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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.

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  • $\begingroup$ I do not think that the title corresponds to the actual question. Please correct the title. $\endgroup$ – Tsuyoshi Ito Mar 27 '11 at 15:48
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    $\begingroup$ 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. $\endgroup$ – Raphael Mar 27 '11 at 16:26
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    $\begingroup$ @Charles: you need to provide some details before this question can be answered. Otherwise it's just vapourware. $\endgroup$ – Dave Clarke Mar 27 '11 at 19:15
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    $\begingroup$ I am downvoting because this is hardly a question, and is most definitely not a question in theoretical computer science. $\endgroup$ – Andrej Bauer Mar 27 '11 at 22:04
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    $\begingroup$ I am closing this post as a "not a real question". $\endgroup$ – Kaveh Mar 28 '11 at 22:56
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Both temporal logic or dynamic logic are two modal logics that can be used to reason about changes in the world over time. 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 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 even exist (I'm sure google will turn up even more).

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    $\begingroup$ I was wondering about temporal logic: it seemed to do exactly what the OP wanted. $\endgroup$ – Suresh Venkat Mar 27 '11 at 17:27
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    $\begingroup$ And there are of course many temporal logics, even temporal logics of action, to explore to find the most appropriate one. $\endgroup$ – Dave Clarke Mar 27 '11 at 17:36
  • $\begingroup$ 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. $\endgroup$ – Charles Moeller Mar 27 '11 at 18:27
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    $\begingroup$ @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. $\endgroup$ – Dave Clarke Mar 27 '11 at 19:12
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    $\begingroup$ Generally one takes into account causation through some sort of substructural logic such as a linear or affine approach: en.wikipedia.org/wiki/Linear_logic $\endgroup$ – sclv Mar 27 '11 at 19:13
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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, or 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.

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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.

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    $\begingroup$ @acp10bda: This is not an answer to the question. $\endgroup$ – Hsien-Chih Chang 張顯之 Mar 28 '11 at 8:29
  • $\begingroup$ 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. $\endgroup$ – acp10bda Mar 28 '11 at 8:45
  • $\begingroup$ 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. $\endgroup$ – acp10bda Mar 28 '11 at 8:49
  • $\begingroup$ 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. $\endgroup$ – acp10bda Mar 28 '11 at 14:54
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    $\begingroup$ @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. $\endgroup$ – Dave Clarke Mar 28 '11 at 19:48

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