<p>Both <a href="http://en.wikipedia.org/wiki/Temporal_logic">temporal logic</a> or <a href="http://en.wikipedia.org/wiki/Dynamic_logic_%28modal_logic%29">dynamic logic</a> are two modal logics that can be used to reason about changes in the world over time. <a href="http://en.wikipedia.org/wiki/F-coalgebra">Coalgebra</a> is a formal approach to dealing with observations of state-based models changing in time. This approach generalises state machines, labelled transition systems, and so forth, which themselves provide behavioural models of systems. <a href="http://arxiv.org/abs/cs.LO/0605128">Coalgebraic modal logics</a> are a generalised approach to defining logics for coalgebras, thus providing a systematic way of generating logics for reasoning about systems that change in time. <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.75.8207&amp;rep=rep1&amp;type=pdf">Logics of causality</a> even exist (I'm sure google will turn up even more).</p>

CC BY-SA 2.5
2011-03-27T17:27:46.870Z
I was wondering about temporal logic: it seemed to do exactly what the OP wanted.
Suresh Venkat
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2011-03-27T17:36:23.470Z
And there are of course many temporal logics, even temporal logics of action, to explore to find the most appropriate one.
Dave Clarke
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2011-03-27T18:27:56.520Z
Dave Clarke, All the "temporal" logics, save for mine, take place in the space-domain. Those conventional systems of logic can only recognize or perform operations that transform or translate values in space. Ordinary logic treats time poorly through the necessity of translating all temporal signs, signals, and effects into the space-domain so as to be suitable for space-only operations. Contemporary logic can easily assign state, but it is powerless to assign cause.
Charles Moeller
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2011-03-27T19:12:02.590Z
@Charles: Please improve your question by providing a reference to your logic and perhaps explain what it's core ingredients are. It is impossible for us to provide any answer to your question without any concrete details.
Dave Clarke
CC BY-SA 2.5
2011-03-27T19:13:55.500Z
Generally one takes into account causation through some sort of substructural logic such as a linear or affine approach: http://en.wikipedia.org/wiki/Linear_logic
sclv
CC BY-SA 2.5
2011-03-27T16:12:51.450Z
2011-03-27T16:12:51.450Z
Dave Clarke
<p>Temporal logic seems to be exactly the thing you are looking for (it allows proving things where the truth of propositions can be qualified by operators such as 'in the next time', 'for always after now', 'at some time after now').</p>

<p>But you may want to look at some other formalisms, namely <a href="http://en.wikipedia.org/wiki/Turing_machine" rel="nofollow">Turing machines</a>, or <a href="http://en.wikipedia.org/wiki/Operational_semantics" rel="nofollow">Operational semantics</a>. These both treat state transitions as changes from one time to the next.</p>

<p>From your comments, it seems that none of these is sufficient for you since they translate the concept of temporal causation into the 'space-domain'. I'm not sure what you mean by this, but you can surely represent temporal causation in any of the above formalisms.</p>

CC BY-SA 2.5
2011-03-27T22:06:30.520Z
2011-03-27T22:06:30.520Z
Mitch
<p>Brush up on your Einstein. Time and space are the same thing. <a href="http://en.wikipedia.org/wiki/Space_time" rel="nofollow">http://en.wikipedia.org/wiki/Space_time</a></p>

<p>Okay so this wikipedia article doesn't do my statement justice as it states that the space-time model doesn't jive with Einstein relativity, but I disagree, Einstein relativity can be modelled in space-time it just requires reference domains based on the velocity of objects.</p>

<p>More to the point, relative time passes the "same" no matter at what speed you travel, however from a universal stand point the faster you go the slower you move through time, this only requires that your "velocity" in a 4D space is constant. </p>

<p>But I digress. Majorly.</p>

CC BY-SA 2.5
2011-03-28T08:29:05.197Z
@acp10bda: This is not an answer to the question.
Hsien-Chih Chang 張顯之
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2011-03-28T08:45:23.137Z
Sure it is, it's saying that the question is a waste of time. Logic as it exists now works fine in space and time, more over logic as it stands is not bound to space or time, it's completely independent of it. It would have to be, because if it weren't that would suggest that without space or time, there is no logic, so from what was the universe derived? You might as well say that it was the flying spaghetti monster, as the absence of any logic would state this is totally plausible.
acp10bda
CC BY-SA 2.5
2011-03-28T08:49:25.377Z
V = !0

P(T = 0) = X

Given V != 0 Then P( T + 1 ) != P ( T )

T = T + 1
P(T) ? 

This is very simplistic, but hey look I just simulated with simple logic that if a object with a position (P) and a Velocity (V) moving through time (T), it's position at one time cannot equal his position at another time. There I just did that. AMAZING.
acp10bda
CC BY-SA 2.5
2011-03-28T14:54:58.400Z
To those who voted this down, please, enlighten me with a reason why logic as it stands is not fit to model logistics in the temporal domain. Further extra credit if you can tell me how time should be handled without using methods that use our old supposedly flawed logics, and "space-domains". Every model I know for time transforms it into a "space-domain." As far as I can tell there is no model for time that isn't somehow related to a extra dimension of space.
acp10bda
CC BY-SA 2.5
2011-03-28T19:48:48.237Z
@acp10bda: I think the burden of proof lies with you and Charles to show us how the state of the art is flawed. Otherwise you both come across sounding unscientific, with high flung unjustified claims.
Dave Clarke
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2011-03-30T07:24:43.993Z
I'm terribly sorry, but how is it my burden of proof to disprove something someone else is claiming without any proof?
acp10bda
CC BY-SA 2.5
2011-03-30T07:32:08.167Z
You've said a bunch of stuff in the original post with no proof backing it up. Why is transforming time into a space domain bad? Cause is represented in logics, it's called an implication. a -> b. where A is true, B is also true, This can be understood as A will cause B. Why exactly is logic restrained to a static domain? We have variables in logic, why don't we use thoes? Also all models of everything we know scientifically is based in these traditional logics. So what are you on about?
acp10bda
CC BY-SA 2.5
2011-03-30T07:41:48.707Z
I was responding more to Moeller saying that we needed a whole new kind of logic and math to handle temporal domains as Conventional logics translates time into space-domain and that's somehow bad, but the problem with that is there is no way to represent time without fitting it to some metric which becomes comparable to space, and thus in the "space-domain". Now there of course exist logics that deal with time like the ones you later posted about, but they don't avoid space-domain. Your post is flawed. There is nothing wrong with logics as it stands.
acp10bda
CC BY-SA 2.5
2011-03-30T07:45:19.997Z
Furthermore to all your cohorts who don't see the fallacies in your arguments, who keep down voting my posts, and slandering the conventional logic which all other logics and science itself derives. Seriously, what the hell is wrong with you?
acp10bda
CC BY-SA 2.5
2011-03-30T07:52:39.647Z
Appologies: I apparently have become confused. Dave Clarke, we seem to actually be arguing the same thing. I don't understand how you came to think I was backing Moeller, but I wasn't. I am a proponent that Conventional Logic is just fine at representing time domain relations, we do not need anyone reinventing logic on some false belief that the space-domain isn't appropriate to express time-domain relationships which is what I've been vocally refuting here.
acp10bda
CC BY-SA 2.5
2011-03-28T08:28:01.277Z
2011-03-28T08:42:11.470Z
2011-03-28T08:42:11.470Z
acp10bda
<p>The conventional systems of logic can only recognize or perform operations that transform or translate values in space. Ordinary logic treats time poorly through the necessity of translating all temporal signs, signals, and effects into the space-domain so as to be suitable for space-only operations. Contemporary logic can easily assign state, but it is powerless to assign cause. Life is typified by activity. Humans have more than static existence: we experience and also cause dynamic processes in both space and time. How does one concisely express and treat the aspects of movement, change, cause and effect, and life with only static formal tools? </p>

<p>I propose a time-domain mathematics and logic, a method of automating physical processes that does not first require translation to the space-domain before applying logic. </p>

CC BY-SA 2.5
2011-03-27T15:48:19.497Z
I do not think that the title corresponds to the actual question.  Please correct the title.
Tsuyoshi Ito
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2011-03-27T16:26:38.817Z
Did you not post a slightly more ambitious question somewhere? Cannot find it anymore. What you write sounds a little bit cranky. How do you expect anyone to comment your approach without having seen it? I suggest proposing it in total to an expert audience of mathematicians and/or computer scientists. You can try submitting it to journals or conferences or use any community platform, as e.g. Stackexchange.
Raphael
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2011-03-27T18:00:41.057Z
Yes, I did. Apparently taken down by the moderators.
Charles Moeller
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2011-03-27T18:01:29.793Z
Could we discuss the concept?
Charles Moeller
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2011-03-27T18:03:10.683Z
If you want the whole thing, you will have to wait for the book.
Charles Moeller
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2011-03-27T19:15:51.480Z
@Charles: you need to provide some details before this question can be answered. Otherwise it's just vapourware.
Dave Clarke
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2011-03-27T22:04:30.770Z
I am downvoting because this is hardly a question, and is most definitely not a question in theoretical computer science.
Andrej Bauer
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2011-03-27T22:12:48.690Z
Fair enough. I withdraw.
Charles Moeller
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2011-03-28T22:56:05.117Z
I am closing this post as a "not a real question".
Kaveh
<lo.logic><automata-theory><formal-modeling>
Time-based mathematics or logic
Charles Moeller
# Time-based mathematics or logic

(Asked by: username_0 on 2011-03-27)

The conventional systems of logic can only recognize or perform operations that transform or translate values in space. Ordinary logic treats time poorly through the necessity of translating all temporal signs, signals, and effects into the space-domain so as to be suitable for space-only operations. Contemporary logic can easily assign state, but it is powerless to assign cause. Life is typified by activity. Humans have more than static existence: we experience and also cause dynamic processes in both space and time. How does one concisely express and treat the aspects of movement, change, cause and effect, and life with only static formal tools?

I propose a time-domain mathematics and logic, a method of automating physical processes that does not first require translation to the space-domain before applying logic.

**Tags:** lo.logic, automata-theory, formal-modeling

> **Comments:**
> **username_1**: I do not think that the title corresponds to the actual question.  Please correct the title.  
> **username_2**: Did you not post a slightly more ambitious question somewhere? Cannot find it anymore. What you write sounds a little bit cranky. How do you expect anyone to comment your approach without having seen it? I suggest proposing it in total to an expert audience of mathematicians and/or computer scientists. You can try submitting it to journals or conferences or use any community platform, as e.g. Stackexchange.  
> **username_0**: Yes, I did. Apparently taken down by the moderators.  
> **username_0**: Could we discuss the concept?  
> **username_0**: If you want the whole thing, you will have to wait for the book.  
> **username_3**: @Charles: you need to provide some details before this question can be answered. Otherwise it's just vapourware.  
> **username_4**: I am downvoting because this is hardly a question, and is most definitely not a question in theoretical computer science.  
> **username_0**: Fair enough. I withdraw.  
> **username_5**: I am closing this post as a "not a real question".  

---

## 3 answers

### Answer by username_3 on 2011-03-27. Score: 16

Both [temporal logic](http://en.wikipedia.org/wiki/Temporal_logic) or [dynamic logic](http://en.wikipedia.org/wiki/Dynamic_logic_%28modal_logic%29) are two modal logics that can be used to reason about changes in the world over time. [Coalgebra](http://en.wikipedia.org/wiki/F-coalgebra) is a formal approach to dealing with observations of state-based models changing in time. This approach generalises state machines, labelled transition systems, and so forth, which themselves provide behavioural models of systems. [Coalgebraic modal logics](http://arxiv.org/abs/cs.LO/0605128) are a generalised approach to defining logics for coalgebras, thus providing a systematic way of generating logics for reasoning about systems that change in time. [Logics of causality](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.75.8207&rep=rep1&type=pdf) even exist (I'm sure google will turn up even more).

> **Comments:**
> **username_6**: I was wondering about temporal logic: it seemed to do exactly what the OP wanted.  
> **username_3**: And there are of course many temporal logics, even temporal logics of action, to explore to find the most appropriate one.  
> **username_0**: username_3, All the "temporal" logics, save for mine, take place in the space-domain. Those conventional systems of logic can only recognize or perform operations that transform or translate values in space. Ordinary logic treats time poorly through the necessity of translating all temporal signs, signals, and effects into the space-domain so as to be suitable for space-only operations. Contemporary logic can easily assign state, but it is powerless to assign cause.  
> **username_3**: @Charles: Please improve your question by providing a reference to your logic and perhaps explain what it's core ingredients are. It is impossible for us to provide any answer to your question without any concrete details.  
> **username_7**: Generally one takes into account causation through some sort of substructural logic such as a linear or affine approach: http://en.wikipedia.org/wiki/Linear_logic  

---

### Answer by username_8 on 2011-03-27. Score: 3

Temporal logic seems to be exactly the thing you are looking for (it allows proving things where the truth of propositions can be qualified by operators such as 'in the next time', 'for always after now', 'at some time after now').

But you may want to look at some other formalisms, namely [Turing machines](http://en.wikipedia.org/wiki/Turing_machine), or [Operational semantics](http://en.wikipedia.org/wiki/Operational_semantics). These both treat state transitions as changes from one time to the next.

From your comments, it seems that none of these is sufficient for you since they translate the concept of temporal causation into the 'space-domain'. I'm not sure what you mean by this, but you can surely represent temporal causation in any of the above formalisms.


---

### Answer by username_9 on 2011-03-28. Score: -4

Brush up on your Einstein. Time and space are the same thing. <http://en.wikipedia.org/wiki/Space_time>

Okay so this wikipedia article doesn't do my statement justice as it states that the space-time model doesn't jive with Einstein relativity, but I disagree, Einstein relativity can be modelled in space-time it just requires reference domains based on the velocity of objects.

More to the point, relative time passes the "same" no matter at what speed you travel, however from a universal stand point the faster you go the slower you move through time, this only requires that your "velocity" in a 4D space is constant.

But I digress. Majorly.

> **Comments:**
> **username_10**: @username_9: This is not an answer to the question.  
> **username_9**: Sure it is, it's saying that the question is a waste of time. Logic as it exists now works fine in space and time, more over logic as it stands is not bound to space or time, it's completely independent of it. It would have to be, because if it weren't that would suggest that without space or time, there is no logic, so from what was the universe derived? You might as well say that it was the flying spaghetti monster, as the absence of any logic would state this is totally plausible.  
> **username_9**: V = !0  P(T = 0) = X  Given V != 0 Then P( T + 1 ) != P ( T )  T = T + 1 P(T) ?   This is very simplistic, but hey look I just simulated with simple logic that if a object with a position (P) and a Velocity (V) moving through time (T), it's position at one time cannot equal his position at another time. There I just did that. AMAZING.  
> **username_9**: To those who voted this down, please, enlighten me with a reason why logic as it stands is not fit to model logistics in the temporal domain. Further extra credit if you can tell me how time should be handled without using methods that use our old supposedly flawed logics, and "space-domains". Every model I know for time transforms it into a "space-domain." As far as I can tell there is no model for time that isn't somehow related to a extra dimension of space.  
> **username_3**: @username_9: I think the burden of proof lies with you and Charles to show us how the state of the art is flawed. Otherwise you both come across sounding unscientific, with high flung unjustified claims.  
> **username_9**: I'm terribly sorry, but how is it my burden of proof to disprove something someone else is claiming without any proof?  
> **username_9**: You've said a bunch of stuff in the original post with no proof backing it up. Why is transforming time into a space domain bad? Cause is represented in logics, it's called an implication. a -> b. where A is true, B is also true, This can be understood as A will cause B. Why exactly is logic restrained to a static domain? We have variables in logic, why don't we use thoes? Also all models of everything we know scientifically is based in these traditional logics. So what are you on about?  
> **username_9**: I was responding more to Moeller saying that we needed a whole new kind of logic and math to handle temporal domains as Conventional logics translates time into space-domain and that's somehow bad, but the problem with that is there is no way to represent time without fitting it to some metric which becomes comparable to space, and thus in the "space-domain". Now there of course exist logics that deal with time like the ones you later posted about, but they don't avoid space-domain. Your post is flawed. There is nothing wrong with logics as it stands.  
> **username_9**: Furthermore to all your cohorts who don't see the fallacies in your arguments, who keep down voting my posts, and slandering the conventional logic which all other logics and science itself derives. Seriously, what the hell is wrong with you?  
> **username_9**: Appologies: I apparently have become confused. username_3, we seem to actually be arguing the same thing. I don't understand how you came to think I was backing Moeller, but I wasn't. I am a proponent that Conventional Logic is just fine at representing time domain relations, we do not need anyone reinventing logic on some false belief that the space-domain isn't appropriate to express time-domain relationships which is what I've been vocally refuting here.  

---

