Propositional Operators Research Articles

On the agent-relativity of ‘ought’

Abstract In the standard theory of deontic modals, ‘ought’ is understood as expressing a propositional operator. However, this view has been called into question by Almotahari and Rabern’s puzzle about deontic ‘ought’, according to which the ethical principle that one ought to be wronged by another person rather than wrong them is intuitively coherent but the standard theory makes it incoherent. In this paper, I take up Almotahari and Rabern’s challenge and offer a refinement of the standard theory to handle the puzzle. I propose that ‘ought’ is evaluated relative to contextual parameters (e.g. Kratzer’s conversational backgrounds, Finlay and Snedegar’s alternative sets) and those contextual parameters are sensitive to agents as well as possible worlds.

On algebraic array theories

Automatic verification of programs manipulating arrays relies on specialised decision procedures. A methodology to classify the theories handled by these procedures is introduced. It is based on decomposition theorems in the style of Feferman and Vaught. The method is applied to obtain an extension of combinatory array logic that is closed under propositional operations and Hoare triples. A classification according to expressiveness of six different fragments studied in the literature is given.

Electrophysiological connectivity of logical deduction: Early cortical MEG study

Complex human reasoning involves minimal abilities to extract conclusions implied in the available information. These abilities are considered “deductive” because they exemplify certain abstract relations among propositions or probabilities called deductive arguments. However, the electrophysiological dynamics which supports such complex cognitive processes has not been addressed yet. In this work we consider typically deductive logico-probabilistically valid inferences and aim to verify or refute their electrophysiological functional connectivity differences from invalid inferences with the same content (same relational variables, same stimuli, same relevant and salient features). We recorded the brain electrophysiological activity of 20 participants (age = 20.35 ± 3.23) by means of an MEG system during two consecutive reasoning tasks: a search task (invalid condition) without any specific deductive rules to follow, and a logically valid deductive task (valid condition) with explicit deductive rules as instructions. We calculated the functional connectivity (FC) for each condition and conducted a seed-based analysis in a set of cortical regions of interest. Finally, we used a cluster-based permutation test to compare the differences between logically valid and invalid conditions in terms of FC. As a first novel result we found higher FC for valid condition in beta band between regions of interest and left prefrontal, temporal, parietal, and cingulate structures. FC analysis allows a second novel result which is the definition of a propositional network with operculo-cingular, parietal and medial nodes, specifically including disputed medial deductive “core” areas. The experiment discloses measurable cortical processes which do not depend on content but on truth-functional propositional operators. These experimental novelties may contribute to understand the cortical bases of deductive processes.

Pre-DP only is a propositional operator at LF: a new argument from ellipsis

Based on the scope possibilities of pre-DP only relative to modals and their interaction with ellipsis, we provide a new argument (following Benbaji 2021) for a theory according to which only is always a propositional operator at LF, despite surface appearance.

Gradient-based mixed planning with symbolic and numeric action parameters

Dealing with planning problems with both logical relations and numeric changes in real-world dynamic environments is challenging. Existing numeric planning systems for the problem often discretize numeric variables or impose convex constraints on numeric variables, which harms the performance when solving problems. In this paper, we propose a novel algorithm framework to solve numeric planning problems mixed with logical relations and numeric changes based on gradient descent. We cast the numeric planning with logical relations and numeric changes as an optimization problem. Specifically, we extend syntax to allow parameters of action models to be either objects or real-valued numbers, which enhances the ability to model real-world numeric effects. Based on the extended modeling language, we propose a gradient-based framework to simultaneously optimize numeric parameters and compute appropriate actions to form candidate plans. The gradient-based framework is composed of an algorithmic heuristic module based on propositional operations to select actions and generate constraints for gradient descent, an algorithmic transition module to update states to next ones, and a loss module to compute loss. We repeatedly minimize loss by updating numeric parameters and compute candidate plans until it converges into a valid plan for the planning problem. In the empirical study, we exhibit that our algorithm framework is both effective and efficient in solving planning problems mixed with logical relations and numeric changes, especially when the problems contain obstacles and non-linear numeric effects.

Parenthesis Notation

The formal language of a logical system usually contains several types of symbols. In infix notation, two different kinds of symbols are used to construct compound formula and to indicate the order of combination. The logical constants such as Ø, Ú are used to construct compound formula, and auxiliary symbols such as ( ) are used to indicate the order of a combination. In Polish notation, there is no need for auxiliary symbols such as ( ), and only one class of symbols, N, C, K, etc., is used as a conjunction to make it function as a parenthesis. Contrary to Polish notation, a new parenthesis notation is put forward in this paper. Parenthesis notation uses only parentheses, and empowers them the function of connectives. More importantly, it is proved in this paper that we can define logical constants such as propositional connectives, quantifiers, modal operators and temporal operators in the same formula by using only parenthesis, which can greatly simplify the initial connectives needed to construct the formal system.

‘Ought’, ownership and agentive ought: Remarks on the semantic meaning of ‘indexed ought’

Bernard Williams in his essay “Ought and moral obligation” (OMO) takes a stand on the proper logical interpretation of ‘ought’ sentences. He claims that ought being central to ethical reflection, that is, ought issuing personal requirements to agents, is to be interpreted like any ordinary ‘ought’ – as a propositional operator that is not indexed to a person. The driving idea behind Williams’s logical point about ‘ought’ seems to be that logical interpretation of ‘ought’ sentences with moral content in terms of indexed ought lacks semantic significance. John Broome disagrees. In a series of his recent writings devoted to an analysis of the notion of normative ought, he defends the view opposite to the one fostered by Williams. According to Broome, indexation of ‘ought’ to agent matters for extra-logical reasons; it is a way of exhibiting that ought has its normative owner, which in turn is important for determining the holder of responsibility for the ought in question. In the paper I argue that Broome may be right, but his arguments do not show that fact. In particular, I claim that he is wrong in thinking that indexation in terms of ownership is useful in the analysis of ‘ought’ sentences with agentive content, and thus nicely applies to moral ought being a paradigmatic example of such sentences. According to my diagnosis, Broome’s positive view about the semantic and ethical significance of interpreting agentive ought as indexed ought, suffers from one central problem. It alludes to an unsuccessful substantive semantics of ‘indexed ought’ that fails to give an accurate explanation of the meaning of the ought in question. I conclude the paper by offering an alternative to Broome’s substantive semantics of ‘indexed ought’, and explain why I think that it fares better in capturing the nature of the agentive ought.

Stoic Sequent Logic and Proof Theory

This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen’s negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall’s connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.

Propositions and the Meaning of ‘Ought’

Matthew Chrisman’s new book, The Meaning of ‘Ought’: Beyond Descriptivism and Expressivism in Metaethics, presents a semantic treatment of the deontic modal operator Ought designed to address the problem of subject-sensitivity: why, for example, “I ought to dance with you” might be true, while “You ought to dance with me” is false. Such sentence-pairs challenge the view that Ought is an operator on propositions—an assumption which is common ground amongst both classical and much contemporary work. Chrisman argues that rather than propositions, the operator Ought takes as its argument a non-propositional formal object called a practition. In this review, I discuss the inspiration and formal features of this treatment. While I argue that the distinction between practitions and propositions is not adequately characterized in Chrisman’s compositional semantics, subject-sensitivity raises interesting questions about the metaethical assumptions at play in the formal semantics—including the worry that treating Ought as a propositional operator illicitly begs the question in favor of broadly consequentialist views.

LOGICS FOR PROPOSITIONAL CONTINGENTISM

AbstractRobert Stalnaker has recently advocated propositional contingentism, the claim that it is contingent what propositions there are. He has proposed a philosophical theory of contingency in what propositions there are and sketched a possible worlds model theory for it. In this paper, such models are used to interpret two propositional modal languages: one containing an existential propositional quantifier, and one containing an existential propositional operator. It is shown that the resulting logic containing an existential quantifier is not recursively axiomatizable, as it is recursively isomorphic to second-order logic, and a natural candidate axiomatization for the resulting logic containing an existential operator is shown to be incomplete.

Confusion of tongues: A theory of normative language by Stephen Finlay

Reviewed by: Confusion of tongues: A theory of normative language by Stephen Finlay James Lenman Confusion of tongues: A theory of normative language. By Stephen Finlay. Oxford: Oxford University Press, 2014. Pp. 288. ISBN 9780199347490. $69 (Hb). We begin with good. Good, Finlay suggests, is an incomplete predicate. When we say something is good we are saying something of this form. (1) It is good for e if p. Where p is some proposition (for good, he argues, is fundamentally expressive of a propositional operator), and e is some end salient in the context. Such salience is a slippery thing. Sometimes Drinking a lot of gin is good might be relative to the end of having fun. But we can still ask Drinking a lot of gin may be fun, but is it good? to raise the question of its conduciveness to some other end comparable in eligibility for salience, such as health perhaps. So we have a form of reductive naturalism that offers to defuse the open question argument. Or at least we do when we complete the reduction in the way F proposes. (2) ‘It is good for e if p’ means that p increases the probability of e. Or more precisely: (3) ‘It is good for e if p’ means that pr(e/p & b) > pr(e/~p & b). Where b specifies certain background conditions. Along the way in discussing good, F has a swipe at the Aristotelian moral functionalist. A sentence saying X is a good K, where K is some functional kind, has both a functional reading, where the end served by the state of affairs denoted by p in which X features as a K is that end eK specified by the function of K; and a nonfunctional reading, where X is something that happens to be a K featuring in some state of affairs good for some contextually salient e. F now claims we can distinguish these readings in the following way. On a nonfunctional reading, X being a good (or bad) K entails X being a K. On a functional reading, it does not. (This test is credited to Shyam Nair.) To be a good (or bad) person, he then proposes, entails being a person. Likewise, to be a good (or bad) human being entails being a good human being. Goodbye moral functionalism. It would be good to see this very quick argument slowed down a bit. As it stands, I am not quite clear what the rules would be for seeking counterexamples. An ice-axe is not a weapon, but it may be a good weapon. Only, if it is a good weapon, it surely is a weapon. Do we take weapon here to mean something made for a martial purpose or something serviceable to one? On the latter understanding, weapon surely still counts as a functional kind, but it is hard here to see how a good weapon could fail to be a weapon. Maybe things get clearer if we accentuate the negative. A postage stamp is a really bad weapon. Indeed, it is (partly for that reason) not a weapon at all. We turn next to ought. F focuses on the instrumental conditional. (4) If Max wants to evade arrest, then he ought to mingle with the crowd. F tackles this in two steps. The first offers an analysis of the more straightforward: (5) If Max is going to evade arrest, then he has to mingle with the crowd. [End Page 1000] This F takes to mean that, in all possibilities consistent with certain background conditions in which Max will evade arrest, Max will mingle with the crowd. F then offers the neat proposal that we read the original wants to as a relevance or ‘biscuit’ conditional akin to 6, which is supposedly elliptical for 7. (6) If you want biscuits, there are some on the table. (7) If you want biscuits, [then this will be relevant information:] there are some on the table. Likewise, for 8 we get the preliminary analysis proposed in 9. (8) If Max wants to evade arrest, then he ought to mingle with the crowd. (9) If Max wants to evade arrest, [then...

Negation in Exclamatives

AbstractThe goal of this article is to provide a principled account of the semantic contribution of the so‐called “expletive” negation, as it occurs in negative exclamatives. This type of negation differs from canonical negation in that it does not contribute its canonical meaning as a propositional operator, occurs higher in the syntactic structure of the clause, does not license negative polarity items, and expresses a bias towards the opposite polarity. In this article, we show that these properties are also found in negative rhetorical questions. We propose that exclamatives, as well as rhetorical questions, denote sets of propositions structured into Boolean algebras, with the negation imposing an order of informativity upon such structures.

The Calculus of Higher-Level Rules, Propositional Quantification, and the Foundational Approach to Proof-Theoretic Harmony

We present our calculus of higher-level rules, extended with propositional quantification within rules. This makes it possible to present general schemas for introduction and elimination rules for arbitrary propositional operators and to define what it means that introductions and eliminations are in harmony with each other. This definition does not presuppose any logical system, but is formulated in terms of rules themselves. We therefore speak of a foundational (rather than reductive) account of proof-theoretic harmony. With every set of introduction rules a canonical elimination rule, and with every set of elimination rules a canonical introduction rule is associated in such a way that the canonical rule is in harmony with the set of rules it is associated with. An example given by Hazen and Pelletier is used to demonstrate that there are significant connectives, which are characterized by their elimination rules, and whose introduction rule is the canonical introduction rule associated with these elimination rules. Due to the availabiliy of higher-level rules and propositional quantification, the means of expression of the framework developed are sufficient to ensure that the construction of canonical elimination or introduction rules is always possible and does not lead out of this framework.

Remarks on the Gupta-Belnap fixed-point property for k-valued clones

Here, I first prove that certain families of k-valued clones have the Gupta-Belnap fixed-point property. This essentially means that all propositional languages that are interpreted with operators belonging to those clones are such that any net of self-referential sentences in the language can be consistently evaluated. I then focus on two four-valued generalisations of the Kleene propositional operators that generalise the strong and weak Kleene operators: Belnap’s clone and Fitting’s clone, respectively. I apply the theorems from the initial part of the paper to analyse the fixed-point property of Belnap’s and Fitting’s clones when some special operators that reflect the semantics are added. The conclusion of the paper is that Fitting’s clone is better suited than Belnap’s to provide self-referential languages with highly expressive resources.

Lost in translation: Language independence in propositional logic – application to belief change

While propositional logic is widely used as a representation framework for many AI applications, the concept of language independence in the propositional setting has not received much attention so far. In this paper, we define language independence for a propositional operator as robustness w.r.t. symbol translation. We motivate the need to focus on symbol translations of restricted types, introduce and study several families of translations of interest, and provide a number of characterization results. We also identify the computational complexity of recognizing symbol translations from those families. Then we investigate the robustness of belief merging, belief revision and belief update operators w.r.t. translations of different types. It turns out that some rational merging/revision/update operators are not guaranteed to offer the most basic (yet non-trivial) form of language independence.

One Ought Too Many

[Extract] Normative branches of philosophy take special interest in claims expressed by sentences of the form ‘S ought to φ', where S is some agent and φ-ing is some action (broadly conceived). We'll call sentences satisfying this description agential. These claims raise many important philosophical issues, such as: Which ones are true? What makes them true? How can we know if they are true? What follows from them? Some philosophers have hoped that light would be cast on these issues if we could get clearer on the exact meaning of this word ‘ought’. Once we start looking to language for philosophical clues we should notice that the word ‘ought’ is also employed in different kinds of sentences. It appears in normative sentences with different grammatical forms, like ‘It ought to be that Charlie was born a girl’, and ‘There ought to be less suffering in the world’. For convenience we'll call sentences like these nonagential.1 (It also has a non-normative epistemic use meaning approximately what is probable, as in ‘The sky ought to get dark soon’.) In these other sentences, ‘ought’ appears to represent a propositional operator, O(p).

Hare and Others on the PropositionDOI: 10.5007/1808-1711.2011v15n1p51

History witnesses alternative approaches to “the proposition”. The proposition has been referred to as the object of belief, disbelief, and doubt: generally as the object of propositional attitudes, that which can be said to be believed, disbelieved, understood, etc. It has also been taken to be the object of grasping, judging, assuming, affirming, denying, and inquiring: generally as the object of propositional actions, that which can be said to be grasped, judged true or false, assumed for reasoning purposes, etc. The proposition has also been taken to be the subject of truth and falsity: generally as the subject of propositional properties, that which can be said to be true, false, tautological, informative, inconsistent, etc. It has also been taken as the subject and object of logical relations, e.g. that which can be said to imply, be implied, contradict, be contradicted, etc. Prima facie, such properties and relations are non-mental and objective. It has also been taken to be the resultants or products of propositional operations, usually mental or linguistic; e.g. judging, affirming, and denying have been held to produce propositions called judgments, affirmations, and negations, respectively. Propositions have also been taken to be certain declarative sentences. Finally, propositions have been taken to be meanings of certain declarative sentences. This essay is an informal, selective, and incomplete survey of alternative approaches to “the proposition” with special attention to the views of the late American philosopher Peter Hare (1935–2008) and of those who influenced him.

The tractability of model checking for LTL

In a seminal paper from 1985, Sistla and Clarke showed that the model-checking problem for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If in contrast, the set of propositional operators is restricted, the complexity may decrease. This article systematically studies the model-checking problem for LTL formulae over restricted sets of propositional and temporal operators. For almost all combinations of temporal and propositional operators, we determine whether the model-checking problem is tractable (in PTIME) or intractable (NP-hard). We then focus on the tractable cases, showing that they all are NL-complete or even logspace solvable. This leads to a surprising gap in complexity between tractable and intractable cases. It is worth noting that our analysis covers an infinite set of problems, since there are infinitely many sets of propositional operators.

A Completeness Proof of Kiczuk’s Logic of Physical Change

In this paper the class of minimal models C ZI for Kiczuk's system of physical change ZI is provided and soundness and completeness proofs of ZI with respect to these models are given. ZI logic consists of propositional logic von Wright's And Then and six specific axioms characterizing the meaning of unary propositional operator "Zm", read "there is a change in the fact that". ZI is intended to be a logic which provides a formal account for describing two kinds of process change: the change from one state of the process to its other state (e.g., transmitting or absorbing energy with greater or less than the usual intensity) and the perishing of the process (e.g., cessation of the energetic activity of the sun).

Peter Hare on the Proposition

History witnesses alternative approaches to proposition. The has been referred to as the object of belief, disbelief, and doubt: generally as the object of propositional attitudes . It has also been taken to be the object of grasping, judging, assuming, affirming, denying, and inquiring: generally as the object of propositional actions . The has also been taken to be the subject of truth and falsity: generally as the subject of propositional properties . It has also been taken as the subject and object of logical propositional relations , e.g. that which can be said to imply, be implied, contradict, be contradicted, etc. It has also been taken to be the resultants or products of propositional operations , usually mental or linguistic; e. g. judging, affirming, and denying have been held to produce propositions called judgments, affirmations, and negations, respectively. Propositions have also been taken to be certain declarative sentences . Finally, the has been taken to be meanings of certain declarative sentences. This essay focuses on alternative approaches to proposition considered by the late American philosopher Peter Hare (1935-2008) and of those who influenced him.