal"), or compound (e.g., \If you are Socrates, you are Plato's friend"), but the major problem ternary syntax creates is term homogeneity. Geach argues: Our distinction between names and predicables enables us to clear up the confusion, going right back to Aristotle, as to whether 30 J.-M. Castro-Manzano and P.-O. Reyes-Cardenas´ there are genuine negative terms: predicables come in contradictory pairs, but names do not, and if names and predicables are both called \terms" there will be a natural hesitation over the question \Are there negative terms?" [17, p.64] Broadly speaking, the worry with term homogeneity is that it does not allow us to preserve the fundamental noun-verb distinction. This incapacity is troublesome because the roles of nouns and predicates are not interchange- able: the function of a noun is naming, whereas the function of a predicate is predicating. Thus, the interchange of subject and predicate terms is a syn- tactical matter that produces an undesired semantic effect, for only a noun can be a logical subject, but a noun cannot maintain its role as a noun if it suddenly becomes a predicate. So, this syntactical issue turns out to be a semantic impossibility: between an Aristotelian term logic and genuine logic, goes Geach [17, p.54], there can only be war! By contrast, genuine logic, namely first order logic, follows the Fregean paradigm that results from dropping terms and adopting a binary grammar of function-argument pairs. These pairs promote a syntax that includes individ- ual constants (a; b; c; : : :) or variables (x; y; z; : : :) as arguments that stand for individual objects as logical subjects, plus relations (A; B; C; : : :) as functions that stand for concepts, not objects, as logical predicates. Thus, for instance, a singular proposition like \Socrates is mortal" could not be understood as a rela- tion between a subject term and a predicate term, but as a function-argument pair were a constant, a saturated and complete element, say s, denotes an object named \Socrates" and works as an argument for the unsaturated and incomplete expression \... is mortal", say Mx, in such a way that the for- mula Ms represents the proposition \Socrates is mortal": clearly, this last representation forbids any term shifting. Moreover, given this binary syntax, propositions like \All men are mortal" and \Every circle is a figure; therefore, anyone who draws a circle draws a figure” cannot be grasped as strings of terms but as strings of quantifiers, variables, and relations, say 8x(Hx ) Mx) and 8x(Cx ) F x) ` 8x((Dx ^ 9y(Cy ^ Rxy)) ) (Dx ^ 9y(F y ^ Rxy))), respectively. Genuine logic, thus, results from discarding the ternary syntax (subject- copula-predicate) in order to favor a binary syntax (function-argument) that promotes the use of first order languages. This syntactical standard is of course familiar to us because we usually follow it when we teach, research, or apply logic: this is the received view of logic. However, it comes as no surprise that this view might very well feel familiar, but it is certainly not natural. Woods comments (emphasis is ours): It is no secret that classical logic and its mainstream variants Term Functor Logic Tableaux 31 aren't much good for human inference as it actually plays out in the conditions of real life|in life on the ground, so to speak. It isn't surprising. Human reasoning is not what the modern orthodox logics were meant for. The logics of Frege and Whitehead & Russell were purpose-built for the pacification of philosophical perturbation in the foundations of mathematics, notably but not limited to the troubles occasioned by the paradox of sets in their application to transfinite arithmetic. [40, p.404] Genuine logic (classical, according to Woods), no doubt, has been funda- mental for the study of inference, but it amazes us that, despite its original pur- pose, it is constantly used as a bona fide tool for representing natural language reasoning. Let us consider, to this effect, \Bar-Hillel's challenge" (emphasis is ours): I challenge anybody here to show me a serious piece of argu- mentation in natural languages that has been successfully evaluated as to its validity with the help of formal logic. I regard this fact as one of the greatest scandals of human existence. Why has this happened? How did it come to be that logic which, at least in the views of some people 2,300 years ago, was supposed to deal with evaluation of argumentation in natural languages, has done a lot of extremely interesting and important things, but not this? [36, p.256] Since the late 60's, Fred Sommers championed a revision of the ternary syntax under the veil of Bar-Hillel's challenge, that is, under the assumption that logic has to deal with natural language reasoning. His project unfolded into three branches: ontology, semantics, and logic (cf. [34]). These branches became, respectively, a theory of categories that uses terms as the foundational elements of language, a theory of truth that employs the properties of terms in order to revive a correspondence theory of truth, and a theory of logic known as Term Functor Logic that takes terms as basic units of predication [32, 33, 35, 11, 14, 15]. This last theory is basically a plus-minus algebra|a \logibra", as Sommers dubbed it|that uses terms rather than first order language elements such as individual variables or quantifiers. However, since the proof methods for this theory are still in the making, in this contribution we offer a tableaux method for it. This goal should be of interest because the plus-minus calculus of Term Functor Logic features a peculiar algebra that, as of today, has not been used 32 J.-M. Castro-Manzano and P.-O. Reyes-Cardenas´ to produce a full tableaux method (cf. [8, 28]).1 To reach this goal we briefly present Term Functor Logic (with special emphasis on syllogistic), then we introduce our contribution and, at the end, we discuss some of its features. 1 Preliminaries 1.1 Syllogistic Syllogistic is a term logic that has its origins in Aristotle's Prior Analytics [1] and deals with inference between categorical propositions. A categorical propo- sition is a proposition composed by two terms, a quantity, and a quality. The subject and the predicate of a proposition are called terms: the term-schema S denotes the subject term of the proposition and the term-schema P denotes the predicate. The quantity may be either universal (All) or particular (Some) and the quality may be either affirmative (is) or negative (is not). These categori- cal propositions have a type denoted by a label (either a (universal affirmative, SaP), e (universal negative, SeP), i (particular affirmative, SiP), or o (partic- ular negative, SoP)) that allows us to determine a mood, that is, a sequence of three categorical propositions ordered in such a way that two propositions are premises and the last one is a conclusion. A categorical syllogism, then, is a mood with three terms one of which appears in both premises but not in the conclusion. This particular term, usually denoted with the term-schema M, works as a link between the remaining terms and is known as the middle term. According to the position of this middle term, four figures can be set up in order to encode the valid syllogistic moods (Table 1).2 First Second Third Fourth Figure Figure Figure Figure aaa eae iai aee eae aee aii iai aii eio oao eio eio aoo eio Table 1: Valid syllogistic moods 1[35, p.183ff] have already advanced a proposal, but its scope is limited to propositional logic. 2For sake of brevity, but without loss of generality, here we omit the syllogisms that require existential import. Term Functor Logic Tableaux 33 1.2 Term Functor Logic Term Functor Logic (TFL) is a plus-minus calculus, developed by Sommers [32, 33, 35] and Englebretsen [11, 14, 15], that deals with syllogistic by using terms rather than first order language elements such as individual