## Time and the law of contradiction

• 2.9k
The law of non-contradiction (LNC) is defined as the impossibility of a proposition being both true and false in the same respect and at the same time. In formal logic this is expressed as ~(p & ~p).

We all understand what we mean by "same respect" - the requirement for definitions or meanings remaining the same through the course of a discussion. The fallacy of equivocation is an example of violating this rule.

As for the requirement of "same time", in previous discussions I've had people on this forum deny this condition as part of the LNC. I, however, think time is crucial to the meaning of the LNC. For example, take the statements H = I'm happy and ~H = I'm not happy. ~H is the negation of H and the situation H & ~H is a contradiction, a violation of the LNC. However, I were to state H at 7 AM in the morning and ~H at 3 PM there would be non contradiction at all since H and ~H express states at different times. But stating H and ~H at the same time, say at 4PM, is a contradiction. Do you find anything wrong with this reading that time is necessary for the LNC to have meaning? If you do read on. Thanks.

Time is a strange creature and I'll refer to Zeno's paradox to illustrate the point, specifically his arrow paradox (AP). In the AP, the arrow moves through space and time but at any one instant or moment in time the arrow is NOT in motion. How do states of motionlessness add up to motion? This, I think, is the crux of the paradox.

From the above we can see that an instant/moment of time is meaningless or, at least, leads to unsolvable paradoxes. I suggest therefore that we give up the notion of an instant of time and always consider time to be an interval - a distance, so to speak, between two points of a clock.

If that is how we must view time then the LNC is meaningless because it requires the notion of an instant of time. Nothing happens in an instant/moment/single point of time.

All phenomena take a non-zero amount of time but the LNC depends on propositions in a timeless (zero-time) world which is impossible.

• 41
One thing to clear up is that formally (i.e. as a rule of logical inference) the LNC has nothing to say about time. It is simply a law that allows you to remove assumptions in an argument by drawing a formal contradiciton from that assumption (plus other assumptions/premises of your argument). One issue then becomes whether, under an interpretation of a formal system, one can have true contradictions, thus apparently rendering the formal rules inapplicable to that domain. Well, fairly early on in Arnold Bennet's novel "The Old Wives' Tale", Sophia's mother receives a visit from Sophia's schoolteacher, and the schoolteacher says something along the lines that she supposed Sophia's mother to be a little surpised by her visit, to which Sophia's mother responds:
"Well, I am and I'm not"
Is that a true contradiction? It seems to have the p & ~p form after all (not exactly, but it could be rephrased very simply to have that form, even if it would sound less natural : 'I am and it is not the case that I am', nobody (except philosophers) actually talks like that). Those who want to preserve the LNC under all interpretations will take the line that we need to make a distinction between what Sophia's mother says and what she implies by what she has said, and that ultimately the LNC will be concerned with the interpretation resulting from the implication (e.g Sophia implies: I am suprised in the sense that I did not expect a schoolteacher to visit me personally, rather than send a letter; I am not surprised in the sense that I knew that Sophia was having issues at school - here we no longer have p & ~p form). If that is going to be the technique to resolve all apparent violations of the LNC in an interpretation, then I think you are right that many many apparent violations will require us to assume that all statements are implicitly time-indexed. Now whether or not the time-index needs to be a specific instant of time, or could be a duration of time, I don't know - I need to see an argument saying that the notion of time-indexing a proposition commits one to specific instances rather than specific durations.
• 2.9k
Now whether or not the time-index needs to be a specific instant of time, or could be a duration of time, I don't know - I need to see an argument saying that the notion of time-indexing a proposition commits one to specific instances rather than specific durations.

I think we're on the same page. Mental objects seem to be, well, timeless. Over the course of a discourse we expect meanings of terms and words to remain constant in time.

So the notion of ''at the same time'' could mean anything from an hour to an entire year. I agree with you but only partially about time NOT being part of the meaning of LNC.

I think time is something that simply can't be ignored. It flows whether you acknowledge it or not and it's part of the definition of almost everything. For instance ''prove'' meant ''test'' in the good old days and that's the reason for much confusion over the adage ''the exception proves the rule''.

Anyway, on the whole, I believe there's an implicit consensus on definitions not changing over the course of some discussion whether it takes a minute or a decade.

Another thing. What do you make of the idea of an instant of time? Does it makes sense?
• 41

Another thing. What do you make of the idea of an instant of time? Does it makes sense?
I'm not sure. I want to say "of course an instant in time makes sense as being some specific point in the temporal continuum" - and to prove that these things exist, I might show you a video of some event or another and then press pause and say - what is represented by that frozen frame is a specific instant in time. Having said that, it does not seem to make sense to view the continuum itself as being made up of such point instants, even if they do exist, since no matter how many such instances you have, since they do not last for any amount of time, you are not going to create a duration of time by adding them together (0+0=0). Could there be a way of taking an instant in time not as a thing itself, but rather as a way of thinking of a duration of time?
• 41
I think time is something that simply can't be ignored. It flows whether you acknowledge it or not and it's part of the definition of almost everything. For instance ''prove'' meant ''test'' in the good old days and that's the reason for much confusion over the adage ''the exception proves the rule''.

I see a point here, but I wonder if it is one that need worry the "LNC is timeless" advocates. There is a notion of a proof as an activity which involves a series of steps, taken one at atime. There is also a notion of a proof as a kind of logical object incorporating all its steps at once. A timeless notion of LNC would presumably commit one to the latter kind of realism about proofs. An anti-realism about proofs - i.e. one that insists on a proof-as-activity conception - might require a temporal conception of the LNC even within formal logic. So, perhaps LNC-as-timeless commits one to the existence of abstract objects (proofs).
• 2.4k

The law of non-contradiction (LNC) is defined as the impossibility of a proposition being both true and false in the same respect and at the same time.

Perhaps time in the definition is an attempt to apply a formal law to our empirical experience of the world. The LNC is a formal rule, but in the empirical world the things we perceive are continually changing, becoming. By saying "in the same respect and at the same time", I think we idealizing the empirical, trying to treat it as if it could stand still.
• 677
The law simply says "in the same way". Time is simply one of the ways that might not be the same.
It is daytime, and it is not daytime can both be true if they are truths about different times of day. The two statements are not 'in the same way', so there is no violation of the law.

Relatvity allows one to say that event A happens before event B in one frame, but in another frame B happens before A. A precedes B and B precedes A, but in different frames, so they're not 'in the same way'. Again, no violation of the law of non-contradiction.
• 28
Well, points are also adimensional, but displayed in continuum succession they make a line, which is dimensional. A line, on the other hand, contains only one dimension, but displayed parallelly in a continuum succession they make an area, which is bidimensional. The same is valid for volume.
• 851
From the above we can see that an instant/moment of time is meaningless or, at least, leads to unsolvable paradoxes. I suggest therefore that we give up the notion of an instant of time and always consider time to be an interval - a distance, so to speak, between two points of a clock

Here is an argument that time is discrete. Consider:
- 1 second of time
- 1 year of time
According to the definition of continuous, both intervals must be graduated to the same precision and thus have the same information content. But 1 year has more information than 1 second - contradiction, so time must be made of discrete moments.

Continuous leads to paradoxes like Zeno's. Discrete is not paradoxical. In the Arrow Paradox, the arrow always has a velocity associated with it and so is never at rest so I don't see that as paradoxical.
• 34

Citing Kant's clear, unbelievably ignored, account on this topic

«Whatever the content of our cognition may be, and however it may be related to the object, the general though to be sure only negative condition of all of our judgments whatsoever is that they do not contradict themselves; otherwise these judgments in themselves (even without regard to the object are nothing. But even if there is no contradiction within our judgment, it can nevertheless combine concepts in a way not entailed by the object, or even without any ground being given to us ei- ther a priori or a posteriori that would justify such a judgment, and thus, for all that a judgment may be free of any internal contradiction, it can still be either false or groundless.
Now the proposition that no predicate pertains to a thing that con- tradicts it is called the principled of contradiction, and is a general though merely negative criterion of all truth, but on that account it also belongs merely to logic, since it holds of cognitions merely as cognitions in general, without regard to their content, and says that contra­ diction entirely annihilates and cancels them.
But one can also make a positive use ofit, i.e., not merely to ban false­ hood and error (insofar as it rests on contradiction), but also to cognize truth. For, if the judgment is analytic, whether it be negative or affir­ mative, its truth must always be able to be cognized sufficiently in ac­ cordance with the principle of contradiction. For the contrary of that which as a concept already lies and is thought in the cognition of the objecta is always correctly denied, while the concept itself must neces-
B191 sarilybeaffirmedofit,sinceitsoppositewouldcontradicttheobject.b Hence we must also allow the principle of contradiction to count as the universal and completely sufficient principle' of all analytic cognition; but its authority and usefulness does not extend beyond this, as a sufficient criterion of truth. For that no cognition can be opposed to it without annihilating itself certainly makes this principled into a con- AI52 ditiosinequanon,butnotintoadetermininggroundofthetruthofour cognition. Since we now really have to do only with the synthetic part of our cognition, we will, to be sure, always be careful not to act con­trary to this inviolable principle, but we cannot expect any advice from it in regard to the truth of this sort of cognition.

There is, however, still one formula of this famous principle, al­though denuded of all content and merely formal, which contains a syn­ thesis that is incautiously and entirely unnecessarily mixed into it. This is: "It is impossible for something to be and not to be at the same time." In addition to the fact that apodictic certainty is superfluously appended to this (by means of the word "impossible"), which must yet be understood from the proposition itself, the proposition is affected by the condition of time, and as it were says: "A thing = A, which is some­ thing = B, cannot at the same time be non-B, although it can easily be both (B as well as non-B) in succession." E.g., a person who is young cannot be old at the same time, but one and the same person can very well be young at one time and not young, i.e., old, at another. Now the principle of contradiction, as a merely logical principle, must not limit its claims to temporal relations.' Hence such a formula is entirely con­ trary to its aim. The misunderstanding results merely from our first ab­stracting a predicate of a thing from its concept and subsequently connecting its opposite with this predicate, which never yields a con­tradiction with the subject, but only with the predicate that is combined with it synthetically, and indeed only when both the first and the second predicate are affirmed at the same time. If I say "A person who is unlearned is not learned," the condition at the same time must hold; for one who is unlearned at one time can very well be learned at another time. But if I say that "No unlearned person is learned," then the proposition is analytic, since the mark (of unlearnedness) is now com­ prised in the concept of the subject, and then the negative proposition follows immediately from the principle of contradiction, without the condition at the same time having to be added. This is also then the cause why I have above so altered the formula of it that the nature of an analytic proposition is thereby clearly expressed.» Critique of Pure Reason(Guyer Wood) B190-193
• 34

I might show you a video of some event or another and then press pause and say - what is represented by that frozen frame is a specific instant in time.

This presupposes the vision of other frames, because if you have a vision of just one instant you can not say anything different except spatially.

it does not seem to make sense to view the continuum itself as being made up of such point instants,

This is correct, but not because continuum is not made up of point of instants, but because the Continuum is a logical measure(in set theory: the cardinality of the set of real number) while instants are physical measure(in reference to an arbitrary chosen velocity).

since they do not last for any amount of time, you are not going to create a duration of time by adding them together (0+0=0).

Here's the confusion between logic and physics: instants have duration to us, arbitrarily chosen in respect to some velocity. Moreover there is a confusion between physics and maths: instants are not numbers, and time is not a series of instants, but it is a physical(thermodynamical) effect due to our interaction with the physical world, in respect to the value of what the Boltzmann parameter designates: eggs break but do not unbreak, because our interaction with what Boltzmann parameter designates results in a value big enough to influence our mode of perceiving things(i.e. a mode which distinguishes irreversible events). It is treated as a series, improperly speaking, in calculus, to calculate velocity, and in this case it presupposes space, and moreover PHYSICAL space, since velocity is the distance run divided by the time employed to run it.

Could there be a way of taking an instant in time not as a thing itself, but rather as a way of thinking of a duration of time?

As I said, Time is not duration(a property of things) but an effect(a mode of perceiving things. Duration presupposes identity, because to establish that something is the same thing, but in a successive instant in time, you need logic, and a subjective criterion(structurally objective) of determining an object. This presupposes a referential structure, in which a subject is distinguished by the object which the former is related to and in such a way the the perceiving maintain this distinction recognizable.
• 93
As for the requirement of "same time", in previous discussions I've had people on this forum deny this condition as part of the LNC. I, however, think time is crucial to the meaning of the LNC.

I don't think temporality is any more implied in the formalized LNC than it is, say, in the proposition that "2+1=3". You could read this to say something like: "if you start with two and add one you get three," and then go on to claim that because you move from two to one time is somehow implicit. While certainly it's not inconsistent with the identity to say something like that, one of the goals and advantages of formal systems is that they strive for efficiency. So if you don't need a conception of time in order to do the calculations (except, of course, practically speaking in the sense that it takes time to mentally calculate), whether mathematical or logical, why introduce it?
• 5.2k
So if you don't need a conception of time in order to do the calculations (except, of course, practically speaking in the sense that it takes time to mentally calculate), whether mathematical or logical, why introduce it?

The answer to this is that time is what grounds the validity logic, making it sound. Sure, you don't need a conception of time to carry out valid logic, but what's the point if your conclusions are not sound? If we do not qualify the LNC with "at the same time" it becomes unsound because it's clearly true that one thing can have a property then not have that property, as time passes. Likewise, "2+1=3" loses its soundness if it is not grounded in something like "if you start with two and add one you get three", that's what makes it true. There is always a temporal connotation in logical principles, and without it logic gets divorced from truth.
• 93

I'm not convinced that's the case. Perhaps some of the issue is that the formalized LNC, -(p & -p), doesn't have anything explicit to say about properties, only propositions, and so is more akin to the math expression. In the case of the math expression, "2+1=3", I really don't seen any temporality at all. You could read it the way I was proposing, but I think a more natural reading just sees the expression as an identity that holds absolutely and without regard to temporal sequence. I don't see how a things identity with itself necessarily implicates time.

If you claim is more of a metaphysical one - that time is inescapably implicit in any claim about anything whatsoever - such that we just can't think anything unless we assume time is present, I guess it's true but probably tautological. It would be like a Kantian category: a condition of thought itself. I wouldn't take that to be a proper subject for propositional logic, though.
• 5.2k
I'm not convinced that's the case. Perhaps some of the issue is that the formalized LNC, -(p & -p), doesn't have anything explicit to say about properties, only propositions, and so is more akin to the math expression.

Even when the LNC is taken to be about propositions, there is a necessary temporal qualification. Opposing propositions cannot be "at the same time" true. As I said, without that it loses its grounding. This is what allows the relativity of simultaneity to work. Depending on the frame of reference, "X occurred at precisely 10:00" might be true or it might be false because "at the same time" is reference dependent.

In the case of the math expression, "2+1=3", I really don't seen any temporality at all. You could read it the way I was proposing, but I think a more natural reading just sees the expression as an identity that holds absolutely and with regard to temporal sequence. I don't see how a things identity with itself necessarily implicates time.

I don't see how 2+1=3 is an expression of identity. You have identified three distinct things, "2", "1", and "3". The "+" symbol says that you add the first two things together, "2" and "1". The "=" symbol says that these things added together are by some standard equivalent to the third thing, "3".

If you claim is more of a metaphysical one - that time is inescapably implicit in any claim about anything whatsoever - such that we just can't think anything unless we assume time is present, I guess it's true but probably tautological. It would be like a Kantian category: a condition of thought itself. I wouldn't take that to be a proper subject for propositional logic, though.

That's right, it is "implicit". But since we're discussing logic, whatever is implied is relevant. So we can't just dismiss the proposition that temporality is implied within logical principles as irrelevant. Even if it is tautological, if it is true, it would be an important ontological principle
• 93
I don't see how 2+1=3 is an expression of identity. You have identified three distinct things, "2", "1", and "3". The "+" symbol says that you add the first two things together, "2" and "1". The "=" symbol says that these things added together are by some standard equivalent to the third thing, "3".

You haven't really identified 3 distinct things. You've only identified one thing, the number 3, and the fact that "2+1" is identical to it, or just another way of describing it. In other words, it's just the claim that the expression "2+1" and the expression "3" are two different ways of saying the same thing. It's similar to Frege's evening star/morning star example, albeit analytical and not empirical in nature. If you don't see how that's an identity statement having learned arithmetic, I'm not sure I can explain it any more clearly.

whatever is implied is relevant

It's also implicit in logic that there are people around who can think, but no one believes that particular implication has any place in the formulation a formal logical system. Similarly, logic is empirical, but you won't find that assumption anywhere in the axioms or rules, although you might find a reference to it in the introductory part of a logic book. That's because the subject logic takes for itself isn't metaphysics, ontology, epistemology or whatever, but the rules that constitute the limits of proper reasoning. The fact that time or space are necessary for doing logic (or anything) at all aren't rules that govern reasoning, they're preconditions of it, and so, in fact, are not relevant to what logicians takes their subject matter to be. (Leaving aside the fact there are temporal logics, which I take to be the exception that proves the rule.)

That all said, there is a paradox here, because when we get to discussion metaphysic, ontology, etc. the assumption I think most people make is that our reasoning should be logical. In that case, whatever we say on a metaphysical level is going to be constrained by what we think logic requires. However, when we work out what logic requires, we do so with a naïve, inexplicit understanding about the nature of time. So there is a circularity which is what you might be worried about, but I guess most people would just say it's not vicious and be content to live with some degree of fundamental paradoxality, especially given the proven practical benefits and results logic produces being just what it is now.
• 198
If that is how we must view time then the LNC is meaningless because it requires the notion of an instant of time. Nothing happens in an instant/moment/single point of time.

The LNC can be expressed without instants of time just fine. Take the statement "I am happy during the interval of 1pm - 2pm". This statement is incompatible with the statement that "I am sad during the interval of 1pm - 2pm" as that would imply that "I am not happy during the interval of 1pm - 2pm". A clear contradiction as both statements overlap one another and require inconsistent ideas to hold during the same period of time. This can all be stated even in a world where time is gunky.
• 5.2k
You haven't really identified 3 distinct things. You've only identified one thing, the number 3, and the fact that "2+1" is identical to it, or just another way of describing it.

No, that's the point 2+1 is not identical to 3. Nor is it a different way of describing the same thing. Do you know what equivalent means? It means have the same value. These two things, "2", and "1", together have the same value as this thing "3".

It's similar to Frege's evening star/morning star example, albeit analytical and not empirical in nature. If you don't see how that's an identity statement having learned arithmetic, I'm not sure I can explain it any more clearly.

No, this case is not analogous, as it is a case of using two different names to refer to the same thing, it is not a case of giving the same value to different things. Do you see the difference?

It's also implicit in logic that there are people around who can think, but no one believes that particular implication has any place in the formulation a formal logical system.

That's clearly false, because I surely believe it. And, believing that human thought is what formulates formal logical systems, I really can't understand how you could truly believe that human thought doesn't have any place in the formulation of formal logical systems

That all said, there is a paradox here, because when we get to discussion metaphysic, ontology, etc. the assumption I think most people make is that our reasoning should be logical. In that case, whatever we say on a metaphysical level is going to be constrained by what we think logic requires. However, when we work out what logic requires, we do so with a naïve, inexplicit understanding about the nature of time. So there is a circularity which is what you might be worried about, but I guess most people would just say it's not vicious and be content to live with some degree of fundamental paradoxality, especially given the proven practical benefits and results logic produces being just what it is now.

I think, that the appearance of "a paradox" is just a function of your extremely naïve and unrealistic understanding of what logic is.

This can all be stated even in a world where time is gunky.

Gunky?
• 198
Gunky?

Means it doesn't have instants, no fundamental units. Gunky space has no points either if you're wondering.
• 5.2k

It's continuous, as opposed to discrete? "Gunky" seems to imply a mixture of both.
• 198
It's continuous, as opposed to discrete? "Gunky" seems to imply a mixture of both.

I don't define the terms. I would've liked to refer to it as continuous, but there are also continuums that contain fundamental units in the form of dimensionless points/instants which should be contrasted with that of gunk.

In any case, this discussion is getting off-topic.
• 6.8k
From the above we can see that an instant/moment of time is meaningless

I agree with that. In my view time is identical to motion or change. The idea of a "frozen slice of motion/change" doesn't make any sense. It's not motion or change if it's frozen.

Motion change of any particular entity is always relative to other motion or change. So it only makes sense to speak of divisions in terms of relative changes.
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