# Can logic be without time?

I think logic is dependent on time.

My reasoning is that all of the basic logic concepts are based on axioms that are observations in time (so basic that they do not require proof). This then leads to the conclusion that logic only works in time, and that if time would not move in a one directional stream but instead would move at random, sometimes forward, sometimes backwards, one would not be able to use these axioms and the whole thing would fall over.

EDIT: For example, if we want to take a measurement of something, we take it while time is moving forward. The result comes out as x. Then suppose we take the same measurement while time is moving backwards. If we get a different result, it would imply that to do the calculation we would need to know in which direction time is moving. (The direction of time would become an input parameter into the function.) Now if we get the same result, no matter in which direction time is moving, it would imply that it is not dependent on time. I do understand that there is no way to move in time backwards to take the measurement, but that's what philosophy is for.

Can anyone come up with an example where logic would work independent of time?

• "all the base logic concepts are based on axioms that are observations in time" : can you explain better what are you meaning ? What kind of observation ? – Mauro ALLEGRANZA May 12 '14 at 13:19
• Sorry, I do not see the connection with your statement ... – Mauro ALLEGRANZA May 12 '14 at 13:31
• @MauroALLEGRANZA these axioms are basis for all the logic. For example if you have a thing 'x' it is equal to itself, but this observation always been done by someone who was making observation while being in time moving forward. – Matas Vaitkevicius May 12 '14 at 13:41
• Assuming that we can "imagine" some kind of "observation" which ground the logical law \$x=x\$, on what basis we imagine that if we move backward in time we can perform a different type of "observations" from which conclude that "x=x" is not valid ? – Mauro ALLEGRANZA May 12 '14 at 13:53

As soon as classical logic tries to talk about the physical world, it needs time or a similar "trick". Is this only a "problem" of classical logic, or will any logic be affected by this "problem"? Jean-Yves Girard invented linear logic (and proof nets) to address this problem. It turns out that linear logic is successful in this respect, but why should we consider it to be a logic? Girard argues convincingly that it is indeed a logic, because it has a complete deduction system and a cut-elimination theorem. (He also mentions that classical logic and intuitionistic logic can be embedded into linear logic, and that linear logic can be embedded into classical logic by using time.)

There is causation in linear logic, but no (linear global) time. So logic can be without time! Can logic also be without causation? Good question... Intuitively I would say no, but I might be wrong.

• morally equivalent trick? or time as a moral concept? – virmaior May 14 '14 at 23:59
• @virmaior Sorry, I inadvertently used mathematical slang. See this article by Eugenia Cheng on morality in the context of mathematics. I guess I should just write "equivalent" instead of "morally equivalent". Let's see whether browsing Eugenia Cheng's article will help me find a more appropriate English expression for this concept. – Thomas Klimpel May 15 '14 at 6:51
• Okay, that makes a lot better sense now (skimming part of the article)... – virmaior May 15 '14 at 7:05
• Can you explain how "causal" is meant in this context? – sjmc May 15 '14 at 15:58
• @sjmc I mean it in the way Jean-Yves Girard uses it when he talks about linear logic. One thing he says is that you can think of implication as a causal process. You put some things in, and some other things come out. But that's not the whole story, and I don't want to misrepresent Girard's views. I will read the corresponding passages again more carefully later, and then try to explain it. – Thomas Klimpel May 15 '14 at 16:51

I don't think this is correct. Formal logic, like mathematics, is typically atemporal, it deals with structural relationships, not progressions. For example, the logical statement IF A THEN B may sound like something that takes place in time, with A happening first and B happening second, but in actuality it just means that in the case that A is true, B must also be true.

• I also often hear people say that the number π goes on forever (almost surely imagining a process of more and more decimal digits appearing). It doesn't go anywhere, it just is. – user132181 May 12 '14 at 15:28
• Logic is causal, or at least a logic "should" have a causal deduction system. This is different from set theory (as practiced today), where "causal" is replaced by "consistent". Of course one can debate about whether a logic must be causal. Think about why Quine argued that second order logic is not a logic at all, and whether you agree with him. – Thomas Klimpel May 15 '14 at 8:05
• You are too fast. Is evaluation of A or B being true - temporal? Requires time? Calculation? You as a being have no timeless experience -> so non of what you say about timelessness should be too creditable. True? – Asphir Dom May 15 '14 at 20:38
• Sure, thats one way of interpreting that statement ('if then..'); and has been very successful in classical logic. But I think its imporant to note that it can be intepreted as time, and then we ought to get a new way of thinking about logic. – Mozibur Ullah Jun 3 '14 at 21:03

Thomas Campbell, a former NASA physicist, who became a consciousness researcher states that time is not fundamental, at least not the time we experience here in this universe. It comes down to this: is the time in a computer game fundamental? No it is not, it can be paused, rewound or reset. The time in the game is the experience of the gamer of the difference between the results of calculations. The game is no continuous fluid process, but it is a iterative process (results of calculations). In this perspective, the hardware and software is more fundamental than the time the gamer experiences. Mr. Campbell states that what we live in is a calculated reality as well, just like that game. Our time is therefore not fundamental, because there are processes going on that do the calculations, just like in the game, that make us experience time. Those processes are more fundamental than the time we experience.

If, what we live in, is a virtual calculated reality. Than the big bang was the moment the game was loaded and began to play. Before the initializing of the game (the big bang) modules were loaded, memory was allocated, processing power was put to use, but our time didn't start yet, not until the big bang / the game was initialized.

So my answer would be: Yes there is logic without time, it depends on which virtualized layer you are at, and at which calculated layer you are looking upon. If you look at the time of the game, that time is not fundamental to you, you can pause the game e.g. so you are more fundamental than the time of the game. Mr Campbell states there are more fundamental processes than the space/time universe we experience. Your consciousness is more fundamental than the time(and space) it experiences here.

This may seem far fetched, it might even be unsettling to think about, as it may make you feel infinitely small, and it may wider your experience-able reality.

• You skip quite fast across different levels of calculated reality. In a first step, our brain "calculates" our individual reality. In of the next steps, our culture "calculates" our observable universe. Some further steps down, the cosmos "calculates" what we can observe. And many steps further down, the cosmos itself is "calculated" by some computer. But why should an individual be able to see much further than its culture, or a culture to see much further than its observable universe? I don't think that it can, so the observable universe is the limit for the individual. – Thomas Klimpel May 15 '14 at 8:36
• @ThomasKlimpel What do define by observable universe is probably the physical universe. But there is more to reality than the physical universe. Physical universe is part of a larger reality, not vica versa. Our brains are merely an interface, they are not the root. If you suffer brain damage, the interface is broken, therefore, it can no longer bring expression to this physical universe. But the other end of the interface, not belonging to this physical universe is still part of the reality you live in. – Mike de Klerk May 15 '14 at 13:13
• @ThomasKlimpel Why would a culture calculate our observable reality? Considering Einstein realitivity, you are always the center of your observable universe due to the speed limit of light. What does our culture has to do with that? It is true that the culture you live in biases (or influences) your perspective on reality. – Mike de Klerk May 15 '14 at 13:15
• I just said "calculate", because you used that terminology. Just like our brain also has eyes, ears, as well as touch and other sensors in addition to its "raw computational power", our culture also provides powerful sensors (like space based telescopes and high energy particle colliders) in addition to its "raw computational power". At least own perception of the observable universe is much more influenced by the "information" provided by my culture, than by the "information" my own brain and its sensors provided to me. So my comment was a way to agree with your answer in a down to earth way. – Thomas Klimpel May 15 '14 at 14:01

Arthur Prior's temporal logic studies the relation of time to logic, a question that even Aristotle studied (future contingents). Thus, your question could be rephrased: "What are statements in temporal logic that are true for all time?"

There is no time or causality in logic or mathematics. So, "if A then B" does not mean that A causes B. Or even that A precedes B in time.

I find it helpful to think of the world of logic and mathematics as a kind of book of randomish letters. Suppose that every "A" is directly followed by a "B". You wouldn't say that A's "cause" the next letter to be a "B". Or that A's "precede" B's in time. You could say that if a letter A is found then the next letter will be B. Or that, you cannot have any letter other than B following an A. These things would also be considered to be true if there were no A's at all in this book.

EDIT 1

For a more "real world" analogy that occurred to me: When analyzing conditional but non-temporal and non-causal relations in real world phenomena, imagine what you might observe in snapshots of that phenomena presented in no particular order, with no indication of the passage of time. Determine what observations are associated with one another from these snapshots alone.

EDIT 2

See my math blog posting, "The Drinker's Paradox" (dated June 3, 2014)

• I agree that mathematics is not causal, but that is handled by set theory. The deduction system of a logic is causal, or at least a sequent calculus with cut-elimination and the subformula property is causal. Higher order logic with standard semantics is not causal, but there is disagreement whether it really is a logic, precisely because of its failure to be causal. – Thomas Klimpel May 20 '14 at 21:27

Just because a statement of the form IF A THEN B, or p > q, does not necessarily describe two events or situations occurring in succession, does that also go for one of the form IF A THEN NOT A, or p > ¬p ? How can the two propositions in this expression be simultaneously true?

This is the crack in formal logic through which we can glimpse the river of time. Nevertheless, the expression is not seen as a nonsense or self-contradictory in standard propositional calculus - check with a truth table. And moreover along with its converse, ¬p > p, it forms part of two basic theorems:

(p > ¬p) = ¬p, (¬p > p) = p

These are sometimes known as the "Paradoxes of implication", but nevertheless have a soundly rational interpretation as a fundamental logical principle. When a proposition is followed by its negative, the connective > ¬ (or reverse) becomes an updating or correcting function. The latest assertion, the one you end up with, is the one you want acknowledged as the final say on the matter, however much at odds it is with what went before. It's the latest, corrected version of your essay you want accepted, not the earliest. When you see a circus poster with a "cancelled" sticker across it, you assume it is indeed cancelled as of now if that negating sticker was appended after the poster was put up.

The relative position of a proposition in a sequence is not normally taken to indicate relative position in time. Once they're written on the surface of a page, all propositions in a formal argument, like all lines in a static diagram or graph are present to the viewer simultaneously. Can we have logic without time? Yes, if we can have a geometry without time. But time in formal logic, as in geometry, is the hidden extra dimension.

There is "space" and "time" so "logically" (according to Kant) there is a sequence...meaning "from one follows the other." Could be a didactic problem..."where one thing another must follow" because even in the negative ("where one thing not the other must not") the ordering is not reversed just the implication. Knowing the latter I would think gave rise to "rhetoric" in Ancient Greece...the "aposite" of logic...meaning the "argument" flowed in the same direction but was always in the negative...critically "allowing the observer to draw their own conclusion."

In other words the fact that someone other than the "auto-didact" gives the answer is the "proof." But the only scientifically observable phenomenon is that there was an "ask to" and an "answer from." I believe in Computer Science it's called a "query" for logic and a "language" for rhetoric.

"Logisticians" then go running off to explain "temporal logic" which I would imagine means "thought in time" but in our modern society I would argue we see this as "the punch line" meaning what's leads us to a laugh, a shout, to cry, etc.

This is not the purpose of either logic or rhetoric however. The purpose of the former (logic) is to focus the mind and the latter(rhetoric) to focus the mouth...but I would need an expert on the Ancient Greek language in order to agree to these suppositions. In other words what is the physical "shape" of the term "logic" in Ancient Greek? What is the physical shape of the term "rhetoric"?

Does the actual " figure of the speech" convey a meaning as well?