# Is there a logic that takes time into account?

I hope this is neither too simple nor off-topic; I apologize if so. I have just noticed that standard sentence logic seems to make assumptions about time in determining the truth/falsity of a statement. Specifically: a statement like "It is dark outside", or "It is Wednesday", seems to assume time is "frozen", i.e., both statements may be true at, say, 10 p.m., or, for the 2nd, on Wednesday. But the truth-value would change as time changed (say at 10 a.m. on a Friday).

I think what you are looking for is Temporal Logic -- from SEP's entry:

[Temporal logic] been broadly used to cover all approaches to the representation of temporal information within a logical framework, and also more narrowly to refer specifically to the modal-logic type of approach introduced around 1960 by Arthur Prior under the name of Tense Logic and subsequently developed further by 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 handling the temporal aspects of the execution of computer programs

Temporal logic tries to account for the affect that time has on the truth value of sentences. It's an extension of classical logic that adds some new operators.

Classical logic is tenseless, meaning that it's statements are independent of time beyond the present. It can make sense of statements like:

"Bachelors are unmarried males."
"If a number is even if and only if it is divisible by two with no remainder."

This kind of logic was developed to help analyze mathematical arguments, so this makes sense. There never was a time that even meant anything other than "divisible by 2", nor will there ever be a time. The words we use to express the concepts might change, but the concepts themselves will not.

However, this isn't true for most of the propositions we encounter. The best example is "It is raining right now." Now, is this true or false? It depends on when you say it, and where (but that's another topic). Another motivation is a desire to express what is happening inside, say a computer. "The value of memory byte X is 00000001" is going to change once you perform an operation on that byte. Temporal logic, or at leasts ideas that stem from it, actually does see some use in computer science for just this reason.

### So that's the motivation. Now, what's the intuition behind the system?

There are two different ways we could go about doing this. The first is to try to create a tenseless logic: You simply add a new predicate that means something like "At time X" and you get sentences like "It was raining at 8:30 on February 13th 2007" The second is to try to create a tensed logic, which comes up with a way to express "in the past" and "in the future". We are going to take the second approach.

We take classical logic and we add two new sentential operators:

GA: It will always the case that A.
HA: It has always been the case that A.

Both tenses are relative to a certain time. We put them out in front of our sentence like a modal operator.

One way to think of it is to imagine an infinite strip of tape that is cut up into different boxes. Each box represents a different slice of time; you could imagine each box marked with a date and time to represent this, but as you will see that's not necessary. In each box are written different propositions, "It is raining", "1+1=2", etc, along with their truth value at that time. Finally, imagine some kind of marker or vehicle that travels along this tape. The current position of this vehicle is the 'present'. Things in front of it are the future, and things it behind it are the past. GA is true if and only if in each box in from of the vehicles present position, A is true at that time. We give a similar definition for HA. We could get more technical and define it in terms of Kripke frames, but the basic intuitions behind the definition are the same.

We can use these two operators to further define two new operators:

PA: -H-A (There was a time such that A, or It is not the case that there was never a time A)
FA: -G-A (There will be a time such that A, or It is not the case that there will never be a time that A.)

Now, let's create a sentence using these new operators and examine if they are true.

L(x): x is alive.

PL(Nathan): There is a time in the past where "Nathan is alive" is true. Yep!
FL(Nathan): There is a time in the future where "Nathan is alive" is true. I hope so.
HL(Nathan): There is never a time in the past where "Nathan is alive" is false. Nope. I was born.
GL(Nathan): There is never a time in the future where "Nathan is alive" is false. :<

That's the basics of it. The rest of temporal logic takes the tape that I outlined and plays around with it. What would happen if there was a point where the tape started? Where it stops? What if the tape loops? What if it splits into two portions? What if there are two tapes, and they merge? Formally, this is done by articulating a Kripke model for the logic and playing around with axioms.

A solid intro to temporal, and more exotic, logics is John Burgess's Philosophical Logic. It's extremely easy to read, and covers a great range of topics. Highly recommended.

• +1 Good answer! A little formatting would go a long way here though. :P Feel free to rollback or improve some formatting changes I made to make it a bit easier to read. Sep 30, 2011 at 22:58
• Nah, they all look fine. Thanks. I'll try to use more markdowns in the future. Oct 1, 2011 at 2:06

If you read Quine’s Philosophy of Logic you might be convinced that normal predicate logic could be used to make temporal assertions; at least if you're willing to combine assertions.

When I think about it though, there seems to be an intrinsic temporal aspect to modal logic. If you say X is usually the case then you are making an assertion about something’s truth value across time. Modals seem to me to be intrinsic claims about something’s truth value over time. But this is a shot in the dark.

• This is a great answer, but is there any chance you could unpack this a bit? Either way, +1... Sep 21, 2011 at 22:29
• Temporal logics are usually studied as a particular kind of modal logic. The box and diamond operators ('necessity' and 'possibility' respectively) of modal logic can be interpreted as 'always' and 'eventually'. See Michael's wikipedia link for details. Sep 21, 2011 at 22:44

Temporal logic is definately the subject you're looking for, this is a good book on intensional logic in which time (and possible world) logic is dealt with: http://www.amazon.com/Logic-Language-Meaning-Intensional-Logical/dp/0226280888

Also, look into the writings of Hans Reichenbach, he has a system of jumping in time that is very interesting.