Tero Tulenheimo

I discuss the views on logic held by three early Nordic neo-Aristotelians — the Swedes Johannes Canuti Lenaeus (1573–1669) and Johannes Rudbeckius (1581–1646), and the Dane Caspar Bartholin (1585–1629). They all studied in Wittenberg (enrolled respectively in 1597, 1601, and 1604) and were exponents of protestant (Lutheran) scholasticism. The works I utilize are Janitores logici bini (1607) and Enchiridion logicum (1608) by Bartholin; Logica (1625) and Controversiae logices (1629) by Rudbeckius; and Logica peripatetica (1633) by Lenaeus. Rudbeckius’s and Lenaeus’s books were published much later than they were prepared. Rudbeckius wrote the first versions of his books in 1606, and the material for Lenaeus’s book had been prepared by 1607. Bartholin calls the treatment of the nature of logic the “first doorkeeper of logic”. To compare the views of the three neo-Aristotelians on this topic, I systematically investigate what they have to say about second notions, the subject of logic, the internal and external goal of logic, and the definition of logic. I also compare their approaches with those of Jacob Martini (teacher of Rudbeckius and Bartholin) and Iacopo Zabarella (an intellectual predecessor of all three).

In this article, I present a formal semantic framework that renders explicit how to reconcile the condition that a proposition about a contingent future event is true at a moment t0 with the idea that at t0, this proposition is ‘truth-maker indeterminate’: a state of affairs making it true will obtain later on, though no such state of affairs obtains at t0. The semantics I formulate employs ‘open temporal models’. They represent the passage of time by a specific component termed time-resource, which acts on durations construed as model-external inputs. A model does not by itself specify which course of events gets actualized in a given duration depending on the latest moment that has already got actualized. A time-resource merely represents schematically the dependence between a moment t and a course of events that gets actualized in a time-span of a given length counted from t; until that much time has indeed passed, it is not fixed which course of events actually extends t. Further, I introduce evaluations as a fine-grained tool for studying truth-conditions of tensed formulas, and I use this tool to define the notion of truth-maker. I define what it means that a truth-maker will obtain but does not, and what it means for a truth-maker to be determinate. It is proven that my semantic analysis retains the desirable link between determinacy and historical necessity—namely, a truth-maker of a proposition being determinate entails that the proposition is historically necessary.

I argue that the semantics of sentences expressing future contingent propositions is best viewed as being based on a clear distinction between a time at which a proposition is true and a time at which a state of affairs that makes it true gets actualized. That a prediction is true here and now means that its truth-maker gets actualized later. This is not to say that if a contingent proposition p concerning the future is true at t, it acquires the truth-value true at t only retrospectively, at a later moment. Nor must this be seen as suggesting that it is a settled, unpreventable fact at t that p is true at t. It just means that the reason for its present truth is something that happens later on: the future happens to evolve in such a way as to make a truth-maker of p obtain. In this case, then, it can be said that at t, p is truth-maker indeterminate, or that it has an indeterminate truth-maker. I develop a formal semantics based on this analysis in the follow-up article ‘A Formal Framework for Future Contingents’. Here, I lay down the conceptual framework and indicate Boethius and Abelard as precursors of the view I wish to defend.

This book develops a novel generalization of possible world semantics, called ‘world line semantics’, which recognizes worlds and links between world-bound objects (world lines) as mutually independent aspects of modal semantics. Addressing a wide range of questions vital for contemporary debates in logic and philosophy of language and offering new tools for theoretical linguistics and knowledge representation, the book proposes a radically new paradigm in modal semantics. This framework is motivated philosophically, viewing a structure of world lines as a precondition of modal talk. The author provides a uniform analysis of quantification over individuals (physical objects) and objects of thought (intentional objects). The semantic account of what it means to speak of intentional objects throws new light on accounts of intentionality and singular thought in the philosophy of mind and offers novel insights into the semantics of intensional transitive verbs.

Johannes Rudbeckius, one of Sweden’s most influential figures in theology, pedagogy, and church-state relations in the 17th century, published two books on logic: Logica ex optimis et præstantissimis autoribus collecta & conscripta and Controversiæ logices vel potius earum epitome. In this paper, I present and critically discuss Rudbeckius’s view on the nature of logic as it can be reconstructed from these two works. This requires, in particular, identifying the larger intellectual framework within which Rudbeckius operated. The single most important philosopher having affected Rudbeckius’s ideas on logic is Jacopo Zabarella, which leads me to consider at some length the views on logic that Zabarella puts forward in his De Natura Logicæ.

Propono hic conversionem in linguam Anglicam conclusionis Collegii Logici, anno 1608–1609 semestri studiorum spatio hiberno a Johanne Rudbeckio Wittenbergæ habiti. Hic commentarius prooemium in conversionem est. Rudbeckius primus Suecus erat, qui librum didacticum de logica publicavit. Maiorem partem libri iam anno 1606 scripserat, cum Mathesis Professor Upsaliensis esset, sed Logica ex optimis et præstantissimis autoribus collecta & conscripta non ante annum 1625 edita sit. Cum Johanne Canuti Lenæo, collega suo, Rudbeckius primas partes agebat in inducenda in Sueciam scholastica Lutherana, cuius rei scopus defensio erat fidei Lutheranæ per logicam et metaphysicam Aristotelicam. In primo capite condiciones conclusionis Rudbeckii commentor. Secundum caput brevem Rudbeckii vitæ descriptionem offert. Tertium quartumque capita de scripto converso et de ipsa conversione observationes quasdam continent.This note is an introduction to the English translation of the concluding speech of the Collegium Logicum that Johannes Rudbeckius taught in Wittenberg during the winter semester 1608–1609. Rudbeckius was the first Swede to publish a textbook on logic; his Logica ex optimis et præstantissimis autoribus collecta & conscripta appeared in 1625. The first version of the textbook was completed already in 1606 when Rudbeckius was professor of mathematics at Uppsala University in Sweden. Together with his colleague Johannes Canuti Lenæus, Rudbeckius played a key role in the introduction of Lutheran scholasticism in Sweden; this was a movement within the Lutheran Church whose aim was to defend the Lutheran faith by making use of Aristotelian logic and metaphysics. In Section 1, I comment on the context of Rudbeckius’s discourse. Section 2 offers a short biography of Rudbeckius. Sections 3 and 4 contain, respectively, some remarks about the text translated and about the translation itself.

The present paper provides novel results on the model theory of Independence friendly modal logic. We concentrate on its particularly well-behaved fragment that was introduced in Tulenheimo and Sevenster (Advances in Modal Logic, 2006). Here we refer to this fragment as ‘Simple IF modal logic’ (IFML s ). A model-theoretic criterion is presented which serves to tell when a formula of IFML s is not equivalent to any formula of basic modal logic (ML). We generalize the notion of bisimulation familiar from ML; the resulting asymmetric simulation concept is used to prove that IFML s is not closed under complementation. In fact we obtain a much stronger result: the only IFML s formulas admitting their classical negation to be expressed in IFML s itself are those whose truth-condition is in fact expressible in ML

Typical applications of Hintikka’s game-theoretical semantics give rise to semantic attributes—truth, falsity—expressible in the $\Sigma^{1}_{1}$-fragment of second-order logic. Actually a much more general notion of semantic attribute is motivated by strategic considerations. When identifying such a generalization, the notion of classical negation plays a crucial role. We study two languages, $L_{1}$ and $L_{2}$, in both of which two negation signs are available: $\rightharpoondown $ and $\sim$. The latter is the usual GTS negation which transposes the players’ roles, while the former will be interpreted via the notion of mode. Logic $L_{1}$ extends independence-friendly logic; $\rightharpoondown $ behaves as classical negation in $L_{1}$. Logic $L_{2}$ extends $L_{1}$, and it is shown to capture the $\Sigma^{2}_{1}$-fragment of third-order logic. Consequently the classical negation remains inexpressible in $L_{2}$

Motivated by constraint satisfaction problems, Feder and Vardi (SIAM Journal of Computing, 28, 57–104, 1998) set out to search for fragments of satisfying the dichotomy property: every problem definable in is either in P or else NP-complete. Feder and Vardi considered in this connection two logics, strict NP (or SNP) and monadic, monotone, strict NP without inequalities (or MMSNP). The former consists of formulas of the form , where is a quantifier-free formula in a relational vocabulary; and the latter is the fragment of SNP whose formulas involve only negative occurrences of relation symbols, only monadic second-order quantifiers, and no occurrences of the equality symbol. It remains an open problem whether MMSNP enjoys the dichotomy property. In the present paper, SNP and MMSNP are characterized in terms of partially ordered connectives. More specifically, SNP is characterized using the logic D of partially ordered connectives introduced in Blass and Gurevich (Annals of Pure and Applied Logic, 32, 1–16, 1986), Sandu and Väänänen (Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 38, 361–372 1992), and MMSNP employing a generalization C of D introduced in the present paper.

G. H. von Wright proposed that a temporal interval exemplifies a real contradiction if at least one part of any division of this interval involves the presence of contradictorily related (though non-simultaneous) states. In connection with intervals, two negations must be discerned: 'does not hold at an interval' and 'fails throughout an interval'. Von Wright did not distinguish the two. As a consequence, he made a mistake in indicating how to use his logical symbolism to express the notion of real contradiction. The present paper aims to reconstruct and philosophically motivate von Wright's argument for the possibility of real contradictions.

Hintikka's second generation epistemic logic introduces a syntactic device allowing to express independence relations between certain logical constants. De re knowledge attributions can be reformulated in terms of quantifier independence, but the reformulation does not extend to non-factive attitudes like belief. There, formulae with independent quantifiers serve to express a new type of attitude, intermediate between de dicto and de re, called ‘de objecto’: in each possible world compatible with the agent's belief, there is an individual with the specified property – in each world the same individual, which or who need not exist actually. The philosophical benefits of our analysis of propositional attitudes include a refined account of the behaviour of proper names as well as of indefinite and definite descriptions in attitude reports. Some remarks about perception and about the hallucination argument are also presented

When discussing knowledge, two relations are of interest: justified doxastic accessibility \ , she is in \ ) and justification equivalence \ exactly the same justified beliefs that she has in \ ). Speaking of compatibility with the agent’s justified beliefs is potentially ambiguous: either of the two relations \ or \ can be meant. I discuss the possibility of identifying the relation of epistemic accessibility \ , she is in \ ) with the union of \ and \ . Neither Gettier’s examples nor the ‘fake barn’ cases contradict this identification. However, the proposal leads to justification equivalent scenarios being symmetric with respect to knowledge: we cannot know a true proposition in a scenario if it is false in a justification equivalent scenario. This analysis may appear to render non-trivial knowledge impossible. This conclusion follows if the extra premise is granted that for all relevant true propositions there is a justification equivalent scenario in which the proposition is false. I provide a meaning-theoretic argument against this premise. I conclude by pointing out problems that would ensue from giving up the proposed connection between \ , \ and \ and allowing asymmetry of justification equivalent scenarios relative to knowledge

The hybrid logic and the independence friendly modal logic IFML are compared for their expressive powers. We introduce a logic IFML c having a non-standard syntax and a compositional semantics; in terms of this logic a syntactic fragment of IFML is singled out, denoted IFML c . (In the Appendix it is shown that the game-theoretic semantics of IFML c coincides with the compositional semantics of IFML c .) The hybrid logic is proven to be strictly more expressive than IFML c . By contrast, and the full IFML are shown to be incomparable for their expressive powers. Building on earlier research (Tulenheimo and Sevenster 2006), a PSPACE -decidable fragment of the undecidable logic is disclosed. This fragment is not translatable into the hybrid logic and has not been studied previously in connection with hybrid logics. In the Appendix IFML c is shown to lack the property of ‘quasi-positionality’ but proven to enjoy the weaker property of ‘ bounded quasi-positionality’. The latter fact provides from the IFML internal perspective an account of what makes the compositional semantics of IFML c possible.

In Prior’s tense-logical analysis, we can avoid mentioning instants in our language by construing them as propositions of a special kind. Instead of qualifying instants by predicates, we may qualify propositions by modalities. Prior shows that by changing the informal interpretation of our modal-like language, we can similarly attempt to avoid ontological commitments to worlds and even to selves and other bona fide individuals. As he notes, the paraphrasing strategy works too generally to be of direct metaphysical use. I wish to speak of several types of entities within one and the same language. This leads to the issue of cross-context identity: how to make sense of speaking of the same entity of one type relative to distinct entities of another type. My analysis employs the notion of world line. I discern a structure of mutually interrelated categories with three informal interpretations: tense, modal and egocentric. They correspond to different metaphysical views on the interrelations of time, logical alternatives and individuals. Prior’s dilemma regarding the possibility of modalizing, not only our talk concerning times and worlds, but even our discourse pertaining to the ‘real world of individuals’ does not vanish but reappears also in my setting.