Catatan Awal tentang Logika Sentensial dan implikasinya dalam diskusi Manunggaling Kawula Gusti dan Trinitas

AbstrakDalam catatan awal ini kami mengajukan argumen bahwa konsep logikasentensial memuat kemungkinan betweenness/neitherness/ bothness yang tidakdikenal dalam logika biner Aristotelian. Kami mengusulkan bahwa konsep logikasentensial akan berguna untuk menjembatani dialog antara pendukung nondualisme seperti mistisisme kaum sufi dan para pendukung mazhab dualisme.Dalam konteks ini, kekristenan menawarkan kerangka berpikir bahwa hanyaYesuslah satu-satunya Sang Manunggaling Kawula Gusti yang sejati, sementarakita sebagai manusia dapat berperan sekaligus sebagai makhluk yang berbedadengan Sang Gusti, namun pada saat yang sama, umat percaya menyatu denganTuhan, meski bukan dengan konsep manunggalnya para sufi. Artinya, logikasentensial/ proposisional memungkinkan kita memahami bahwa manusiaserempak disatukan dengan Sang Khalik, namun pada saat yang sama tetapberbeda dengan Sang Khalik. Artinya non-dualisme dan dualisme pada saat yangsama. Tentunya diperlukan kajian yang lebih mendalam mengenai topik ini, yangakan kami tuliskan dalam artikel lain kemudian. AbstractIn this introductory exploration, we argue that the concept of sentential logiccontains a possibility of betweenness / neitherness / bothness unknown toAristotelian binary logic. We propose that the concept of sentential logic will beuseful for bridging the dialogue between supporters of non-dualism such as themysticism of the Sufis and the supporters of the dualism schools. In this context,Christianity offers a framework of thinking that only Jesus is the trueManunggaling Kawula Gusti, while we humans can simultaneously act ascreatures that are different from the Gusti (God), but at the same time, believersare one with God, although not the same with the concept of divine unity of theSufis. That is, sentential / propositional logic makes it possible that humans aresimultaneously united with the Creator, but at the same time remain differentfrom the Creator. It means non-dualism and dualism at the same time. For sure, amore in-depth exploration is required, and we plan to present it in another article.

SEVENTEENTH-CENTURY SCHOLASTIC SYLLOGISTICS. BETWEEN LOGIC AND MATHEMATICS?

The seventeenth century can be viewed as an era of (closely related) innovation in the formal and natural sciences and of paradigmatic diversity in philosophy (due to the coexistence of at least the humanist, the late scholastic, and the early modern tradition). Within this environment, the present study focuses on scholastic logic and, in particular, syllogistic. In seventeenth-century scholastic logic two different approaches to logic can be identified, one represented by the Dominicans Báñez, Poinsot, and Comas del Brugar, the other represented by the Jesuits Hurtado, Arriaga, Oviedo, and Compton. These two groups of authors can be contrasted in three prominent features. First, in the role of the theory of validity, which is either a common basis for all particular theories (in this case, sentential logic and syllogistic), or a set of observations regarding a particular theory (in this case, syllogistic). Second, in the view of syllogistic, which is either an implication of a general theory of validity and a semantics of terms, or an algebra of structured objects. Third, in the role of the scholastic analysis of language in terms of suppositio, which either is a semantic underpinning of syllogistic, or it is replaced by a semantics of propositions.

Two-Sided Trees for Sentential Logic, Predicate Logic, and Sentential Modal Logic

This paper will present two contributions to teaching introductory logic. The first contribution is an alternative tree proof method that differs from the traditional one-sided tree method. The second contribution combines this tree system with an index system to produce a user-friendly tree method for sentential modal logic.

Many-valued logics, philosophical issues in

The first philosophically-motivated use of many-valued truth tables arose with Jan Łukasiewicz in the 1920s. What exercised Łukasiewicz was a worry that the principle of bivalence, ‘every statement is either true or false’, involves an undesirable commitment to fatalism. Should not statements about the future whose eventual truth or falsity depends on the actions of free agents be given some third status – ‘indeterminate’, say – as opposed to being (now) regarded as determinately true or determinately false? To implement this idea in the context of the language of sentential logic (with conjunction, disjunction, implication and negation), we need to show – if the usual style of treatment of such connectives in a bivalent setting is to be followed – how the status of a compound formula is determined by the status of its components. Łukasiewicz’s decision as to how the appropriate three-valued truth-functions should look is recorded in truth tables in which (determinate) truth and falsity are represented by ‘1’ and ‘3’ respectively, with ‘2’ for indeterminacy (see tables in the main body of the entry). Consider the formula A∨B (‘A or B’), for example, when A has the value 2 and B has the value 1. The value of A∨B is 1, reasonably enough, since if A’s eventual truth or falsity depends on how people freely act, but B is determinately true already, then A∨B is already true independently of such free action. There are no constraints as to which values may be assigned to propositional variables. The law of excluded middle is invalidated in the case of indeterminacy: if p is assigned the value 2, then p∨ ¬p also has the value 2. This reflects Łukasiewicz’s idea that such disjunctions as ‘Either I shall die in a plane crash on January 1, 2030 or I shall not die in a plane crash on January 1, 2030’ should not be counted as logical truths, on pain of incurring the fatalistic commitments already alluded to. Together with the choice of designated elements (which play the role in determining validity played by truth in the bivalent setting), Łukasiewicz’s tables constitute a (logical) matrix. An alternative three-element matrix, the 1-Kleene matrix, involves putting 2→2=2, leaving everything else unchanged. And a third such matrix, the 1,2-Kleene matrix, differs from this in taking as designated the set of values {1,2} rather than {1}. The 1-Kleene matrix has been proposed for the semantics of vagueness. In the case of a sentence applying a vague predicate, such as ‘young’, to an individual, the idea is that if the individual is a borderline case of the predicate (not definitely young, and not definitely not young, to use our example) then the value 2 is appropriate, while 1 and 3 are reserved for definite truths and falsehoods, respectively. Łukasiewicz also explored, as a technical curiosity, n-valued tables constructed on the same model, for higher values of n, as well as certain infinitely many-valued tables. Variations on this theme have included acknowledging as many values as there are real numbers, with similar applications to vagueness and approximation in mind.

Tense and temporal logic

A special kind of logic is needed to represent the valid kinds of arguments involving tensed sentences. The first significant presentation of a tense logic appeared in Prior (1957). Sentential tense logic, in its simplest form, adds to classical sentential logic two tense operators, P and F. The basic idea is to analyse past and future tenses in terms of prefixes ‘It was true that’ and ‘It will be true that’, attached to present-tensed sentences. (Present-tensed sentences do not need present tense operators, since ‘It is true that Jane is walking’ is equivalent to ‘Jane is walking’.) Translating the symbols into English is merely a preliminary to a semantics for tense logic; we may translate ‘P’ as ‘it was true that’ but we still have the question of the meaning of ‘it was true that’. There are at least two versions of the tensed theory of time – the minimalist version and the maximalist version – that can be used for the interpretation of the tense logic symbols. The minimalist version implies that there are no past or future particulars, and thus no things or events that have properties of pastness or futurity. What exists are the things, with their properties and relations, that can be mentioned in certain present-tensed sentences. If ‘Jane is walking’ is true, then there is a thing, Jane, which possesses the property of walking. ‘Socrates was discoursing’, even if true, does not contain a name that refers to a past thing, Socrates, since there are no past things. The ontological commitments of past and future tensed sentences are merely to propositions, which are sentence-like abstract objects that are the meanings or senses of sentences. ‘Socrates was discoursing’ merely commits us to the proposition expressed by the sentence ‘It was true that Socrates is discoursing’. The maximalist tensed theory of time, by contrast, implies that there are past, present and future things and events; that past items possess the property of pastness, present items possess the property of presentness, and future items possess the property of being future. ‘Socrates was discoursing’ involves a reference to a past thing, Socrates, and implies that the event of Socrates discoursing has the property of being past.

Knot and Tonk: Nasty connectives on many-valued truth-tables for classical sentential logic

Prior’s Tonk is a famously horrible connective. It is defined by its inference rules. My aim in this paper is to compare Tonk with some hitherto unnoticed nasty connectives, which are defined in semantic terms. I first use many-valued truth-tables for classical sentential logic to define a nasty connective, Knot. I then argue that we should refuse to add Knot to our language. And I show that this reverses the standard dialectic surrounding Tonk, and yields a novel solution to the problem of many-valued truth-tables for classical sentential logic. I close by outlining the technicalities surrounding nasty connectives on many-valued truth-tables.Published in Analysis.