In fact, the truth-or-falsity of a proposition often be expressed into dynamical transition, then a continuous true-valued function must be needed in logic, hence we require a logic to explain this situation.

If I have understood right, it depends on the question. There are many questions which one can answer clearly for example is it raining today? And there are also questions which one cannot answer clearly just by a probability: Is this girl beautyful?

The first type of questions is well known and the second ones can be studied via the fuzzy logic.

I can give some hints via category theory:

The theory of elementary topoi has geometric aspects but it gives also models of intuitionistic logic. The subobject classifyer (see wikipedia) Omega can consist of two points {true, false} but it can also be a segment of reals [0,1] (where 1=true and 0=false) as in intuitionistic fuzzy logic.

Some spaces may contain one point only though they are not reduced to a terminal object (See the book Johnstone, Stone spaces) In this case there is a single truth value true. This would need deeper investigations though.

When thinking about it I wondered about a logic where the set of truth values would be a circle with a distinguished point true. What sort of logic would correspond to such a subobject classifyer?

Dear Marc Carvallo ,

I just re-read your question. I will add some aspects, as a novice in philosophy:

I wonder if you would think with Heraclitus that there are two fighting principles or with Parmenides that finally everything stays the same and is only "deformed".

Presocratic thinkers have proposed different answers to similar questions to yours.

Dear all,

Thanks to all answers and suggestions. Marc.

Sorry to be arch-but it does come down to the meanings of truth and falsehood. After all, they are never straightforward. It is true the sun shines as that is a common statement in that sun has equivalents elsewhere as does shine. The sun glares is really a metaphor on one level but not another. If this experience is shared elsewhere, then it has truth, if it isn't necessarily true.

There exist also fuzzy logic or tri-state logic. The truth is more complex than right or wrong.

... and there are also superimposed states, between truth and falsehood, like the cat of Schrödinger in the box. And statements are also which are not true and not false, see the problem of Gödel. The question of the truth is very complex.

As far as we are working in intuitionistic logic the notion of Kripke semantics is adequate and complete for provability. When we work in the setting of an elementary topos ( I guess there are some containing the category of hermitian manifolds ) the notion of Kripke semantics has been extended to the notion of Kripke-Joyal semantics. If you need some references consider MacLane-Moerdijk, Sheaves in Geometry and Logic, A first introduction to topos theory. This is a somewhat technical answer and I apologize for this to those not used to this language.

Let me express also my interest in the question of language semantics which has much more subtle aspects than only technical ones.

Consider for instance the famous paradoxes like the liar paradox...

An additional aspect is the following one: Problems of truth or falsehood are often related to the question how you treat double negations: e.g. 'It is not true that I said no', would not be equivalent to have said yes. I could have said nothing at all...

Some people say the trees are standing and others say the trees are moving and everybody are right. What is the truth then?

An experiment, try it out: put lukewarm water into a glass. First hold your hand in cold water. How do you feel the lukewarm water. Then hold your hand in hot water for a while. And now how is the lukewarm water?

Truth might be observer-dependant. A sort of relation between the observer and the object. But mostly we exspect truth to objective and not related to a point of view.

There is a duality between observer and observed in any case.

Dear Dr. Krantz,

I must be arguing with you. The truth mostly depends on the observer - at least in everyday life. I have a personal experience about it: earlier I was very slim. I saw my friend fat. Then I gained 20 kilos and I saw her very slim but she did nor change. If someone would have asked me - according to the fuzzy logic - to get a statistical line, how I see my friend, I would have answered him in different ways in different times.

Dear Dr. Szöke,

There is also an adequate and complete Kripke-semantics for temporal logic if I remember well.

To get back to the duality subject-object, I believe truth or opinion can only be known through a sort of duality bracket question-answer with values in the space of logical truth values. The observable may change and also the observed. The truth value could evolve according to certain principles (information gain e.g.) For instance getting an answer to a question fixes somehow things. When you open the box you see in which state is Schroedingers cat e.g. and this fixes somehow its state, right?

The observer and the observed get linked though the common knowledge of what is the answer to the question asked (or the observation made).

There should be special immutable observables corresponding to classical truth.

Dear Dr. Krantz,

In the story of the Schrödinger's cat there is a little radioactive material in the box near to the cat, which starts a process to kill the cat (toxic gas, knife over the cat or I do not know what). This little radioactive material decomposes randomly and it is very-very sensitive. If you open the box, you let in it some air, some light, etc. which changes the decomposition process of this material and you will get maybe a different result as if you had not opened the box. However, in the world of quantummechanic, every measurement is an intervention into the state of the object and it is irreversible. It means that if you would like to know the place of a flying electron you cannot measure it twice. So one assumes that each electron is the same and one uses many electrons to get the distribution function of the place of a single one. Or one can use more the same detectors to get a probability about the place of the electron. In everyday life one might not accept that all people (like detectors) are the same if one ask them about a question. So in each answer there is a duality between the object studied and the answerer and a verdict about a person or a thing can be never authentic.

I think classical intuitions will not lead in the case of quantum mechanics to a valuable representation. I explain as far as I am able as a non professional physicist. Please feel free to correct if I am mistaken or mix up some things.

1) Young's experiment with material particles like electrons proved that they exactly behave as light waves to some respect: they produce interference, which cannot be explained by a classical model.

2) Schrödinger's experiment with the cat is intended to prove that also macroscopic objects can be in "superposition" of different states.

3) With respect to me this state is not already determined and I only just don't know it yet, or it could change when I open the box. Its reality is that it is in a superposition of states like a wave is composed of different vibrations corresponding to eigenfrequencies so to say.

4) There is Einstein-Rosen-Podolski's experiment with entangled particles moving far one from the other. When you measure the spin of one you immediately know the spin of the other, though they might be far from each other. Some tried to explain this by the presence of hidden variables. Alain Aspects experiments and a statistical analysis proved that the idea of hidden variables had to be rejected: The particles are really in a superposition of states until a measure is done.

5) In a funny way this does not contradict the interdiction of transporting information quicker that light speed. It can be used though for cryptographic uses.

6) the subobject classifier {true, false} would correspond to classical mechanics and and a complex truth value with norm below 1 to quantum mechanics...

To make a suggestion, without any proof that what I am saying is correct, I suggest that there could be a logic with only one truth value: true, and any other statement would be undetermined. I fact I believe this is essentially intuitionistic logic. (see wikipedia)

(It makes me think of the idea that evil in the world is sometimes defined to be the absence of good.)

Dear Dr. Krantz,

I am a teacher, not a physicist but:

1. It is right

2. False, a macroscopic object cannot be in a superposed state.

(What do we call macroscopic? - a fuzzy question again) The cat is always in a concrete state but we do not know in which one. (E.g. a cat will never interfere with itself.) The problem is in the cat's case that if we open the box we cannot decide if the cat was alive or dead before the opening. This problem shows how astounding can be a behaviour in the macroscopic world which is natural among microobjects.

3. If we measure a microobject to know the shape its probability function we destroy even this state by the measurement - it is a paradox.

4. Sorry, I do not know too much about the EPR paradox, but I think you are right. It shows that there is not any hidden parameter behind the QM.

5. I have heard about particles which are faster than the light: the tachions. One must match the theory about the information propagating faster than the light with the relativity theory.

6. Yes, I think it is right. However, inside of classical mechanic there is an area: the statistical mechanic and thermodynamic which also deals with probabilities. The Coppenhagen interpretation of the quantummechanic is based even on this area of the classical mechanic. (E.g. by measuring many electrons we get the state function of a single one.)

7. In the fuzzy logic the probability function is given and one does not ask how one has had it. It can also come from inaccuracy of the measurements.

I am sorry I still have to contradict (partially). I apologize to the forum if this is maybe a bit outside the range of the initial question.

"Macroscopic" objects can be in a state of superposition:

For small ones consider Bose-Einstein condensates. Ok they are are small, but I heard about experimental proof of superposed currents in superconductor rings which are quite large.

So far so good, you would argue that these are not macroscopic. I think you are right about the fact that one cannot observe a superposition of states. This is due to the fact that any observation fixes the state and so the superposition is destroyed. This is similar to electrons around a nucleus: As long as you don't observe them they are in a state op superposition but when you do you measure either their position or their speed or spin etc. you fix the state.

One thing that is true, and it is may-be what you mean, is that the Planck constant is very small so that the product of the incertaincies of the position and the speed of macroscopic material waves could be considered neglectable, but Schrödinger's thought experiment proves that the effects can be macroscopic...

I should learn something about statistical QM also.

Dear all,

Thanks for all answers so far. Marc.

Dear Dr. Krantz,

You are right. In a Bose condensate there are coherent atoms in the same lowest energy state which can interfere. Now it is possible to produce a condensate from more than 500000 atoms and the number of atoms can be increased. These are not localized because their momentum is very close to zero, so the diameter of the condensate is 10 micrometer, so it is "big". One must use magnetic trap to get a compact condensate. When I learned about it, it was possible to make just from a few hundred atoms a condensate. Now the experiences are nearer to the macroscopic objects. It is very interesting that one has measured the refractive index of a condensate and one had got the value 17 600 000 what means a light speed about 15-16m/s in the condensate.

In macroscopic objects e. g. among 10^23 atoms the microscopic states are different and the conditions of the interference are not satisfied. For example if we would like to have an interference pattern by a usual light beam (the photon is boson, too) it is possible only if we share the beam and then we unite it again. If we use different lamps we do not experience any interference. The laser beam behaves differently because the laserlight is almost monochromatic and so it is coherent.

I have found a video in the youtube (about the condensate). At its end scientists are talking about quantumcomputers where the two logic states: zero and one can occur at the same time. It is very interesting, but I do not know too much about quantumcomputers.

https://youtu.be/EK6HxdUQm5s

Dear Eva,

Your answer leads me to ask you 2 questions:

(i) is the truth or the falsehood?

(ii) is identical to identity or not?

Thanks for your answer upon (i) and (ii). Marc.

In mathematics fields play a special role:

You know the field of the real or the complex numbers, there is also the field F_2 with two elements {0,1}. But there is also the one element field which behaves more generally and is a sort of generalization of the notion of field. I should look up this again, there is a technical text on wikipedia if you look up

"Field with one element"...

Scientists have much fun when they do their research, engineers have much proud when they turn it into applications. New inventions, before the enthusiasm has boiled down, often lead to much damage, because one does not know how to behave with. Legal texts don't keep up with the track etc. etc.

Dear Dr. Carvallo,

i) Yes. It is truth and falsehood, see the work of János Neumann.

ii) I just guess that it is not identity but it means that the quantum object is in a mixed state.

{0,1} means the binary system. If there is an electric sign in the circuit or there is not.

Dear Dr. Krantz,

Could you show an example for one element field and for its application?

F_1 is a bit unusual. If you want to represent it by sets it has in fact two elements:

0+0=0, 1+0=0+1=1;

0*0=0, 0*1=1*0=0, 1*1=1

and the unusual 1+1=1 (in F_2, it is 1+1=0)

I hope this is correct, this is also new to me...

Thank you, to me it is also new.

I listened to a series of conferences on the subject by Alain Connes in the video archive of the College de France (in french).

https://www.college-de-france.fr/site/en-alain-connes/course-2017-01-05-14h30.htm

There are other conferences by him available in english also.

And papers on ArXiv by Alain Connes and Catharina Consani.

Article Homological algebra in characteristic one

If you like F_2 and F_1 are both based on the set {0,1} but the operations are different:

in F_2 "+" corresponds to the exclusive disjunction XOR, and "*" to the conjugation AND.

in F_1 "+" corresponds to the inclusive disjunction OR, and "*" to the conjugation AND.

source: http://www.univ-rouen.fr/LMRS/Persopage/Lescot/algabsodef.pdf

It is interesting but very abstract to me.

Alain Connes says in his introduction to his lecture roughly: non-commutative geometry in the Field F_1 is particularly adapted to studying the structure of space-time at most small scale.

1) Now it is very-very interesting. I have heard about topology, it is like gum-geometry. Logic is thinking about truth and falsehood. But how can one combine these two topics? It is topos, am I right?

2) How does it connect to the spacetime?

Still another representation of F_1 would be the following: Consider the set P({1}) of parts of the set {1}. So P({1})={{},{1}}. "+" would be the union and "*" the intersection.

Dear Éva Szőke ,

Yes, topos theory does not study individual geometric objects but categories of objects meaning the objects and the homomorphisms between the objects.

A topos is a sort of generalization of the category of sets. The category of sets is indeed a particular topos.

Another topos would be the following:

Objects: finite graphs, homomorphisms: mappings respecting the graph structure i.e. mapping edges on edges.

Another topos is the topos of presheaves over a topological space T.

Example of an object: Functor associating to every open set X the set of smooth functions over X.

Best borrow or buy the book of MacLane and Moerdijk for an introduction. And also:

Conceptual Mathematics: A First Introduction to Categories, by Lawvere and Shanuel for a still more easy introduction.

Consider the following category:

Objects: set of parts of a set E closed by finite union and finite intersection.

Morphism: a mapping respecting union and intersection of parts.

I conjecture that F_1 is in this category the counterpart of {0,1} for the topos of sets i.e. a subobject classifier and that the category is indeed a topos.

In ontology this subject (Lattices) is most important.

I make explicit the subobject classifier:

The terminal object is the one point set together with the one-element set {1}.

The classifying object Omega is F_1.

'true' corresponds to the mapping from {1} to F_1 associating to the element 1 the element 1.

Dear Dr. Krantz,

Thank you for the explanation.

Dear Éva Szőke,

You are welcome, but please make sure you are convinced of the details before communicating about it. I should still verify.

So, I have never heard about topos, thank you that you have explained to me. Something I have understood and anything I have not. I still do not know what is the relationship between topos and the spacetime. I know that the spacetime can be connected to topology, because gravitation and mass deform the space(time), so several things remain invariant. These features of the spacetime can be dealt by topology. But I do not see how one can connect the spacetime to

the truth and falsehood or to categories.

Consider a topological space T.

To any open set U for T you can associate a set F(U) e.g. the R-valued continuous functions on U: F(U) := C^0(U,R). R could be the discrete space {0,1} or the topological space of the real numbers etc.

Now if U subset V then any function on V can be restricted to a function on U : we obtain a mapping F(V) -> F(U).

F is what we call a (contravariant) functor from the category T to the category of sets.

Now the topological space is in some sense identified to F. It is the "same" to consider the R-valued continuous functions on T or the space T. (I simplify a bit).

I just defined what is called a presheaf.

Given from one presheaf on T to another on T (think of representations of a Lie group over a differential manifold if you want) form a category. You can compose them and have an identity.

Now the presheaves on T form a topos and any topos has a classifying object.

The subobjects of a subobject classifyer represent the truth-values.

In case you have F_1 you will have a unique truth value, I don't know if it is the case for any topological space T. I still have to work on the details....

But in the great lines this should be ok.

PS. I am just wondering about the truth values for a topological space-presheaf, this should be related to points of the topological space. I guess to any point corresponds a truth value (to be checked).

After thinking again about it, you need first a sort of duality: an integration on the topological space with values in an adequate space of truth-values. The duality bracket could correspond to pointwise evaluation for instance. etc.

Dear Dr. Krantz,

Thank you for writing down the full explanation to me. I have thought it will be a bit simplier. I must read it more times to understand it. In my point of view the topic of topos will be very useful for Dr. Carvallo. Your explanation about topos and spacetime is very interesting. If i am thinking after, everything can be deduced to 0 and 1, e.g. to truth and falsehood.

Concerning you final remark, I am not quite sure about this. There is a very active research domain about algebraic geometry in the field F_1. It depends what you mean by "deduction to 0 and 1".

Another remark I would like to make is an interesting article I came across from J. Lambek:

The Influence of Heraclitus on Modern Mathematics, (google it to find the pdf).

It could be of interest to this discussion.

I have thought on coincidences. If two events meet in the spacetime, we can interprete it as truth. For example in topology the structure of the coordinate grid remains invariant (e.g. we do not lose any cross point).

I think a topological space is the surface of a ball for example. On

a subset of this is a real valued function F defined which can map into the set {0,1}. What does it mean? Do you know something about its physical content maybe?

Well I used Capital letters for a functor and small letters for morphisms (functions). Please look up the meaning of functor.

A continuous function f from a topological space into {0,1} is necessarily constant on the connected components. So it is simply a characteristic function of the open set.

A continuous function f from a topological space into the set of reals is to be related to a multiple of the probability measure on the measure space generated by the open sets. I simplify again a bit, this is Radon-Nykodim-Lebesgue theorem.

(I apologize to any mathematician who would read this.) But replacing continuous function with measurable function it is not too far from the truth. Anyhow any measurable function can be approximated by continuous functions. There was some time mathematicians discussed the question if a discontinuous function was really a function...

So finally integrating this function on the closure of an open set would give the probability to find e.g. an electron in this portion of space when you measure it.

I learned a long time ago about functions and about measure theory but I have already forgot it. Just now I have explained to my mom (80 years old) what can be the topology. She asked me what does happen if the rubber sheet breaks away or cork-skrews...or if we heat it...and we finished at the question: can be vacuum inside the Earth?

Returning to the initial problem, I do not know what is a functor. I have thought a real valued function can be the lenght on the surface or the degrees of the angles. What kind of meaning can have a functor with values 0 or 1?

A functor is a mapping from one category to another (possibly the same), sending objects to objects and morphisms to morphisms, identity to identity and composition of morphisms to the composition of their images.

An example of a functor would be: Sending a topological space to its fundamental group, or sending a vector space to its underlying set, or sending a topological space to the set of real-valued functions on the space etc.

If the rubber sheet breaks away you add open and closed sets to your topology. Algebraic topology cuts the space into simple pieces in order to compare topological spaces and to calculate topological invariants. By removing a point from a sphere you get an open disc plus a point. You can calculate de Rham cohomology, sheaf homology, etc. etc. By calculating the fundamental group of the complement of a knot you can find out about if the knot can be undone in R^3 or wherever.

Thank you for your answer. I think I have understood what a functor is but the topic is too difficult to me. It can be the top of the problem about truth and falsehood.

(One could "knot" an Einstein-Bose-condensate in order to trap light?)

By the way an atom is a sort of knot, where the electrons are trapped I guess.

About a logic that would be good to speak about is also linear logic.

I am very sure that there is a direct link to F_1.

Check http://girard.perso.math.cnrs.fr/Synsem.pdf if you like.

Yes, you are right. Electric force keeps the electron in the atom there. The find probability of the electron is the greatest on the

first electron trick in the hydrogen atom.

One can trap light maybe by a reflective surface inside the condensate.

Gravitation bends light, but also interaction with other particles...

I wonder if one can produce different refraction indexes inside a condensate?

Sorry this is again far out of the scope of the initial question(?)

Our thoughts of truth and falsehood can be fuzzy and overlap each other, but our decision of action rests upon one choice. Non-Aistotelian logic (fuzzy sets) is possible theoretically, but in practice we deal with separate precise action