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Unit 6: Formalized Symbolic Logics
‡ P ↔¬ ¬ ; P Notes
P ↔¬ ¬ . P
‡
Thus it is possible that it will rain today if and only if it is not necessary that it will not rain
today; and it is necessary that it will rain today if and only if it is not possible that it will not rain
today. Alternative symbols used for the modal operators are “L” for necessarily and “M” for
Possibly.
Self Assessment
State whether the following statements are true or false:
19. A informal modal logic represents modalities using modal operators.
20. Modal logic is a type of formal logic primarily developed in the 1980s.
6.11 Temporal Logic
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.
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 a temporal 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.
Self Assessment
State whether the following statements are true or false:
21. Temporal logic has found an important application in formal verification.
22. Temporal logic is used to describe any system of rules and symbolism for representing,
and reasoning about, propositions qualified in terms of time.
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