{"history":[],"article":{"id":"35176cc0-c564-42e7-9f1a-49bc3f1f2b41","title":"Temporal logic","slug":"temporal-logic","content":":::summary\n:::\n\n# \nIn [logic](https://bliptext.com/articles/logic), **temporal [logic](https://bliptext.com/articles/logic)** is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of [time](https://bliptext.com/articles/time) (for example, \"I am always hungry\", \"I will eventually be hungry\", or \"I will be hungry until I eat something\").\nIt is sometimes also used to refer to **tense logic**, a [modal logic](https://bliptext.com/articles/modal-logic)-based system of temporal logic introduced by [Arthur Prior](https://bliptext.com/articles/arthur-prior) in the late 1950s, with important contributions by [Hans Kamp](https://bliptext.com/articles/hans-kamp).\nIt has been further developed by [computer scientists](https://bliptext.com/articles/computer-scientists), notably [Amir Pnueli](https://bliptext.com/articles/amir-pnueli), and [logicians](https://bliptext.com/articles/logician).\nTemporal logic has found an important application in [formal verification](https://bliptext.com/articles/formal-verification), where it is used to state requirements of hardware or software systems.\nFor instance, one may wish to say that whenever a request is made, access to a resource is eventually granted, but it is never granted to two requestors simultaneously.\nSuch a statement can conveniently be expressed in a temporal logic.\n\n# Motivation\nConsider the statement \"I am hungry\".\nThough its meaning is constant in time, the statement's truth value can vary in time.\nSometimes it is true, and sometimes false, but never simultaneously true and false.\nIn a temporal logic, a statement can have a truth value that varies in time—in contrast with an atemporal logic, which applies only to statements whose truth values are constant in time.\nThis treatment of truth-value over time differentiates temporal logic from [computational verb logic](https://bliptext.com/articles/computational-verb-logic).\nTemporal logic always has the ability to reason about a timeline.\nSo-called \"linear-time\" logics are restricted to this type of reasoning.\nBranching-time logics, however, can reason about multiple timelines.\nThis permits in particular treatment of environments that may act unpredictably.\nTo continue the example, in a branching-time logic we may state that \"there is a possibility that I will stay hungry forever\", and that \"there is a possibility that eventually I am no longer hungry\".\nIf we do not know whether or not I will ever be fed, these statements can both be true.\n\n# History\nAlthough [Aristotle](https://bliptext.com/articles/aristotle)'s logic is almost entirely concerned with the theory of the [categorical syllogism](https://bliptext.com/articles/categorical-syllogism), there are passages in his work that are now seen as anticipations of temporal logic, and may imply an early, partially developed form of [first-order](https://bliptext.com/articles/first-order-logic) temporal modal [bivalent](https://bliptext.com/articles/bivalent-logic) logic.\nAristotle was particularly concerned with the [problem of future contingents](https://bliptext.com/articles/problem-of-future-contingents), where he could not accept that the [principle of bivalence](https://bliptext.com/articles/principle-of-bivalence) applies to statements about future events, i.e. that we can presently decide if a statement about a future event is true or false, such as \"there will be a sea battle tomorrow\".\nThere was little development for millennia, [Charles Sanders Peirce](https://bliptext.com/articles/charles-sanders-peirce) noted in the 19th century:\nTime has usually been considered by logicians to be what is called 'extralogical' matter.\nI have never shared this opinion.\nBut I have thought that logic had not yet reached the state of development at which the introduction of temporal modifications of its forms would not result in great confusion; and I am much of that way of thinking yet.\nSurprisingly for [Peirce](https://bliptext.com/articles/charles-sanders-peirce), the first system of temporal logic was constructed, as far as we know, in the first half of 20th century.\nAlthough [Arthur Prior](https://bliptext.com/articles/arthur-prior) is widely known as a founder of temporal logic, the first formalization of such logic was provided in 1947 by Polish logician, [Jerzy Łoś](https://bliptext.com/articles/jerzy-o).\nIn his work Podstawy Analizy Metodologicznej Kanonów Milla (The Foundations of a Methodological Analysis of Mill’s Methods) he presented a formalization of [Mill's canons](https://bliptext.com/articles/mill-s-canons).\nIn [Łoś](https://bliptext.com/articles/jerzy-o)' approach, emphasis was placed on the time factor.\nThus, to reach his goal, he had to create a logic that could provide means for formalization of temporal functions.\nThe logic could be seen as a byproduct of [Łoś](https://bliptext.com/articles/jerzy-o)' main aim, albeit it was the first positional logic that, as a framework, was used later for [Łoś](https://bliptext.com/articles/jerzy-o)' inventions in [epistemic logic](https://bliptext.com/articles/epistemic-logic).\nThe logic itself has syntax very different than Prior's tense logic, which uses modal operators.\nThe language of [Łoś](https://bliptext.com/articles/jerzy-o)' logic rather uses a realization operator, specific to positional logic, which binds the expression with the specific context in which its truth-value is considered.\nIn [Łoś](https://bliptext.com/articles/jerzy-o)' work this considered context was only temporal, thus expressions were bound with specific moments or intervals of time.\nIn the following years, research of temporal logic by [Arthur Prior](https://bliptext.com/articles/arthur-prior) began.\nHe was concerned with the philosophical implications of [free will](https://bliptext.com/articles/free-will) and [predestination](https://bliptext.com/articles/predestination).\nAccording to his wife, he first considered formalizing temporal logic in 1953.\nResults of his research were first presented at the conference in [Wellington](https://bliptext.com/articles/wellington) in 1954.\nThe system Prior presented, was similar syntactically to [Łoś](https://bliptext.com/articles/jerzy-o)' logic, although not until 1955 did he explicitly refer to [Łoś](https://bliptext.com/articles/jerzy-o)' work, in the last section of Appendix 1 in Prior’s Formal Logic.\n[Prior](https://bliptext.com/articles/arthur-prior) gave lectures on the topic at the [University of Oxford](https://bliptext.com/articles/university-of-oxford) in 1955–6, and in 1957 published a book, Time and Modality, in which he introduced a [propositional](https://bliptext.com/articles/propositional-logic) modal logic with two temporal connectives ([modal operators](https://bliptext.com/articles/modal-operator)), F and P, corresponding to \"sometime in the future\" and \"sometime in the past\".\nIn this early work, Prior considered time to be linear.\nIn 1958 however, he received a letter from [Saul Kripke](https://bliptext.com/articles/saul-kripke), who pointed out that this assumption is perhaps unwarranted.\nIn a development that foreshadowed a similar one in computer science, Prior took this under advisement, and developed two theories of branching time, which he called \"Ockhamist\" and \"Peircean\".\nBetween 1958 and 1965 Prior also corresponded with [Charles Leonard Hamblin](https://bliptext.com/articles/charles-leonard-hamblin), and a number of early developments in the field can be traced to this correspondence, for example [Hamblin implications](https://bliptext.com/articles/hamblin-implications).\nPrior published his most mature work on the topic, the book Past, Present, and Future in 1967.\nHe died two years later.\nAlong with **tense logic,** [Prior](https://bliptext.com/articles/arthur-prior) constructed a few systems of positional logic, which inherited their main ideas from [Łoś](https://bliptext.com/articles/jerzy-o).\nWork in positional temporal logics was continued by [Nicholas Rescher](https://bliptext.com/articles/nicholas-rescher) in the 60s and 70s.\nIn such works as Note on Chronological Logic (1966), On the Logic of Chronological Propositions (1968), Topological Logic (1968), and Temporal Logic (1971) he researched connections between [Łoś](https://bliptext.com/articles/jerzy-o)' and [Prior](https://bliptext.com/articles/arthur-prior)'s systems.\nMoreover, he proved that [Prior](https://bliptext.com/articles/arthur-prior)'s tense operators could be defined using a realization operator in specific positional logics.\n[Rescher](https://bliptext.com/articles/nicholas-rescher), in his work, also created more general systems of positional logics.\nAlthough the first ones were constructed for purely temporal uses, he proposed the term **topological logics** for logics that were meant to contain a realization operator but had no specific temporal axioms—like the clock axiom.\nThe binary temporal operators Since and Until were introduced by [Hans Kamp](https://bliptext.com/articles/hans-kamp) in his 1968 Ph.D. thesis, which also contains an important result relating temporal logic to [first-order logic](https://bliptext.com/articles/first-order-logic)—a result now known as [Kamp's theorem](https://bliptext.com/articles/kamp-s-theorem).\nTwo early contenders in formal verifications were [linear temporal logic](https://bliptext.com/articles/linear-temporal-logic), a linear-time logic by [Amir Pnueli](https://bliptext.com/articles/amir-pnueli), and [computation tree logic](https://bliptext.com/articles/computation-tree-logic) (CTL), a branching-time logic by [Mordechai Ben-Ari](https://bliptext.com/articles/mordechai-ben-ari), [Zohar Manna](https://bliptext.com/articles/zohar-manna) and [Amir Pnueli](https://bliptext.com/articles/amir-pnueli).\nAn almost equivalent formalism to CTL was suggested around the same time by [E. M. Clarke](https://bliptext.com/articles/edmund-m-clarke) and [E. A. Emerson](https://bliptext.com/articles/e-allen-emerson).\nThe fact that the second logic can be [decided](https://bliptext.com/articles/decision-problem) [more efficiently](https://bliptext.com/articles/computational-complexity) than the first does not reflect on branching- and linear-time logics in general, as has sometimes been argued.\nRather, Emerson and Lei show that any linear-time logic can be extended to a branching-time logic that can be decided with the same complexity.\n\n# Łoś's positional logic\n[Łoś](https://bliptext.com/articles/jerzy-o)’s logic was published as his 1947 master’s thesis Podstawy Analizy Metodologicznej Kanonów Milla (The Foundations of a Methodological Analysis of Mill’s Methods).\nHis philosophical and formal concepts could be seen as continuations of those of the [Lviv–Warsaw School of Logic](https://bliptext.com/articles/lviv-warsaw-school-of-logic), as his supervisor was [Jerzy Słupecki](https://bliptext.com/articles/jerzy-s-upecki), disciple of [Jan Łukasiewicz](https://bliptext.com/articles/jan-ukasiewicz).\nThe paper was not translated into English until 1977, although [Henryk Hiż](https://bliptext.com/articles/henryk-hi) presented in 1951 a brief, but informative, review in the [Journal of Symbolic Logic](https://bliptext.com/articles/journal-of-symbolic-logic).\nThis review contained core concepts of [Łoś](https://bliptext.com/articles/jerzy-o)’s work and was enough to popularize his results among the logical community.\nThe main aim of this work was to present [Mill's canons](https://bliptext.com/articles/mill-s-canons) in the framework of formal logic.\nTo achieve this goal the author researched the importance of temporal functions in the structure of Mill's concept.\nHaving that, he provided his axiomatic system of logic that would fit as a framework for [Mill's canons](https://bliptext.com/articles/mill-s-canons) along with their temporal aspects.\n\n## Syntax\nThe language of the logic first published in Podstawy Analizy Metodologicznej Kanonów Milla (The Foundations of a Methodological Analysis of Mill’s Methods) consisted of:\nThe set of terms (denoted by S) is constructed as follows:\nThe set of formulas (denoted by For) is constructed as follows:\n\n## Original Axiomatic System\n\n# Prior's tense logic (TL)\nThe sentential tense logic introduced in Time and Modality has four (non-[truth-functional](https://bliptext.com/articles/truth-function)) [modal operators](https://bliptext.com/articles/modal-operator) (in addition to all usual [truth-functional](https://bliptext.com/articles/truth-function) operators in [first-order propositional logic](https://bliptext.com/articles/propositional-calculus)).\nThese can be combined if we let π be an infinite path:\nFrom P and F one can define G and H, and vice versa:\n\n## Syntax and semantics\nA minimal syntax for TL is specified with the following [BNF grammar](https://bliptext.com/articles/backus-naur-form):\nwhere a is some [atomic formula](https://bliptext.com/articles/atomic-formula).\n[Kripke models](https://bliptext.com/articles/kripke-semantics) are used to evaluate the truth of [sentences](https://bliptext.com/articles/sentence-mathematical-logic) in TL.\nA pair (T, <) of a set T and a [binary relation](https://bliptext.com/articles/binary-relation) < on T (called \"precedence\") is called a **frame**.\nA **model** is given by triple (T, <, V) of a frame and a function V called a **valuation** that assigns to each pair (a, u) of an atomic formula and a time value some truth value.\nThe notion \"ϕ is true in a model U=(T, <, V) at time u\" is abbreviated U[⊨](https://bliptext.com/articles/double-turnstile)ϕ[u].\nWith this notation,\nGiven a class F of frames, a sentence ϕ of TL is\nMany sentences are only valid for a limited class of frames.\nIt is common to restrict the class of frames to those with a relation < that is [transitive](https://bliptext.com/articles/transitive-reduction), [antisymmetric](https://bliptext.com/articles/antisymmetric-relation), [reflexive](https://bliptext.com/articles/reflexive-relation), [trichotomic](https://bliptext.com/articles/trichotomy-mathematics), ir[reflexive](https://bliptext.com/articles/reflexive-relation), [total](https://bliptext.com/articles/total-order), [dense](https://bliptext.com/articles/dense-order), or some combination of these.\n\n## A minimal axiomatic logic\nBurgess outlines a logic that makes no assumptions on the relation <, but allows for meaningful deductions, based on the following axiom schema:\nwith the following rules of deduction:\nOne can derive the following rules:\n\n## Translation to predicate logic\nBurgess gives a Meredith translation from statements in TL into statements in [first-order logic](https://bliptext.com/articles/first-order-logic) with one free variable x0 (representing the present moment).\nThis translation M is defined recursively as follows:\nwhere A^+ is the sentence A with all variable indices incremented by 1 and a^* is a one-place predicate defined by.\n\n# Temporal operators\nTemporal logic has two kinds of operators: [logical operators](https://bliptext.com/articles/logical-operator) and modal operators.\nLogical operators are usual truth-functional operators.\nThe modal operators used in linear temporal logic and computation tree logic are defined as follows.\nAlternate symbols:\nUnary operators are [well-formed formulas](https://bliptext.com/articles/well-formed-formula) whenever\nB(φ)\nis well-formed.\nBinary operators are well-formed formulas whenever\nB(φ)\nand\nC(φ)\nare well-formed.\nIn some logics, some operators cannot be expressed.\nFor example, **N** operator cannot be expressed in [temporal logic of actions](https://bliptext.com/articles/temporal-logic-of-actions).\n\n# Temporal logics\nTemporal logics include:\nA variation, closely related to temporal or chronological or tense logics, are modal logics based upon \"topology\", \"place\", or \"spatial position\".","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:09:59.769Z","updated_at":"2025-01-25T19:09:59.769Z","search_vector":"'logic':2 'tempor':1","search_content":" \n \n\n# \nIn [logic](https://bliptext.com/articles/logic), **temporal [logic](https://bliptext.com/articles/logic)** is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of [time](https://bliptext.com/articles/time) (for example, \"I am always hungry\", \"I will eventually be hungry\", or \"I will be hungry until I eat something\").\nIt is sometimes also used to refer to **tense logic**, a [modal logic](https://bliptext.com/articles/modal-logic)-based system of temporal logic introduced by [Arthur Prior](https://bliptext.com/articles/arthur-prior) in the late 1950s, with important contributions by [Hans Kamp](https://bliptext.com/articles/hans-kamp).\nIt has been further developed by [computer scientists](https://bliptext.com/articles/computer-scientists), notably [Amir Pnueli](https://bliptext.com/articles/amir-pnueli), and [logicians](https://bliptext.com/articles/logician).\nTemporal logic has found an important application in [formal verification](https://bliptext.com/articles/formal-verification), where it is used to state requirements of hardware or software systems.\nFor instance, one may wish to say that whenever a request is made, access to a resource is eventually granted, but it is never granted to two requestors simultaneously.\nSuch a statement can conveniently be expressed in a temporal logic.\n\n# Motivation\nConsider the statement \"I am hungry\".\nThough its meaning is constant in time, the statement's truth value can vary in time.\nSometimes it is true, and sometimes false, but never simultaneously true and false.\nIn a temporal logic, a statement can have a truth value that varies in time—in contrast with an atemporal logic, which applies only to statements whose truth values are constant in time.\nThis treatment of truth-value over time differentiates temporal logic from [computational verb logic](https://bliptext.com/articles/computational-verb-logic).\nTemporal logic always has the ability to reason about a timeline.\nSo-called \"linear-time\" logics are restricted to this type of reasoning.\nBranching-time logics, however, can reason about multiple timelines.\nThis permits in particular treatment of environments that may act unpredictably.\nTo continue the example, in a branching-time logic we may state that \"there is a possibility that I will stay hungry forever\", and that \"there is a possibility that eventually I am no longer hungry\".\nIf we do not know whether or not I will ever be fed, these statements can both be true.\n\n# History\nAlthough [Aristotle](https://bliptext.com/articles/aristotle)'s logic is almost entirely concerned with the theory of the [categorical syllogism](https://bliptext.com/articles/categorical-syllogism), there are passages in his work that are now seen as anticipations of temporal logic, and may imply an early, partially developed form of [first-order](https://bliptext.com/articles/first-order-logic) temporal modal [bivalent](https://bliptext.com/articles/bivalent-logic) logic.\nAristotle was particularly concerned with the [problem of future contingents](https://bliptext.com/articles/problem-of-future-contingents), where he could not accept that the [principle of bivalence](https://bliptext.com/articles/principle-of-bivalence) applies to statements about future events, i.e. that we can presently decide if a statement about a future event is true or false, such as \"there will be a sea battle tomorrow\".\nThere was little development for millennia, [Charles Sanders Peirce](https://bliptext.com/articles/charles-sanders-peirce) noted in the 19th century:\nTime has usually been considered by logicians to be what is called 'extralogical' matter.\nI have never shared this opinion.\nBut I have thought that logic had not yet reached the state of development at which the introduction of temporal modifications of its forms would not result in great confusion; and I am much of that way of thinking yet.\nSurprisingly for [Peirce](https://bliptext.com/articles/charles-sanders-peirce), the first system of temporal logic was constructed, as far as we know, in the first half of 20th century.\nAlthough [Arthur Prior](https://bliptext.com/articles/arthur-prior) is widely known as a founder of temporal logic, the first formalization of such logic was provided in 1947 by Polish logician, [Jerzy Łoś](https://bliptext.com/articles/jerzy-o).\nIn his work Podstawy Analizy Metodologicznej Kanonów Milla (The Foundations of a Methodological Analysis of Mill’s Methods) he presented a formalization of [Mill's canons](https://bliptext.com/articles/mill-s-canons).\nIn [Łoś](https://bliptext.com/articles/jerzy-o)' approach, emphasis was placed on the time factor.\nThus, to reach his goal, he had to create a logic that could provide means for formalization of temporal functions.\nThe logic could be seen as a byproduct of [[Łoś](https://bliptext.com/articles/jerzy-o)](https://bliptext.com/articles/jerzy-o)' main aim, albeit it was the first positional logic that, as a framework, was used later for [[Łoś](https://bliptext.com/articles/jerzy-o)](https://bliptext.com/articles/jerzy-o)' inventions in [epistemic logic](https://bliptext.com/articles/epistemic-logic).\nThe logic itself has syntax very different than Prior's tense logic, which uses modal operators.\nThe language of [Łoś](https://bliptext.com/articles/jerzy-o)' logic rather uses a realization operator, specific to positional logic, which binds the expression with the specific context in which its truth-value is considered.\nIn [Łoś](https://bliptext.com/articles/jerzy-o)' work this considered context was only temporal, thus expressions were bound with specific moments or intervals of time.\nIn the following years, research of temporal logic by [Arthur Prior](https://bliptext.com/articles/arthur-prior) began.\nHe was concerned with the philosophical implications of [free will](https://bliptext.com/articles/free-will) and [predestination](https://bliptext.com/articles/predestination).\nAccording to his wife, he first considered formalizing temporal logic in 1953.\nResults of his research were first presented at the conference in [Wellington](https://bliptext.com/articles/wellington) in 1954.\nThe system Prior presented, was similar syntactically to [[Łoś](https://bliptext.com/articles/jerzy-o)](https://bliptext.com/articles/jerzy-o)' logic, although not until 1955 did he explicitly refer to [[Łoś](https://bliptext.com/articles/jerzy-o)](https://bliptext.com/articles/jerzy-o)' work, in the last section of Appendix 1 in Prior’s Formal Logic.\n[Prior](https://bliptext.com/articles/arthur-prior) gave lectures on the topic at the [University of Oxford](https://bliptext.com/articles/university-of-oxford) in 1955–6, and in 1957 published a book, Time and Modality, in which he introduced a [propositional](https://bliptext.com/articles/propositional-logic) modal logic with two temporal connectives ([modal operators](https://bliptext.com/articles/modal-operator)), F and P, corresponding to \"sometime in the future\" and \"sometime in the past\".\nIn this early work, Prior considered time to be linear.\nIn 1958 however, he received a letter from [Saul Kripke](https://bliptext.com/articles/saul-kripke), who pointed out that this assumption is perhaps unwarranted.\nIn a development that foreshadowed a similar one in computer science, Prior took this under advisement, and developed two theories of branching time, which he called \"Ockhamist\" and \"Peircean\".\nBetween 1958 and 1965 Prior also corresponded with [Charles Leonard Hamblin](https://bliptext.com/articles/charles-leonard-hamblin), and a number of early developments in the field can be traced to this correspondence, for example [Hamblin implications](https://bliptext.com/articles/hamblin-implications).\nPrior published his most mature work on the topic, the book Past, Present, and Future in 1967.\nHe died two years later.\nAlong with **tense logic,** [Prior](https://bliptext.com/articles/arthur-prior) constructed a few systems of positional logic, which inherited their main ideas from [Łoś](https://bliptext.com/articles/jerzy-o).\nWork in positional temporal logics was continued by [Nicholas Rescher](https://bliptext.com/articles/nicholas-rescher) in the 60s and 70s.\nIn such works as Note on Chronological Logic (1966), On the Logic of Chronological Propositions (1968), Topological Logic (1968), and Temporal Logic (1971) he researched connections between [Łoś](https://bliptext.com/articles/jerzy-o)' and [Prior](https://bliptext.com/articles/arthur-prior)'s systems.\nMoreover, he proved that [Prior](https://bliptext.com/articles/arthur-prior)'s tense operators could be defined using a realization operator in specific positional logics.\n[Rescher](https://bliptext.com/articles/nicholas-rescher), in his work, also created more general systems of positional logics.\nAlthough the first ones were constructed for purely temporal uses, he proposed the term **topological logics** for logics that were meant to contain a realization operator but had no specific temporal axioms—like the clock axiom.\nThe binary temporal operators Since and Until were introduced by [Hans Kamp](https://bliptext.com/articles/hans-kamp) in his 1968 Ph.D. thesis, which also contains an important result relating temporal logic to [first-order logic](https://bliptext.com/articles/first-order-logic)—a result now known as [Kamp's theorem](https://bliptext.com/articles/kamp-s-theorem).\nTwo early contenders in formal verifications were [linear temporal logic](https://bliptext.com/articles/linear-temporal-logic), a linear-time logic by [Amir Pnueli](https://bliptext.com/articles/amir-pnueli), and [computation tree logic](https://bliptext.com/articles/computation-tree-logic) (CTL), a branching-time logic by [Mordechai Ben-Ari](https://bliptext.com/articles/mordechai-ben-ari), [Zohar Manna](https://bliptext.com/articles/zohar-manna) and [Amir Pnueli](https://bliptext.com/articles/amir-pnueli).\nAn almost equivalent formalism to CTL was suggested around the same time by [E. M. Clarke](https://bliptext.com/articles/edmund-m-clarke) and [E. A. Emerson](https://bliptext.com/articles/e-allen-emerson).\nThe fact that the second logic can be [decided](https://bliptext.com/articles/decision-problem) [more efficiently](https://bliptext.com/articles/computational-complexity) than the first does not reflect on branching- and linear-time logics in general, as has sometimes been argued.\nRather, Emerson and Lei show that any linear-time logic can be extended to a branching-time logic that can be decided with the same complexity.\n\n# Łoś's positional logic\n[Łoś](https://bliptext.com/articles/jerzy-o)’s logic was published as his 1947 master’s thesis Podstawy Analizy Metodologicznej Kanonów Milla (The Foundations of a Methodological Analysis of Mill’s Methods).\nHis philosophical and formal concepts could be seen as continuations of those of the [Lviv–Warsaw School of Logic](https://bliptext.com/articles/lviv-warsaw-school-of-logic), as his supervisor was [Jerzy Słupecki](https://bliptext.com/articles/jerzy-s-upecki), disciple of [Jan Łukasiewicz](https://bliptext.com/articles/jan-ukasiewicz).\nThe paper was not translated into English until 1977, although [Henryk Hiż](https://bliptext.com/articles/henryk-hi) presented in 1951 a brief, but informative, review in the [Journal of Symbolic Logic](https://bliptext.com/articles/journal-of-symbolic-logic).\nThis review contained core concepts of [Łoś](https://bliptext.com/articles/jerzy-o)’s work and was enough to popularize his results among the logical community.\nThe main aim of this work was to present [Mill's canons](https://bliptext.com/articles/mill-s-canons) in the framework of formal logic.\nTo achieve this goal the author researched the importance of temporal functions in the structure of Mill's concept.\nHaving that, he provided his axiomatic system of logic that would fit as a framework for [Mill's canons](https://bliptext.com/articles/mill-s-canons) along with their temporal aspects.\n\n## Syntax\nThe language of the logic first published in Podstawy Analizy Metodologicznej Kanonów Milla (The Foundations of a Methodological Analysis of Mill’s Methods) consisted of:\nThe set of terms (denoted by S) is constructed as follows:\nThe set of formulas (denoted by For) is constructed as follows:\n\n## Original Axiomatic System\n\n# Prior's tense logic (TL)\nThe sentential tense logic introduced in Time and Modality has four (non-[truth-functional](https://bliptext.com/articles/truth-function)) [modal operators](https://bliptext.com/articles/modal-operator) (in addition to all usual [truth-functional](https://bliptext.com/articles/truth-function) operators in [first-order propositional logic](https://bliptext.com/articles/propositional-calculus)).\nThese can be combined if we let π be an infinite path:\nFrom P and F one can define G and H, and vice versa:\n\n## Syntax and semantics\nA minimal syntax for TL is specified with the following [BNF grammar](https://bliptext.com/articles/backus-naur-form):\nwhere a is some [atomic formula](https://bliptext.com/articles/atomic-formula).\n[Kripke models](https://bliptext.com/articles/kripke-semantics) are used to evaluate the truth of [sentences](https://bliptext.com/articles/sentence-mathematical-logic) in TL.\nA pair (T, <) of a set T and a [binary relation](https://bliptext.com/articles/binary-relation) < on T (called \"precedence\") is called a **frame**.\nA **model** is given by triple (T, <, V) of a frame and a function V called a **valuation** that assigns to each pair (a, u) of an atomic formula and a time value some truth value.\nThe notion \"ϕ is true in a model U=(T, <, V) at time u\" is abbreviated U[⊨](https://bliptext.com/articles/double-turnstile)ϕ[u].\nWith this notation,\nGiven a class F of frames, a sentence ϕ of TL is\nMany sentences are only valid for a limited class of frames.\nIt is common to restrict the class of frames to those with a relation < that is [transitive](https://bliptext.com/articles/transitive-reduction), [antisymmetric](https://bliptext.com/articles/antisymmetric-relation), [reflexive](https://bliptext.com/articles/reflexive-relation), [trichotomic](https://bliptext.com/articles/trichotomy-mathematics), ir[reflexive](https://bliptext.com/articles/reflexive-relation), [total](https://bliptext.com/articles/total-order), [dense](https://bliptext.com/articles/dense-order), or some combination of these.\n\n## A minimal axiomatic logic\nBurgess outlines a logic that makes no assumptions on the relation <, but allows for meaningful deductions, based on the following axiom schema:\nwith the following rules of deduction:\nOne can derive the following rules:\n\n## Translation to predicate logic\nBurgess gives a Meredith translation from statements in TL into statements in [first-order logic](https://bliptext.com/articles/first-order-logic) with one free variable x0 (representing the present moment).\nThis translation M is defined recursively as follows:\nwhere A^+ is the sentence A with all variable indices incremented by 1 and a^* is a one-place predicate defined by.\n\n# Temporal operators\nTemporal logic has two kinds of operators: [logical operators](https://bliptext.com/articles/logical-operator) and modal operators.\nLogical operators are usual truth-functional operators.\nThe modal operators used in linear temporal logic and computation tree logic are defined as follows.\nAlternate symbols:\nUnary operators are [well-formed formulas](https://bliptext.com/articles/well-formed-formula) whenever\nB(φ)\nis well-formed.\nBinary operators are well-formed formulas whenever\nB(φ)\nand\nC(φ)\nare well-formed.\nIn some logics, some operators cannot be expressed.\nFor example, **N** operator cannot be expressed in [temporal logic of actions](https://bliptext.com/articles/temporal-logic-of-actions).\n\n# Temporal logics\nTemporal logics include:\nA variation, closely related to temporal or chronological or tense logics, are modal logics based upon \"topology\", \"place\", or \"spatial position\".","revisionCount":0},"hasMore":false,"nextCursor":null}