Valentine’s Day is a romantic holiday celebrated each year on February Learn about St. Valentine, Valentine's Day quotes and the history of Valentine's Day St. Luke gave a sermon in Alushta, in Crimea, on August 12, , of which an audio recording exists. Various verses from Ephesians and excerpts of this sermon have been loosely translated and paraphrased into Greek and English, and the resulting passage is often incorrectly described as St Luke's last words in many Greek-language and English-language websites and publications [3] [4] [5] The Valentin submarine factory is a protective shelter on the Weser River at the Bremen suburb of Rekum [de; nds], built to construct German U-boats during World War blogger.com factory was under construction from to March using forced labour, but was damaged by air-raids and unfinished by the end of the war
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The term Temporal Logic has been broadly used to cover all approaches to reasoning about time and temporal information, as well as their formal representation, within a logical framework, and also more narrowly to refer specifically to the modal-logic type of approach introduced around by Arthur Prior under the name Tense Logic and subsequently developed further by many logicians and computer scientists.
Applications of Temporal Logic include its use as a formalism for clarifying philosophical issues about time, as a framework within which to define the semantics of temporal expressions in natural language, as a language for encoding temporal knowledge in artificial intelligence, and as a tool for specification, formal analysis, and verification of the executions of computer programs and systems.
Here we provide a broadly representative, yet concise and inevitably incomplete, overview of the rich variety of temporal models and logics introduced and studied over the past 50 years.
Discussions of temporality and reasoning about time go back to antiquity, and examples can be found even in the Bible Boyd Much of the early temporal discussion, however, centered around the problem of future contingentsthat is, the question whether statements about future events that are neither necessary nor impossible can have definite truth values.
The most widely known and probably most single st. valentin example is the sea-fight scenario discussed by Aristotle in On Interpretation Chapter 9. Rescher and UrquhartChapter XVII and the entry on future contingents. Philosophical discussions concerning time and the contingent future continued in the Middle Ages, where the theme was taken up by writers such as Peter Aureole, William of Ockham, and Luis Single st. valentin. Ockham, for example, embraced the idea of a true or actual future, holding that future contingent statements are either true or false even though only God knows their truth values.
According to Ockham, this is not to say, single st. valentin, however, that future contingents are necessary, meaning that there are alternative possibilities for humans to choose from. Later, several philosophers and logicians engaged in the problem of relating temporality with free will, indeterminism, and single st.
valentin open future, proposing various different solutions. Peirce objected to the idea that future contingents can have definite truth values. He advanced the view that only the present and the past are actual whereas the future is the realm of possibility and necessity.
In a similar spirit, J. Łukasiewicz devised a three-valued logic, treating the truth values of future contingent statements as undetermined. For a more recent philosophical discussion on free will, indeterminism, and the open future, see e.
Belnap et al. The modern era of formal temporal logic was initiated by the seminal work of Arthur N. Prior, single st. valentin, with important precursors such as H. Reichenbach, single st. valentin, J. Findlay, single st. valentin, J. Łukasiewicz, and J. Prior was convinced that a proper logical approach could help to clarify and solve such philosophical problems. He introduced temporal operators, studied metric tense logic, was a pioneer in hybrid temporal logic, devised two versions of branching time temporal logic, which he took to reflect the views of Ockham and Peirce, respectively, etc.
His work paved the way for the development of the vast and diverse field of temporal logic, with numerous important applications not only in philosophy, single st.
valentin, but also in computer science, artificial intelligence, and linguistics. A comprehensive overview of the history of temporal reasoning and logics is provided in Øhrstrøm and Hasle See also Øhrstrøm and Hasle and Dyke and BardonPart I.
The ontological nature and properties of time give rise to fundamental philosophical questions, which find their expression in the rich variety of formal models of time employed in temporal logics.
For example, is time instant-based or interval-based? Is it discrete, dense, single st. valentin, or continuous?
Does time have a beginning or an end? Is it linear, branching, single st. valentin, or circular? Before we turn to the formal languages of temporal logics and their semantics, we briefly introduce below the two most basic types of formal models of time together with some of their pertinent properties: instant-based and interval-based models.
In instant-based models the primitive temporal entities are points in time, single st. valentin, viz. time instantsand the basic relationship between them besides equality is temporal precedence. There are some basic properties which can naturally be imposed on instant-based flows single st. valentin time. The temporal precedence relation is usually required to be a strict partial ordering, that is, an irreflexive, transitive, and hence asymmetric relation. Sometimes, however, it is assumed to be reflexive, and then the antisymmetry condition is added.
The relevant properties are listed below. One fundamental distinction in the realm of instant-based models of time is the distinction between linear models, where the flow of time is depicted as a line, single st.
valentin, and backward-linear models, which allow a tree-like representation, supporting the view that the past is fixed and hence linear while the future may be open branching into multiple possible futures.
In either case, the temporal ordering may or may not contain minimal or maximal single st. valentin, corresponding to first or last instants in time, respectively. Another important distinction is between discrete models of time, which are prevalent in computer science, single st.
valentin, and dense or continuous ones, which are more common in natural sciences and philosophy. In forward-discrete backward-discrete models, each time instant that has a successor predecessor always has a corresponding immediate successor immediate predecessor.
In dense models, by contrast, between any two subsequent time instants, single st. valentin, there is another instant.
Key examples of properties of instant-based models of time that cannot be expressed by first-order sentences, but require a single st. valentin language with quantification over sets, are continuitywell-orderingsingle st. valentin, and the finite interval property. Continuity demands that there be no gaps in the temporal order.
Not only must the temporal order be dense, it must also be Dedekind completei. An example is the ordered set of real numbers, while a non-example is the ordering of the rational numbers: consider e, single st. valentin. the set of all rational numbers whose square is less than 2.
An instant-based model of time is well-ordered if every non-empty, linear set of time instants has a least element, and it has the finite interval property if between any two subsequent time instants there are at most finitely many instants.
The natural numbers are well-ordered and have single st. valentin finite interval property, the negative integers are not well-ordered but still have the single st. valentin interval property, single st. valentin, and the positive rationals or reals are neither well-ordered nor do they have single st.
valentin finite interval property. As we will see in Section 3. Instant-based models of time are often not suitable for reasoning about events with duration, which are better modeled if the underlying temporal ontology uses time single st.
valentini. periods rather than instants, as the primitive entities. The roots of interval-based temporal reasoning can be traced back to Zeno and Aristotle Øhrstrøm and Hasle Interval-based models usually presuppose linear time.
Still, they are ontologically richer than instant-based models, as there are many more possible relationships between time intervals than between time instants. In an influential early work on the formal study of interval-based temporal ontology and reasoning in AI, Allen considered the family of all binary relations that can arise between two intervals in a linear order, subsequently called Allen relations.
These 13 relations, displayed in Table 1are mutually exclusive and jointly exhaustive, i. Moreover, they turn out to be definable in terms of only two of them, viz. Table 1: Allen relations between time intervals and the corresponding Halpern-Shoham modal operators see Section 6.
Given an abstract structure defined by a certain set of interval relations of arbitrary arity that are required to satisfy certain properties, the question arises whether it can be represented by means of a concrete interval-based model over linear time.
Answers are provided by various representation theorems, see e. van Benthem ; Ladkin ; and Venema The choice between instants and intervals as the primary objects of temporal ontology has single st. valentin a highly debated philosophical theme since the times of Zeno and Aristotle. Technically, the two types of temporal ontologies are closely related, and they are reducible to each single st.
valentin on the one hand, time intervals can be defined by pairs of time instants beginning and end ; on the other hand, a time instant can be construed as a degenerate interval, viz. as a point-interval whose beginning and end points coincide. Still, the technical reductions do single st. valentin resolve the semantic question whether sentences are to be evaluated with respect to instants or with respect to intervals, and one may argue that both instants and intervals are needed as mutually complementary.
Two-sorted point-interval models were studied in e. Balbiani et al. minutes, hours, days, years, etc. Here we discuss both instant-based and interval-based temporal single st. valentin. For further discussion on single st. valentin ontological primacy of instants versus intervals in temporal logics, see Hamblin ; Kamp ; Humberstone ; Galton ; as well as van Benthem for a detailed comparative exploration of both approaches. A more philosophical and historical overview is provided in e.
Øhrstrøm and Hasle ; ; Dyke and Bardon ; and Meyer In this section, we discuss the language, semantics, and axiomatization of the basic tense logic TL introduced by Prior ; ;the founding father of temporal logic.
Technically, this was achieved by the introduction of temporal operators into the language, single st. valentin, which were given a modal-logic type of semantics, single st. valentin. In view of the pivotal role that tense is playing in his framework, Prior himself referred to his account as Tense Logicwhereas nowadays the more general expression Temporal Logic is prevailing. The respective past and future operators are duals of each other, i.
For example:. Hamblin and Prior showed that, in models of linear time, single st. valentin, any sequence of temporal operators reduces to a sequence of at most two operators. In total, they identified 15 different such combinations — or tensesas they call them — that can be expressed in TL over linear flows of time see PriorChapter III. Even though these combinations seem to surpass the number of verbal tenses in e. For details, see Kuhn and Portner and the entry on tense and aspect.
The standard semantics of TL is essentially a Kripke-style semantics, familiar from modal logic. In modal logic, sentences are evaluated over so-called Kripke frames consisting of a non-empty set of possible worlds and an accessibility relation between them. In temporal logic, the possible worlds are time instants, and the accessibility relation has a concrete interpretation in terms of temporal precedence.
Note that so far no conditions, single st. valentin, like transitivity, irreflexivity, etc. true at all time instants in all temporal models. As in the case of modal logic, the language and semantics of TL can be translated into classical first-order logic see e. van Benthem Not every first-order formula has a correspondent in TL, single st.
valentin. For instance, single st. valentin, the example above can be re-written as. Thus, validity of a formula of TL in a temporal model is a first-order property. Validity in a temporal frame, on the other hand, turns out to be a second-order property as it involves quantification over valuations. The standard translation of TL into first-order logic enables a systematic treatment of various aspects of temporal logic with the tools single st.
valentin techniques of classical logic see e. Blackburn et al. A non-trivial correspondence between temporal logic and first-order logic as alternative single st. valentin for describing properties of time emerges, single st. valentin. Yet, there are crucial differences between the two approaches to formalizing the logic of time.
Temporal Logic (Stanford Encyclopedia of Philosophy)
Valentine’s Day is a romantic holiday celebrated each year on February Learn about St. Valentine, Valentine's Day quotes and the history of Valentine's Day The Valentin submarine factory is a protective shelter on the Weser River at the Bremen suburb of Rekum [de; nds], built to construct German U-boats during World War blogger.com factory was under construction from to March using forced labour, but was damaged by air-raids and unfinished by the end of the war St. Luke gave a sermon in Alushta, in Crimea, on August 12, , of which an audio recording exists. Various verses from Ephesians and excerpts of this sermon have been loosely translated and paraphrased into Greek and English, and the resulting passage is often incorrectly described as St Luke's last words in many Greek-language and English-language websites and publications [3] [4] [5]
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