The Buridanian Account of Inferential Relations between Doubly Quantified Propositions: a Proof of Soundness

On the basis of passages from John Buridan's Summula Suppositionibus and Sophismata, E. Karger has reconstructed what could be called the ‘Buridanian theory of inferential relations between doubly quantified propositions’, presented in her 1993 article ‘A theory of immediate inference contained in Buridan's logic’. In the reconstruction, she focused on the syntactical elements of Buridan's theory of modes of personal supposition to extract patterns of formally valid inferences between members of a certain class of basic categorical propositions. The present study aims at offering semantic corroboration—a proof of soundness—to the inferential relations syntactically identified by E. Karger, by means of the analysis of Buridan's semantic definitions of the modes of personal supposition. The semantic analysis is done with the help of some modern logical concepts, in particular that of the model. In effect, the relations of inference syntactically established are shown to hold also from a semantic point of view, which means thus that this fragment of Buridan's logic can be said to be sound.     

Propositions,Properties and Relations: Wittgenstein's “Notes on Logic” and the Tractatus

Frege famously argued that truth is not a property or relation. In the “Notes on Logic” Wittgenstein emphasised the bi‐polarity of propositions which he called their sense. He argued that “propositions by virtue of sense cannot have predicates or relations.” This led to his fundamental thought that the logical constants do not represent predicates or relations. The idea, however, has wider ramifications than that. It is not just that propositions cannot have relations to other propositions but also that they cannot have relations to anything at all. The paper explores the consequences of this insight for the way in which we should read the Tractatus. In the “Notes on Logic” the insight led to Wittgenstein's emphasis on “facts” in any attempt to understand the nature of symbolism. This emphasis is continued in the Tractatus. It is central to his view that propositions are facts which picture facts which prevent us from construing such picturing as a relation between what pictures and what is pictured. It illuminates the importance of context principle with regard to the distinction between showing and saying to which Wittgenstein attached so much importance and it underlies the non‐relational view of psychological propositions which he advocates. Finally, if propositions by virtue of sense cannot have predicates or relations the paradox at the end of a work which consist largely of propositions about propositions becomes intelligible.     

Medieval Disputationes de obligationibus as Formal Dialogue Systems

Formal dialogue systems model rule-based interaction between agents and as such have multiple applications in multi-agent systems and AI more generally. Their conceptual roots are in formal theories of natural argumentation, of which Hamblin’s formal systems of argumentation in Hamblin (Fallacies. Methuen, London, 1970, Theoria 37:130–135, 1971) are some of the earliest examples. Hamblin cites the medieval theory of obligationes as inspiration for his development of formal argumentation. In an obligatio, two agents, the Opponent and the Respondent, engage in an alternating-move dialogue, where the Respondent’s actions are governed by certain rules, and the goal of the dialogue is establishing the consistency of a proposition. We implement obligationes in the formal dialogue system framework of Prakken (Knowl Eng Rev 21(2):163–188, 2006) using Dynamic Epistemic Logic (van Ditmarsch et al. in Dynamic epistemic logic, Synthese Library Series. Springer, Berlin, 2007). The result is a new type of inter-agent dialogue, for consistency-checking, and analyzing obligationes in this way also sheds light on interpretational and historical questions concerning their use and purpose in medieval academia.     

Against Probabilistic Measures of Coherence

It is shown that the probabilistic theories of coherence proposed up to now produce a number of counter-intuitive results. The last section provides some reasons for believing that no probabilistic measure will ever be able to adequately capture coherence. First, there can be no function whose arguments are nothing but tuples of probabilities, and which assigns different values to pairs of propositions {A, B} and {A, C} if A implies both B and C, or their negations, and if P(B)=P(C). But such sets may indeed differ in their degree of coherence. Second, coherence is sensitive to explanatory relations between the propositions in question. Explanation, however, can hardly be captured solely in terms of probability.     

From Wittgenstein’s N-operator to a New Notation for Some Decidable Modal Logics

Wittgenstein’s N-operator is a ‘primitive sign’ which shows every complex proposition is the result of the truth-functional combination of a finite number of component propositions, and thus provides a mechanical method to determine logical truth. The N-operator can be interpreted as a generalized Sheffer stroke. In this paper, I introduce a new ‘primitive sign’ that is a hybrid of generalized Sheffer stroke and modality, and give a uniform expression for modal formulas. The general form of modal formula in the new notation is [A₀···A_n?1; B₀···B_m?1], which is semantically equivalent to ¬A₀∨···∨¬ A_n?1∨? (¬B₀∨···∨¬B_m?1). Based on this new notation, I propose several analytic axiomatic systems for some decidable modal logics. Every axiom of these analytic systems is an ‘Atomic-Sheffer’, which is the result of the combination of a finite number of component propositions. The inferential rules are analytic in that the set of elementary propositions that are combined in the premiss overlaps the set of elementary propositions combined in the conclusion, in virtue of which every complex proposition can be reduced to an ‘Atomic-Sheffer’ at the ultimate level. The analytic modal systems have the same classical inferential rules. Different modal systems can be built by adding special modal inferential rules. In an analytic system for modal logic L, valid formulas on L-models can be proved by a purely mechanical method.     

Knowing linguistic conventions

A linguistic convention is a principle or norm that has been adopted by a person or linguistic community about how to use, and therefore what the meaning is of, a specific term. Examples of such norms or principles are those expressed by propositions that express the laws of logic or those that express implicit definitions. Arguments about the epistemic status of linguistic conventions, very broadly, fall into two camps: the one holds that the basis of linguistic conventions is objective and that, therefore, such conventions are genuinely truth-bearing. Knowledge of such conventions is therefore knowledge of the objective truth of such conventions. This camp might be labelled ‘realist’. The other holds that there is no known basis to linguistic conventions and that conventions are, to the best of our knowledge, not objective, but invented. Their truth is stipulated by the relevant speaker or linguistic community. Such truth is, consequently, trivial and knowledge of such conventions simply knowledge of the stipulation. This camp might be labelled ‘anti-realist’. These are three standard accounts of the epistemic status of linguistic conventions, which all play into the first camp: (1) knowledge by intuition, (2) inferential a priori knowledge and (3) a posteriori knowledge. I give reasons why these accounts should be rejected. I then argue that linguistic conventions, if conceived of as trivial truths, are knowable non-inferentially a priori. Such an epistemic account provides support for the second camp. In this regard, I marshal support from some recent work by Wright and Hale.     

Modality and its Conversational Backgrounds in the Reconstruction of Argumentation

The paper considers the role of modality in the rational reconstruction of standpoints and arguments. The paper examines in what conditions modal markers can act as argumentative indicators and what kind of cues they provide for the reconstruction of argument. The paper critically re-examines Toulmin’s hypothesis that the meaning of the modals can be analyzed in terms of a field-invariant argumentative force and field-dependent criteria in the light of the Theory of Relative Modality developed within linguistic semantics, showing how this theory can provide a more adequate model for exploiting the modals as indicators. The resulting picture confirms Toulmin’s intuition only in part: on the one hand the modals are always relational in nature and dependent on a contextual conversational background of propositions; on the other hand only epistemic-doxastic modals directly express a speech-act level inferential relation between a set of premises and a standpoint. Other modalities express relations (e.g. causal or final relations) better seen as part of the content of the argument whose argumentative relevance depends on the argumentation scheme employed. Thus non-epistemic modals function as argumentative indicators only indirectly.

Against Propositionalism

‘Propositionalism’ is the widely held view that all intentional mental relations—all intentional attitudes—are relations to propositions or something proposition‐like. Paradigmatically, to think about the mountain is ipso facto to think that it is F, for some predicate ‘F’. It seems, however, many intentional attitudes are not relations to propositions at all: Mary contemplates Jonah, adores New York, misses Athens, mourns her brother. I argue, following Brentano, Husserl, Church and Montague among others, that the way things seem is the way they are, and that propositionalism must be abandoned.     

The necessary a posteriori: A response to Tichý

Conclusion Some of Tichý's conclusions rest on an assumption about substitutivity which Kripke would not accept. If we grant the assumption, then Tichý successfully shows that we can discover true identity statements involving names a priori, but not that we can discover a priori what properties things have essentially. Many of Tichý's arguments require an implausible rejection of the possibility of indirect belief as described in Section III. 25Are there necessary a posteriori propositions? I have argued that we certainly can discover necessary propositions a posteriori, but have left it an open question whether there are necessary propositions which we can only discover a posteriori.What effect do the considerations here presented have on the positivist doctrine that the a priori and the necessary coincide? My explanation of how we discover necessary propositions a posteriori involves our believing them indirectly, in virtue of believing contingent propositions. I would argue that Kripke's examples of the contingent a priori involve, similarly, our believing the contingent propositions in directly, in virtue of believing necessary propositions.This suggests that a reformulation of the positivist thesis along something like the following lines may well be correct. Let us say that someone directly believes a proposition just in case he could not fail to believe it without being in a different cognitive state. Then perhaps one can directly believe a proposition on the basis of a priori evidence only if it is necessary, and can directly believe a proposition on the basis of a posteriori evidence only if it is contingent.     

A Note on Saying Nothing and Saying More in the Tractatus

On the basis of an analysis of the relevant parts of Tractatus logico-philosophicus, a definition of the property (of propositions) of saying something, and of the obviously correlated property of saying nothing, is given. By applying that definition, both tautologies and contradictions are sanctioned as saying nothing, as lacking sense, in full agreement with Wittgenstein's explicit statements. On the other hand, a recent systematic attempt by A. Negro to extract from the Tractatus a criterion for sense containment and a criterion for saying more, that is, for comparing the amount of sense of propositions, leads to the conclusion that the sense of any proposition is contained in the sense of a contradiction and that contradictions say more than any other propositions. The problem faced in this paper is that of restoring consistency between Negro's criteria, which on the whole give a correct interpretation of the text of the Tractatus, and Wittgenstein's thesis that, not only tautologies, but also contradictions say nothing.     

Creative reasoning in formal discussion

Systems of formal dialectics articulate methods of conflict resolution. To this end they provide norms to regulate verbal exchanges between the Proponent of a thesis and an Opponent. These regulated exchanges constitute what are known as formal discussions.One may ask what moves, if any, in formal discusions correspond to arguing for or against the thesis. It is claimed that certain moves of the Proponent's are properly designated as arguing for the thesis, and that certain moves of the Opponent purport to criticize the tenability or the relevance of the reasons advanced. Thus the usefulness of formal dialectic systems as models for reasonable argument is vindicated.It is then proposed to make these systems more realistic by incorporating in them a norm of Creative Reasoning that removes the severe restrictions to which the Proponent's arguing was hitherto subject. As a consequence, a certain type of irrelevant reason is no longer automatically excluded. Therefore, it is proposed to extend the Opponent's rights to exert relevance criticism. The new dialectic systems are shown to be strategically equivalent to the original ones. Finally, it is stressed that the Opponent's criticism should not be designated as arguing against the thesis. The Opponent criticizes, but does not argue.     

On Stalnaker’s Simple Theory of Propositions

Robert Stalnaker recently proposed a simple theory of propositions using the notion of a set of propositions being consistent, and conjectured that this theory is equivalent to the claim that propositions form a complete atomic Boolean algebra. This paper clarifies and confirms this conjecture. Stalnaker also noted that some of the principles of his theory may be given up, depending on the intended notion of proposition. This paper therefore also investigates weakened constraints on consistency and the corresponding classes of Boolean algebras.

     

‘My Future Son is Possibly Alive’. Existential Presupposition and Empty Terms in Abelard's Modal Logic

The aim of this paper is to investigate the problem of existential import in Abelard's modal logic, and to ask whether the system of logical relationships that he proposes for modal propositions maintains its validity when some of the terms included in these propositions are empty. In the following, I first argue that, just as in the case of non-modal propositions, Abelard interprets modal propositions as having existential import, so that it is a necessary condition for the truth of propositions like ‘It is possible for my son to be alive’ or ‘it is necessary that all men are animals’ that their subjects’ referents exist. Then, I present the schemata of inferences that Abelard proposes to describe the logical behaviour of de rebus modal propositions. I argue that these systems of relations are valid only as long as all the terms contained in the formulas have an existing referent. I also claim that Abelard was aware of this difficulty (at least in the Logica Ingredientibus), and, accordingly, he explicitly decided to restrict the validity of his modal system to propositions that do not contain empty terms.     

Belief merging and the discursive dilemma: an argument-based account to paradoxes of judgment aggregation

The aggregation of individual judgments on logically interconnected propositions into a collective decision on the same propositions is called judgment aggregation. Literature in social choice and political theory has claimed that judgment aggregation raises serious concerns. For example, consider a set of premises and a conclusion where the latter is logically equivalent to the former. When majority voting is applied to some propositions (the premises) it may give a different outcome than majority voting applied to another set of propositions (the conclusion). This problem is known as the discursive dilemma (or paradox). The discursive dilemma is a serious problem since it is not clear whether a collective outcome exists in these cases, and if it does, what it is like. Moreover, the two suggested escape-routes from the paradox—the so-called premise-based procedure and the conclusion-based procedure—are not, as I will show, satisfactory methods for group decision-making. In this paper I introduce a new aggregation procedure inspired by an operator defined in artificial intelligence in order to merge belief bases. The result is that we do not need to worry about paradoxical outcomes, since these arise only when inconsistent collective judgments are not ruled out from the set of possible solutions.     

No Justificatory Closure without Truth

It is well-known that versions of the lottery paradox and of the preface paradox show that the following three principles are jointly inconsistent: (Sufficiency) very probable propositions are justifiably believable; (Conjunction Closure) justified believability is closed under conjunction introduction; (No Contradictions) propositions known to be contradictory are not justifiably believable. This paper shows that there is a hybrid of the lottery and preface paradoxes that does not require Sufficiency to arise, but only Conjunction Closure and No Contradictions; and it argues that, given any plausible solution to this paradox, if one is not ready to deny Conjunction Closure (and analogous consistency principles), then one must endorse the thesis that justified believability is factive.     

Setting the Facts Straight

Substantial facts (or states of affairs) are not well-understood entities. Many philosophers object to their existence on this basis. Yet facts, if they can be understood, promise to do a lot of philosophical work: they can be used to construct theories of property possession and truthmaking, for example. Here, I give a formal theory of facts, including negative and logically complex facts. I provide a theory of reduction similar to that of the typed λ-calculus and use it to provide identity conditions for facts. This theory validates truthmaker maximalism: it provides truthmakers for all truths. I then show how the usual truth-in-a-model relation can be replaced by two relations: one between models and facts, saying that a given fact obtains relative to the model, and the other between facts and propositions: the truthmaking relation.     

Proper Names,Contingency A Priori and Necessity A Posteriori

After a brief review of the notions of necessity and a priority, this paper scrutinizes Kripke's arguments for supposedly contingent a priori propositions and necessary a posteriori propositions involving proper names, and reaches a negative conclusion, i.e. there are no such propositions, or at least the propositions Kripke gives as examples are not such propositions. All of us, including Kripke himself, still have to face the old question raised by Hume, i.e. how can we justify the necessity and universality of general statements on the basis of sensory or empirical evidence?     

‘We Can't Whistle It Either’: Legend and Reality

Abstract: There is a famous quip of F.P. Ramsey's, which is my second epigraph. According to a widespread legend, the quip is a criticism of Wittgenstein's treatment in the Tractatus of what cannot be said. The remark is indeed Ramsey's, but he didn't mean what he is taken to mean in the legend. His quip, looked at in context, means something quite different. The legend is sometimes taken to provide support for a reading of the Tractatus according to which the nonsensical propositions of the book were intended to convey what cannot be said. But, since the legend has no basis in reality, it provides no evidence in favor of any such reading of the Tractatus. The quip has great interest if it is read in the context of Ramsey's discussion of generality; it is closely related to issues of importance in the development of Wittgenstein's thought.     

On the Propositional Treatment of Anatmavāda in Early Buddhism and Ātmavāda in Hinduism

As propositions, Anatmavāda and ātmavāda are simply negations of one another. Thus whatever serves as a criterion for truth of the one must serve as a criterion for the other. When we treat them both as a priori propositions, I claim that we are unable to determine their truth value. But if we treat them both as a posteriori propositions, I argue, we are only able to determine their truth value if we attain unqualified omniscience. Because the Hindu account of knowing is far more conducive to the idea of unqualified omniscience, we might be tempted to assert that the empirical verification of these doctrines taken as propositions is far more likely in the Hindu tradition than the early Buddhist one. However, 'empirical omniscience' carries us very far from received views, thus I conclude that it makes no sense to treat these doctrines as truth-valued propositions.     

Inferential Intensionality

The paper is a study of properties of quasi-consequence operation which is a key notion of the so-called inferential approach in the theory of sentential calculi established in [5]. The principal motivation behind the quasi-consequence, q-consequence for short, stems from the mathematical practice which treats some auxiliary assumptions as mere hypotheses rather than axioms and their further occurrence in place of conclusions may be justified or not. The main semantic feature of the q-consequence reflecting the idea is that its rules lead from the non-rejected assumptions to the accepted conclusions.First, we focus on the syntactic features of the framework and present the q-consequence as related to the notion of proof. Such a presentation uncovers the reasons for which the adjective inferential is used to characterize the approach and, possibly, the term inference operation replaces q-consequence. It also shows that the inferential approach is a generalisation of the Tarski setting and, therefore, it may potentially absorb several concepts from the theory of sentential calculi, cf. [10]. However, as some concrete applications show, see e.g.[4], the new approach opens perspectives for further exploration.The main part of the paper is devoted to some notions absent, in Tarski approach. We show that for a given q-consequence operation W instead of one W-equivalence established by the properties of W we may consider two congruence relations. For one of them the current name is kept preserved and for the other the term W-equality is adopted. While the two relations coincide for any W which is a consequence operation, for an arbitrary W the inferential equality and the inferential equivalence may differ. Further to this we introduce the concepts of inferential extensionality and intensionality for q-consequence operations and connectives. Some general results obtained in Section 2 sufficiently confirm the importance of these notions. To complete a view, in Section 4 we apply the new intensionality-extensionality distinction to inferential extensions of a version of the ukasiewicz four valued modal logic.