In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. In a temporal logic we can then express statements like "I am always hungry", "I will eventually be hungry", or "I will be hungry until I eat something". Temporal logic is sometimes also used to refer to tense logic, a particular modal logic-based system of temporal logic introduced by Arthur Prior in the late 1950s, and important results obtained were by Hans Kamp. Subsequently it has been developed further by computer scientists, notably Amir Pnueli, and logicians.

Temporal logic has found an important application in formal verification, where it is used to state requirements of hardware or software systems. For 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. Such a statement can conveniently be expressed in a temporal logic.

Motivation[link]

Consider the statement: "I am hungry." Though its meaning is constant in time, the truth value of the statement can vary in time. Sometimes the statement is true, and sometimes the statement is false, but the statement is never true and false simultaneously. In a temporal logic, statements can have a truth value which can vary in time. Contrast this with an atemporal logic, which can only discuss statements whose truth value is constant in time. This treatment of truth values over time differentiates temporal logic from computational verb logic.

Temporal logic always has the ability to reason about a time line. So called linear time logics are restricted to this type of reasoning. Branching logics, however, can reason about multiple time lines. This presupposes an environment that may act unpredictably. To continue the example, in a branching logic we may state that "there is a possibility that I will stay hungry forever." We may also state that "there is a possibility that eventually I am no longer hungry." If we do not know whether or not I will ever get fed, these statements are both true some times.

History[link]

Although Aristotle's logic is almost entirely concerned with the theory of the 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 temporal modal binary logic. Aristotle was particularly concerned with the problem of future contingents, where he could not accept that the 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".^[1]

There was little development for millennia, Charles Sanders Peirce noted in the 19th century:^[2]

Arthur Prior was concerned with the philosophical matters of free will and predestination. According to his wife, he first considered formalizing temporal logic in 1953. He gave lectures on the topic at the University of Oxford in 1955-6, and in 1957 published a book, Time and Modality, in which he introduce a propositional modal logic with two temporal connectives (modal operators), F and P, corresponding to “sometime in the future” and “sometime in the past”. In this early work, Prior considered time to be linear. In 1958 however, he received a letter from Saul Kripke, who pointed out that this assumption is perhaps unwarranted. In 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”.^{[2][clarification needed]} Between 1958 and 1965 Prior also corresponded with Charles Leonard Hamblin, and a number of early developments in the field can be traced to this correspondence, for example Hamblin implications. Prior published his most mature work on the topic, the book Past, Present, and Future in 1967. He died two years later.^[3]

The binary temporal operators Since and Until were introduced by Hans Kamp in his 1968 Ph. D. thesis,^[4] which also contains an important result relating temporal logic to first order logic—a result now known as Kamp's theorem.^[5][2][6]

Two early contenders in formal verifications were Linear Temporal Logic (a linear time logic by Amir Pnueli) and Computation Tree Logic, a branching time logic by E.M. Clarke and E.A. Emerson. The fact that the second logic is more efficient than the first does not reflect on branching and linear logics in general, as has sometimes been argued. Rather, Emerson and Lei show that any linear logic can be extended to a branching logic that can be decided with the same complexity.

Temporal operators[link]

Temporal logic has two kinds of operators: logical operators and modal operators [1]. Logical operators are usual truth-functional operators (Failed to parse (Missing texvc executable; please see math/README to configure.): \neg,\or,\and,\rightarrow ). The modal operators used in Linear Temporal Logic and Computation Tree Logic are defined as follows.

U Failed to parse (Missing texvc executable; please see math/README to configure.): \psi

holds at the current or a future position, and Failed to parse (Missing texvc executable; please see math/README to configure.): \phi
has to hold until that position. At that position Failed to parse (Missing texvc executable; please see math/README to configure.): \phi
does not have to hold any more.

bar:p color:red width:10 align:left fontsize:S
from:1 till:3

bar:q color:red width:10 align:left fontsize:S
from:3 till:5

bar:pUq color:red width:10 align:left fontsize:S
from:1 till:5

R Failed to parse (Missing texvc executable; please see math/README to configure.): \psi

releases Failed to parse (Missing texvc executable; please see math/README to configure.): \psi
if Failed to parse (Missing texvc executable; please see math/README to configure.): \psi
is true until the first position in which Failed to parse (Missing texvc executable; please see math/README to configure.): \phi
is true (or forever if such a position does not exist).

bar:p color:red width:10 align:left fontsize:S
from:2 till:4
from:6 till:8

bar:q color:red width:10 align:left fontsize:S
from:1 till:3
from:5 till:6
from:7 till:8

bar:pRq color:red width:10 align:left fontsize:S
from:1 till:3
from:7 till:8

has to hold at the next state. (X is used synonymously.)

bar:p color:red width:10 align:left fontsize:S
from:2 till:3
from:5 till:6

bar:Np color:red width:10 align:left fontsize:S
from:1 till:2
from:4 till:5

eventually has to hold (somewhere on the subsequent path).

bar:p color:red width:10 align:left fontsize:S
from:2 till:3
from:4 till:5

bar:Fp color:red width:10 align:left fontsize:S
from:0 till:5

has to hold on the entire subsequent path.

bar:p color:red width:10 align:left fontsize:S
from:1 till:3
from:4 till:6

bar:Gp color:red width:10 align:left fontsize:S
from:4 till:6

has to hold on all paths starting from the current state.

holds.

Alternate symbols:

• operator R is sometimes denoted by V
• The operator W is the weak until operator: Failed to parse (Missing texvc executable; please see math/README to configure.): f W g

is equivalent to Failed to parse (Missing texvc executable; please see math/README to configure.): f U g \or G f

Unary operators are well-formed formulas whenever B(Failed to parse (Missing texvc executable; please see math/README to configure.): \phi ) is well-formed. Binary operators are well-formed formulas whenever B(Failed to parse (Missing texvc executable; please see math/README to configure.): \phi ) and C(Failed to parse (Missing texvc executable; please see math/README to configure.): \phi ) are well-formed.

In some logics, some operators cannot be expressed. For example, N operator cannot be expressed in Temporal Logic of Actions.

Temporal logics[link]

Temporal logics include

• Interval temporal logic (ITL)
• μ calculus. which includes as a subset
  • Hennessy-Milner logic (HML)
  • CTL*, which includes as a subset
    • Computational tree logic (CTL)
    • Linear temporal logic (LTL)

A variation, closely related to Temporal or Chronological or Tense logics, are Modal logics based upon "topology", "place", or "spatial position". ^[7][8] One might also take note that in the Russian language, verbs have an aspect, based commonly upon time, but position also.

See also[link]

Logic portal

• Computational verb logic
• HPO formalism
• Kripke structure
• Automata theory
• Chomsky grammar
• State transition system
• Duration calculus (DC)
• Hybrid logic
• Temporal logic in finite-state verification
• Temporal logic of actions (TLA)
• Important publications in formal verification (including the use of temporal logic in formal verification)
• Reo Coordination Language
• Modal logic
• Research Materials: Max Planck Society Archive

Notes[link]

References[link]

• Mordechai Ben-Ari, Zohar Manna, Amir Pnueli: The Temporal Logic of Branching Time. POPL 1981: 164-176
• Amir Pnueli: The Temporal Logic of Programs FOCS 1977: 46-57
• Venema, Yde, 2001, "Temporal Logic," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.
• E. A. Emerson and C. Lei, modalities for model checking: branching time logic strikes back, in Science of Computer Programming 8, p 275-306, 1987.
• E.A. Emerson, Temporal and modal logic, Handbook of Theoretical Computer Science, Chapter 16, the MIT Press, 1990
• Moshe Y. Vardi (2008). "From Church and Prior to PSL". In Orna Grumberg, Helmut Veith. 25 years of model checking: history, achievements, perspectives. Springer. ISBN 978-3-540-69849-4.  preprint Historical perspective on how seemingly disparate ideas came together in computer science and engineering. (The reference to Church is to a little known 1957 in which he proposed a way to perform hardware verification.)

Further reading[link]

• Peter Øhrstrøm; Per F. V. Hasle (1995). Temporal logic: from ancient ideas to artificial intelligence. Springer. ISBN 978-0-7923-3586-3. 

External links[link]

• Stanford Encyclopedia of Philosophy: "Temporal Logic" -- by Anthony Galton.
• Temporal Logic by Yde Venema, formal description of syntax and semantics, questions of axiomatization. Treating also Kamp's dyadic temporal operators (since, until)
• Notes on games in temporal logic by Ian Hodkinson, including a formal description of first-order temporal logic
• CADP - provides generic model checkers for various temporal logic
• PAT is a powerful free model checker, LTL checker, simulator and refinement checker for CSP and its extensions (with shared variable, arrays, wide range of fairness).

George Sotiros Pappas (born 1942) is a professor of philosophy at Ohio State University.^[1] Pappas specializes in epistemology, the history of early modern philosophy, philosophy of religion, and metaphysics. He is of Greek and English origin.

He is the author of the Stanford Encyclopedia of Philosophy entry on Internalist versus Externalist conceptions of epistemic justification.^[2]

He was co-editor (with Marshall Swain) of Essays on Knowledge and Justification (1978), regarded as a key anthology of essays relating to the Gettier problem^[3] and used as a core text in undergraduate epistemology courses.^[4]

George Pappas is an editorial consultant of Berkeley Studies.^[5]

Studies in Berkeley's Philosophy[link]

George S. Pappas is known to be a leading Berkeley scholar; his essay “Berkeley and Scepticism”^[6] was in 1993 awarded the International Berkeley Prize.^[7] Professor Pappas is a regular participant of International Berkeley Conferences.^[8] At one such conference, celebrating the 300th anniversary of George Berkeley’s birth, Mr. Pappas propounded a new approach to the relationship between Berkeley’s anti-abstractionism and "esse est percipi"^[9] principle. On Pappas reading, Berkeley’s two theses — that there are no abstract ideas and that sensible objects must be perceived in order to exist — entail one another.^[10]

Pappas' interpretation of Berkeley's ‘esse is percipi’ thesis has sparked much discussion.^[11][12] In 1989, the Garland Publishing Company brought out a 15-volume collection of major works on Berkeley; Pappas' paper “Abstract ideas and the 'esse is percipi' thesis” was included in the third volume,^[13] as it was considered to be a significant contribution to Berkeley scholarship.

Pappas developed his treatment of Berkeley’s “esse est percipi” principle^[14] to repudiate the "inherence interpretation of Berkeley", upon which Edwin E. Allaire, among others, elaborated^[15][16][17]

“That account is put forward to answer an extremely perplexing question in the history of philosophy: Why did Berkeley embrace idealism, i. e., why did he hold that esse est percipi, that to be is to be perceived? (Hausman, Alan. Op. cit. Pp. 421–422.)

After emerging in the early 1960s, the “inherence account” attracted numerous proponents and became an influential element of contemporary Berkeley scholarship. In his paper “Ideas, minds, and Berkeley”^[18] Pappas revealed some discrepancies between fountain-head evidences and Allaire’s approach to a reconstruction of Berkeley’s idealism. Pappas' critical examination of the “inherence account” is greatly appreciated by Berkeley scholars. Pappas’ penetrating remarks compelled Edwin B. Allaire to revise and improve his conception.^[19] Even those who share Allaire’s account of Berkeley’s idealism acknowledge Pappas’ article to be “an excellent review and critique of the IA [inherence account].”^[20]

In 2000 George Pappas published his monograph Berkeley's thought in which some parts were based on earlier papers of his. While writings by A. A. Luce or Geoffrey Warnock are long out dated, the book Berkeley's thought written by Dr Pappas is often included in lists of recommended literature on Berkeley’s philosophy.^[21]

Publications[link]

• Pappas, George S. Berkeley’s Thought. Ithaca and London: Cornell University Press, 2000. ISBN 0-8014-3700-8 ISBN 9780801437007

See also[link]

• American philosophy
• List of American philosophers
• George Berkeley

References[link]