Daniel Rönnedal

Tidslogik är en gren av logiken som handlar om temporala begrepp, satser, argument och system. Inom denna gren av logiken undersöker man t.ex. uttryck såsom "Det kommer alltid vara fallet att", "Det kommer någon gång i framtiden vara fallet att", "Det har alltid varit fallet att", "Det var någon gång i det förflutna fallet att". Logiska relationer mellan satser som innehåller temporala begrepp studeras och giltigheten hos argument som består av sådana satser analyseras. Ett multimodalt språk är ett modalt språk som inkorporerar flera olika modala operatorer. En multimodal semantik är en modal semantik som innehåller flera olika s.k. tillgänglighetsrelationer som svarar mot de olika modala operatorerna. Och ett multimodalt system är ett modalt system (av teorem) som är baserat på ett multimodalt språk. Multimodal logik är en gren av modallogiken som handlar om multimodala språk, semantiska teorier och system. I den här uppsatsen visar jag hur tidslogiken kan betraktas som en del av den multimodala logiken. Jag utvecklar ett antal så kallade semantiska tablåsystem och bevisar att dessa är sunda och fullständiga i relation till deras semantik.

This paper introduces a new aporia, the aporia of omniscience. The puzzle consists of three propositions: (1) It is possible that there is someone who is necessarily omniscient and infallible, (2) It is necessary that all beliefs are historically settled, and (3) It is possible that the future is open. Every sentence in this set is intuitively reasonable and there are prima facie plausible arguments for each of them. However, the whole set {(1), (2), (3)} is inconsistent. Therefore, it seems to be that case that at least one of the propositions in this set must be false. I discuss some possible solutions to the problem and consider some arguments for and against these solutions.

The Moral Law is fulfilled iff everything that ought to be the case is the case, and The Good is realised in a possible world w at a time t iff w is deontically accessible from w at t. In this paper, I will introduce a set of temporal modal deontic systems with propositional quantifiers that can be used to prove some interesting theorems about The Moral Law and The Good. First, I will describe a set of systems without any propositional quantifiers. Then, I will show how these systems can be extended by a couple of propositional quantifiers. I will use a kind of TxW semantics to describe the systems semantically and semantic tableaux to describe them syntactically. Every system will include a constant · that stands for The Good. ‘·’ is read as ‘The Good is realised’. All systems that contain the propositional quantifiers will also include a constant '*' that stands for The Moral Law. '*' is read as ‘The Moral Law is fulfilled’. I will prove that all systems (without the propositional quantifiers) are sound and complete with respect to their semantics and that all systems (including the extended systems) are sound with respect to their semantics. It is left as an open question whether or not the extended systems are complete.

Den här uppsatsen handlar om allmänna eller generella eller universella normer och strukturen hos normativa system. Allmänna normer är normer som uttalar sig om alla entiteter eller individer eller fenomen av ett visst slag. Men vilken logisk form har de? Kan de användas för att härleda andra generella normer och normer som handlar om enskilda individer? Det tycks förekomma åtminstone två olika typer av föreskrifter av denna typ: normer där vi kvantifierar över handlingar eller beteenden och normer där vi kvantifierar över personer, människor eller levande eller medvetna varelser. I den här uppsatsen undersöker jag den logiska formen hos dessa. Jag koncentrerar mig på den senare typen och visar hur det är naturligt att använda en kvantifierad deontisk logik för att symbolisera föreskrifter av detta slag och förstå deras logiska form. Jag beskriver hur det är möjligt att använda allmänna normer för att härleda andra allmänna eller partikulära normer och hur man med hjälp av en eller flera allmänna föreskrifter kan bygga upp ett helt normativt system.

FN:s allmänna förklaring om de mänskliga rättigheterna innehåller en katalog över ett antal mänskliga fri- och rättigheter. I den här uppsatsen argumenterar jag för att det krävs en kvantifierad deontisk logik för att förstå den logiska formen hos flera av de normer som uttrycks i denna förklaring. Jag kommer att gå igenom ett antal argument som intuitivt är giltiga, men som inte kan bevisas i klassisk logik. Därefter kommer jag att visa hur dessa argument kan formaliseras och bevisas med hjälp av kvantifierad deontisk logik. Diskussionen ger stöd åt uppfattningen att vi behöver en kvantifierad deontisk logik för att analysera många typer av normativa uttryck och kastar också förhoppningsvis nytt ljus över hur vissa centrala utsagor i FN:s allmänna förklaring bör tolkas och vilken logisk form de har.

In this paper, I introduce a new aporia, the aporia of perfection. This aporia includes three claims: (1) Ought implies possibility, (2) We ought to be perfect, and (3) It is not possible that we are perfect. All these propositions appear to be plausible when considered in themselves and there are interesting arguments for them. However, together they entail a contradiction. Hence, at least one of the sentences must be false. I consider some possible solutions to the puzzle and discuss some pros and cons of these solutions. I conclude that we can avoid the contradiction that follows from (1) – (3) and still hold on to our basic intuitions, if we instead of (1) – (3) accept some slightly different propositions.

Det sägs att den gamla grekiska tänkaren Diodorus Cronus argumenterade för uppfattningen att någonting är möjligt endast om det är eller kommer att vara sant. Hans argument går under benämningen ”Mästerargumentet”. I den här uppsatsen tittar jag närmare på detta. Jag tar upp två möjliga tolkningar och går igenom några argument för utgångspunkterna. Jag visar hur det är möjligt att acceptera alla premisser i argumentet, givet att de tolkas på ett visst sätt, samtidigt som man förkastar slutsatsen. Det här innebär positiva nyheter för alla som tycker att det ligger någonting i Diodorus Cronus resonemang men samtidigt tror att det finns möjligheter som aldrig realiseras.

This paper will introduce and explore a set of quantified counterfactual temporal alethic-deontic systems, that is, systems that combine counterfactual temporal alethic-deontic logic with predicate logic. I will consider three types of systems: constant, variable and constant and variable domain systems. Every system can be combined with either necessary or contingent identity. All logics are described both semantically and proof theoretically. I use a kind of possible world semantics, inspired by the so-called T x W semantics, to characterise them semantically and semantic tableaux to characterise them proof theoretically. Our models contain several different accessibility relations and a similarity relation between possible worlds, which are used in the definitions of the truth conditions for the various operators. Soundness results are obtained for every tableau system and completeness results for a subclass of these.

In Symbolic Logic (1932), C. I. Lewis developed five modal systems S1 − S5. S4 and S5 are so-called normal modal systems. Since Lewis and Langford’s pioneering work many other systems of this kind have been investigated, among them the 32 systems that can be generated by the five axioms T, D, B, 4 and 5. Lewis also discusses how his systems can be augmented by propositional quantifiers and how these augmented logics allow us to express some interesting ideas that cannot be expressed in the corresponding quantifier-free logics. In this paper, I will develop 64 normal modal semantic tableau systems that can be extended by propositional quantifiers yielding 64 extended systems. All in all, we will investigate 128 different systems. I will show how these systems can be used to prove some interesting theorems and I will discuss Lewis’s so-called existence postulate and some of its consequences. Finally, I will prove that all normal modal systems are sound and complete and that all systems (including the extended systems) are sound with respect to their semantics. It is left as an open question whether or not the extended systems are complete.

In this paper, I will develop a set of boulesic-doxastic tableau systems and prove that they are sound and complete. Boulesic-doxastic logic consists of two main parts: a boulesic part and a doxastic part. By ‘boulesic logic’ I mean ‘the logic of the will’, and by ‘doxastic logic’ I mean ‘the logic of belief’. The first part deals with ‘boulesic’ concepts, expressions, sentences, arguments and theorems. I will concentrate on two types of boulesic expression: ‘individual x wants it to be the case that’ and ‘individual x accepts that it is the case that’. The second part deals with ‘doxastic’ concepts, expressions, sentences, arguments and theorems. I will concentrate on two types of doxastic expression: ‘individual x believes that’ and ‘it is imaginable to individual x that’. Boulesic-doxastic logic investigates how these concepts are related to each other. Boulesic logic is a new kind of logic. Doxastic logic has been around for a while, but the approach to this branch of logic in this paper is new. Each system is combined with modal logic with two kinds of modal operators for historical and absolute necessity and predicate logic with necessary identity and ‘possibilist’ quantifiers. I use a kind of possible world semantics to describe the systems semantically. I also sketch out how our basic language can be extended with propositional quantifiers. All the systems developed in this paper are new.

In this paper, I will discuss some examples of the so-called contrary-to-duty paradox, a well-known puzzle in deontic logic. A contrary-to-duty obligation is an obligation telling us what ought to be the case if something forbidden is true, for example: ‘If she is guilty, she should confess’. Contrary-to-duty obligations are important in our moral and legal thinking. Therefore, we want to be able to find an adequate symbolisation of such obligations in some logical system, a task that has turned out to be difficult. This is shown by the so-called contrary-to-duty paradox. I will investigate and evaluate one kind of solution to this problem that has been suggested in the literature, which I will call the ‘counterfactual solution’. I will use some recent systems that combine not only counterfactual logic and deontic logic, but also temporal logic, in my analysis of the paradox. I will argue that the counterfactual solution has many attractive features and that it can give a fairly satisfactory answer to some examples of the contrary-to-duty paradox, but that it nevertheless has some serious problems. The conclusion is that, notwithstanding the many attractive features of the solution, there seem to be other approaches to the paradox that are more promising.

In this paper, I develop a new set of doxastic logical systems and I show how they can be used to solve several well-known problems in doxastic logic, for example the so-called problem of logical omniscience. According to this puzzle, the notions of knowledge and belief that are used in ordinary epistemic and doxastic symbolic systems are too idealised. Hence, those systems cannot be used to model ordinary human or human-like agents' beliefs. At best, they can describe idealised individuals. The systems in this paper can be used to symbolise not only the doxastic states of perfectly rational individuals, but also the beliefs of finite humans. Proof-theoretically, I will use a tableau technique. Every system is combined with predicate logic with necessary identity and ‘possibilist’ quantifiers and modal logic with two kinds of modal operators for relative and absolute necessity. The semantics is a possible world semantics. Finally, I prove that every tableau system in the paper is sound and complete with respect to its semantics.

A contrary-to-duty obligation (sometimes called a reparational duty) is a conditional obligation where the condition is forbidden, e.g. “if you have hurt your friend, you should apologise”, “if he is guilty, he should confess”, and “if she will not keep her promise to you, she ought to call you”. It has proven very difficult to find plausible formalisations of such obligations in most deontic systems. In this paper, we will introduce and explore a set of temporal alethic dyadic deontic systems, i.e., systems that include temporal, alethic and dyadic deontic operators. We will then show how it is possible to use our formal apparatus to symbolise contrary-to-duty obligations and to solve the so-called contrary-to-duty (obligation) paradox, a problem well known in deontic logic. We will argue that this response to the puzzle has many attractive features. Semantic tableaux are used to characterise our systems proof theoretically and a kind of possible world semantics, inspired by the so-called T× W semantics, to characterise them semantically. Our models contain several different accessibility relations and a preference relation between possible worlds, which are used in the definitions of the truth conditions for the various operators. Soundness results are obtained for every tableau system and completeness results for a subclass of them.

The purpose of this paper is to describe a set of counterfactual temporal alethic-deontic systems, i.e. systems that include counterfactual, temporal, alethic and deontic operators. All systems are described both semantically and proof theoretically. We use a kind of possible world semantics, inspired by the so-called T x W semantics, to characterise our systems semantically and semantic tableaux to characterise them proof theoretically. Our models contain several different accessibility relations and a similarity relation between possible worlds, which are used in the definitions of the truth conditions for the various operators. Soundness results are obtained for every tableau system and completeness results for a subclass of them.

We often use sentences that seem conditional in nature when we reason about normative issues, e.g. ‘If you have promised to do something, you should keep your promise’ and ‘If you have done something bad, you should apologize’. We seem to think that promise-making in some sense commits us to promise-keeping and that acting bad in some sense creates an obligation to apologize. It is, however, not obvious how we should symbolize such sentences in a formal language. The purpose of this essay is to investigate some different possible formalizations of different conditional obligation sentences. I consider seven different interpretations of the concept of commitment or conditional obligation and I say something about the logical properties of these different interpretations.

The purpose of this paper is to describe a set of temporal alethic–deontic systems, i.e. systems that include temporal, alethic and deontic operators. All in all we will consider 2,147,483,648 systems. All systems are described both semantically and proof theoretically. We use a kind of possible world semantics, inspired by the so-called T x W semantics, to characterize our systems semantically and semantic tableaux to characterize them proof theoretically. We also show that all systems are sound and complete with respect to their semantics.

The purpose of this paper is to develop a class of semantic tableau systems for some dyadic deontic logics. We will consider 16 different pure dyadic deontic tableau systems and 32 different alethic dyadic deontic tableau systems. Possible world semantics is used to interpret our formal languages. Some relationships between our systems and well known dyadic deontic logics in the literature are pointed out and soundness results are obtained for every tableau system. Completeness results are obtained for all 16 pure dyadic deontic systems and for 16 alethic dyadic deontic systems.

The purpose of this paper is to investigate the relationships between the so-called golden rule and the so-called platinum rule. According to the golden rule (GR) you ought to treat others as you want to be treated by them; and according to the platinum rule (PR), you ought to treat others as they want to be treated by you. In this essay I argue for the following propositions. (1) (GR) and (PR) are logically independent. (2) (PR) can be derived from (GR) given that you want to be treated by others as they want to be treated by you. (3) (GR) is derivable from (PR) given that others want to be treated by you as you want to be treated by them. And (4) (GR) and (PR) are equivalent given that you want to be treated by others as they want to be treated by you, and others want to be treated by you as you want to be treated by them. I consider some possible interpretations of (GR) and (PR), and I also address the question of whether (GR) is more plausible than (PR) or vice versa, or whether they are complementary. I discuss one argument for the view that (PR) is better than (GR) and mention several arguments for the view that (GR) is better than (PR). The discussion suggests that (GR) and (PR) are complementary.

The purpose of this paper is to describe a set of quantified temporal alethic-deontic systems, i.e., systems that combine temporal alethicdeontic logic with predicate logic. We consider three basic kinds of systems: constant, variable and constant and variable domain systems. These systems can be augmented by either necessary or contingent identity, and every system that includes identity can be combined with descriptors. All logics are described both semantically and proof theoretically. We use a kind of possible world semantics, inspired by the so-called T × W semantics, to characterize them semantically and semantic tableaux to characterize them proof theoretically. We also show that all systems are sound and complete with respect to their semantics.

In Paradoxa Stoicorum, the Roman philosopher Cicero defends six important Stoic theses. Since these theses seem counterintuitive, and it is not likely that the average person would agree with them, they were generally called "paradoxes". According to the third paradox, (P3), (all) transgressions (wrong actions) are equal and (all) right actions are equal. According to one interpretation of this principle, which I will call (P3′), it means that if it is forbidden that A and it is forbidden that B, then not-A is as good as not-B; and if it is permitted that A and it is permitted that B, then A is as good as B. In this paper, I show how it is possible to prove this thesis in dyadic deontic logic. I also try to defend (P3′) against some philosophical counterarguments. Furthermore, I address the claim that (P3′) is not a correct interpretation of Cicero’s third paradox and the assertion that it does not matter whether (P3′) is true or not. I argue that it does matter whether (P3′) is true or not, but acknowledge that (P3′) is perhaps a slightly different principle than Cicero’s thesis. The upshot is that (P3′) seems to be a plausible principle, and that at least one part of paradox III in Cicero’s Paradoxa Stoicorum appears to be defensible.

Many interesting philosophical principles include two kinds of modalities, e.g. epistemic and doxastic, alethic and epistemic, or alethic and deontic modalities.The purpose of this essay is to describe a set of bimodal systems, i.e. systems that include two kinds of modal operators, in which it is possible to investigate some formalizations of such principles. All in all we will consider 4,194,304 logics. All logics are described semantically and proof theoretically. We use possible world semantics to characterize the logics semantically, and both axiomatic systems and semantic tableaux to characterize them proof theoretically. We show that all systems are sound and complete with respect to their semantics and we consider some relationships between the various systems.

In this paper I show that one version of motivational internalism follows from the so-called ideal observer theory. Let us call the version of the ideal observer theory used in this essay (IOT). According to (IOT), it is necessarily the case that it ought to be that A if and only if every ideal observer wants it to be the case that A. We shall call the version of motivational internalism that follows from (IOT) (moral) conditional belief motivational internalism (CBMI). According to (CBMI), it is necessarily the case that, for every x: if x were an ideal observer, it would be the case that x believes that it ought to be that A only if x wants it to be the case that A. Given that it is necessarily true that no ideal observer has any false beliefs, (IOT) entails (CBMI). Or, so I shall argue.