On 12 April 2017 at 20:59, Bruno Marchal <[email protected]> wrote:
>
> On 11 Apr 2017, at 18:21, David Nyman wrote:
>
> On 10 April 2017 at 18:32, Bruno Marchal <[email protected]> wrote:
>
>>
>> On 10 Apr 2017, at 12:58, David Nyman wrote:
>>
>> Over the years there have been many references to various modal logics
>> deployed in support of the comp theory, in particular for the analysis of
>> categorical distinctions between third-person and first-person logical
>> consequences. Trouble is, when Bruno refers to these logics in explanation
>> of his points, the presentation is so technical that I for one have never
>> been able to follow these technicalities sufficiently well for them to
>> become intuitively obvious. Hence I've had to come up with my own amateur
>> versions.
>>
>> As David Hilbert famously said "A mathematical theory is not to be
>> considered complete until you have made it so clear that you can explain it
>> to the first man whom you meet on the street.". I wonder whether it would
>> be possible, Bruno, for you to contrive some sort of "man in the street"
>> presentation of the key logics deployed in your arguments and why indeed
>> you regard them as so central. I suspect that this is closely related to
>> the process you describe as interviewing the machine.
>>
>>
>>
>> Propositional modal logic is classical propositional logic with one
>> symbol more, usually, written with a box "’[]". It denotes an unary
>> operation. This means that if A is some formula, like (p -> p), []A is also
>> a "grammatically correct formula", meaning that [](p -> p) is a formula.
>>
>> Originally, modal logic was conceived, like logic itself, by Aristotle,
>> although he did not use any special symbol, but he used the word
>> "necessary" and "possibly", and used it to make his famous "Aristotelian
>> square":
>>
>>
>> Necessary -------------- not-necessary
>>
>>
>> necessary-not ---------- not-necessary-not
>>
>> Now, not-necessary-not is the same as possibly. It is not necessary that
>> man is not rational is the same as "it is possible that man is rational".
>> "Possibly" is the dual of necessary, and is usually abbreviated by the
>> symbol diamond "<>". By definition it is ~[]~. We could have used <> as
>> primitive, and define [] by ~[]~, and write Arstotle's square in the
>> following way:
>>
>> Not-possibly-not -------------- possibly-not
>>
>> Not-possibly ------------ possibly
>>
>> When the box [] and diamond <> are used to denote "necessity " and
>> "possibly" in some metaphysical, sense, we say that it is alethic modal
>> logic. Leibniz, much later, will provide a sort of semantic for it by
>> interpreting the necessity by "truth in all possible world", and the
>> possibility by "truth in at least one world". This can help to agree that
>> alethic modal logic, see as a theory (set of axioms), can admit as axioms
>> the following formula:
>>
>> []p -> p (if p is necessary, then it is true)
>> []p -> [][]p (if p is necessary then it is necessary that it is necessary)
>> <>p -> []<>p (if p is possible, then it is necessary that it is possible)
>>
>> Now a theory is not just a set of axioms. It is a set of axioms together
>> with inference or deduction rules. Most logic have the modus ponens rule,
>> from a proof of A and a proof of A -> B, you can deduce B.
>>
>> The so-called *normal* modal logic have the modus ponens rule and the
>> necessitation rule, which says that if you have a proof of A, you can
>> deduce []A. They have also (by definition of normal modal logic, the axiom
>> [](p -> q) -> ([]p -> []q). In Leibniz theory,/semantics, you can verify
>> that if (p -> q) is true in all words, and if p is true in all worlds, then
>> q is true in all worlds. OK?
>>
>
> OK
>
>
>>
>> Different modal notion will have different axioms, and sometimes
>> different inference rules.
>>
>> Now, modal logic is used in the "machine interview" to simplify a lot the
>> situation. The real difficulty, which is more demanding in term of lengthy
>> formalities, is the provability logic. Gödel succeeded in translating "A is
>> provable", with A put for some arithmetical formula (like "s(0) = 0") in an
>> arithmetical formula. That is longer to explain, I will proceed later.
>>
>> Tell me if you are OK up to now. We might also need to revise a bit
>> "simple" classical propositional logic, and to illustrate the difference
>> between
>> - this theory proves A (for A some being a classical proposition formula,
>> like (p -> q))
>> - this model satisfies A.
>>
>
> OK so far.
>
>
>>
>> The idea that we can explain things to the man in the street is a bit
>> inapt in this context, because the difficulty of logic is that it is
>> necessary to NOT understand the formula, and to see that they are
>> manipulated formally only.
>>
>
> Yes, I do see the difference. Important point.
>
>
>> So logicians start from what the first man in the street already know,
>> and he makes it incomprehensible.
>>
>
> Good, only the incomprehensible is worth our time and effort!
>
>
>
>> For example take the formula (p -> p). the man in the street will be OK
>> with that formula being a tautology, that is an always true formula. (p ->
>> p) seems obviously true whatever proposition is represented by p. If p =
>> "it rains", it seems obvious that if it rains then it rains, etc. OK?
>>
>
> Yes, I think this is equivalent to Tarski's criterion of correspondence
> with the facts, or what I called perceptual correspondence.
>
>
>
> Actually I was just saying that it is obvious that (p -> p), but only by
> alluding to the truth table, or that anyone will accept that IF it rains
> THEN it rains, which is not clearly something we can percept, or perhaps
> (but I was not thinking to Tarski here).
>
> For the propositional calculus, truth, or a model, is defined by a
> function from the set of propositional letters to the set {0, 1}. So a
> model, in the sense of logician is an assignment of 0 or 1 to each of p, q,
> r, p1, q1, r1, p2, q2, ...
>
> So truth is an abstract notion. To give a concrete example, we can
> interpret p by Hillary Clinton won the 2016 election, q by "Donald Trump"
> won the 2016 election. Then we will say that (p & q), for example, is true
> or satisfied, in the model where both Hillary won and Donald won. At that
> level, in the concrete illustration, we will refer to some perceptual
> judgment indeed. I am just no sure how we could perceive an implication
> except by using the equivalence between (p->q) and (~p V q), but that
> assumes either the truth table or some intuition, and the proof of (p->p)
> was given to illustrate a way to proceed which do not refer to any
> intuition, nor any notion of truth, but only formal rule.
> In fact, you cannot perceive the truth of (p -> p), because p is abstract,
> and eventually intended for all proposition. You would need to perceive all
> proposition like If there is a unicorn on the planet venus then there is a
> unicorn on the planet venus, and this for all imaginary and real animals,
> on all planets in the whole multiverse.
>
>
>
>
>
> Yet, if the current theory is the giving of the two axioms:
>>
>> A1 p -> (q -> p)
>> A2 (p -> (q -> r) ) -> ((p -> q) -> (p -> r))
>>
>> With the inference rules modus ponens, and some substitution rule, it
>> will be rather difficult to find a proof of (p -> p).
>>
>> But here, all what the man in the street is asked is in 1) understanding
>> that this is difficult, and 2) being able to verify if a proof is indeed a
>> proof, that is, a sequence of formula which starts from some axiom, and use
>> only axiom, or formula derived from the axioms using only the given
>> inference rule. Even that can be very complex, and usually we add comment
>> to help (like the comment in a program).
>>
>
> Yes, this is important. Failing to do this, even in informal reasoning,
> leads directly to question begging, as I'm fond of pointing out.
>
>
> The informal reasoning is at the level of the comment. We can only hope to
> be enough clear. The formal reasoning iis at the object level: it is the
> deep engine. No biologist will ever claim that you have to write a book in
> biology using only strings of A, T, G, and C. The study of those is done
> formally in english.
>
> When you see someone begging the question, usually it is just because they
> made an non valid informal reasoning.
>
> Don't worry, I am aware I have not been clear enough, on this subtle
> point. Russell and Whitehead missed it too, and physicists never
> understand. In fact, biologist have less problem in general.
>
> And just to make pleasure to John Clark, that is a point missed by the
> greeks, and even the earlier modern logician.
>
>
>
>
> All the more pernicious when (as is usual) the smuggling in of auxiliary
> assumptions is almost always tacit and unrecognised.
>
>
>> Here is a proof of (p -> p):
>>
>> 1) (p -> ((p -> p) -> p) Axiom A1, with q substituted by (p -> p).
>> 2) (p -> ((p -> p) -> p) -> ((p -> (p -> p)) -> (p -> p)) Axiom A2
>> applied on "1)", that is axiom A2 with q substituted by (p ->p) and r by p.
>> 3) ((p -> (p -> p)) -> (p -> p)) (by modus ponens on 1) and 2) ),
>> 4) (p -> (p -> p) (Axiom A1 with q substituted by p),
>> 5) (p -> p) By modus ponens on 4) and 3).
>>
>> You might ask: why do you logician makes things obvious looks becoming so
>> complicated? Why not use a truth table and settle the matter in less than a
>> second?
>>
>> The answer is that in most theories there are no simple method to find a
>> proof or to show that a model satisfy a formula, and the modus ponens will
>> be the main engine that we will have to describe to translate "provable"
>> later in arithmetic.
>>
>
> Yes, and also because it calibrates or tests that the formal mechanism
> actually shadows or mirrors the informal correspondence.
>
>
> In simple case, luckily. But later we will see that no machine can ever
> proves that such a correspondence is satisfied. The machine will still be
> able to hope, pray, ...
>
> *We* will able to believe in such correspondence for simpler (than us)
> machine, because we do have some intuition on numbers that we share, but we
> lost the "obviousness" of the correspondence for machines whose complexity
> or probability power will match our's.
>
>
>
>
> We could perhaps say that Theatatus criterion of justified+true (i.e.
> believes correctly that it is true + it is true) is reconciled with the
> Tarski criterion of (perceptual) correspondence with the facts (e.g. where
> snow being white is perceived as "analytically" true in face of the facts).
>
>
> Yes. It enforces it, and we can make sense of it for the simple machines
> we trust.
>
>
>
>> The proof 1-5 given above is not given to convince anyone that (p->p) is
>> a tautology. It is given to illustrate a proof, and to see that some very
>> simple theory embedded in some machine can prove it.
>>
>
> Calibration again?
>
>
> Not sure. See above. And don't get nervous, logic is a science which has
> the most complex beginning. I see that you still want to interpret the
> logical formula (which is normal giving the role they will have). But the
> point is that they are not interpreted at all. They have intended
> interpretation, of course, that is why we use symbol reminding the goal,
> but that is why people introduced to logic can miss the first step. You
> need to look at (p->p) like you would look at O9ç!.è§.
>
>
>
>
>
>>
>> You must look at such proof as like it was a piece of DNA, and it will be
>> the main object that we will have to represent in arithmetic later, to
>> define provability in arithmetic, and see what this or that theories can
>> prove about it, and if that is not captured by some modal logic (and you
>> know that the answer will be affirmative, as we will get G and G*, two very
>> special modal logics describing what machine can say about themselves, in
>> the 3p ways first, and later, why we will have that although G* proves
>> (([]p & p) <-> []p), yet G does not, making the "knower" existing, in
>> some sense, yet not being *any* machine from its points of view. But I
>> guess now, I have accelerate too much.
>>
>
> Yes, but this is the nub of my question, so let's not forget it.
>
>
> I am not. I am just trying to make myself sure that you do not try to
> understand the syntax, except for the procedural modus ponens rule.
>
> You need only to agree that the following is a valid proof of O9ç!.è§.
>
> axiom a
> ##&!çç -> O9ç!.è§
>
> axiom b
> ##&!çç
>
> Proof of O9ç!.è§
>
> 1) ##&!çç (by axiom a)
>
> 2) ##&!çç -> O9ç!.è§ (by axiom b)
>
> O9ç!.è§ (by the modus ponens rule applied on line 1) and 2).
>
> To be sure, when we have proved (p -> p) we have used another rule
> (substitution), but let us forget it for now.
>
>
>
>
>
> Perhaps, in my language, G* "knows" truths relating to "perceptual" facts
> (where perceptual stands for informally true in this context); IOW truth
> here, informally, is whatever can be seen to correspond with those facts.
> Then the capacity of the machine, in some sense, to navigate by means of G*
> renders it an informal knower with respect to the "facts" it perceives. But
> the informality of this logic (and hence unavailability of formal proof
> procedures) means that G* cannot recapitulate it mechanically and
> consequently cannot appreciate itself as a machine in this sense.
>
>
> G and G* knows truth about the machine. In the case of G, the machine can
> prove them. But G* extends it with the whole truth (limited to some special
> statement). G* knows what G knows, but incompleteness makes G* minus G non
> empty (and quite large). G* knows G and the truth in the corona G* minus G.
>
> We will believe/prove that such truth corresponds to informal arithmetical
> "fact" that we are willing to take as true, indeed. Each one has to ask
> himself if they really believe things like the irrationality of sqrt(2),
> which is equivalent with ~(ExEy(x*x = 2 *y*y). The legend is that
> Pythagorus sacirficed many cows to celebrate the discovery. And that was
> kept secret; it seems a pythagorean get killed for having explain this out
> of the circle of initiate ....
>
> The "informal" will be captured "meta-formally" by S4Grz, and G* will
> "see" that this makes sense. The machine will not see this. Neither its
> 3-self (which will be described by G and G*) nor its 1-self, which will be
> captured by S4Grz, will ever make complete sense, except the day she bet on
> computationnalism and also self-correctness at the meta-level.
>
> G* will prove ([]p) <-> ([]p & p)
>
> But neither G, nor S4Grz will ever be able to prove that. Here the machine
> will discover that there are truth about her that would make her
> inconsistent if she took such truth as axiom.
>
> So you are close to right: G will justify later that some truth, from the
> 1 and 3 view of the machine can only remain informal. Ineffable even.
>
> But for this, I have to explain more about PA's provability. It will work
> for all mechanist extension of PA, as long as they are arithmetically
> sound.
>
>
>
>
>
>
>>
>> G will be the normal modal logic with the Löb formula []([]p -> p) ->
>> []p. It axiomatizes completely the logic of any mechanical extension of PA.
>> G* will be the non normal modal logic, loosing the necessitation rule,
>> extending G, + the formula []p -> p.
>>
>> Likewise, S4Grz, the "universal soul", the logic of a new box [o]p
>> defined by ([]p & p), will be axiomatised by a normal modal logic with the
>> (strange looking) formula:
>>
>> []([](p -> []p) -> p) -> p
>>
>> due to Grzegorczyk. Gödel incompleteness makes "provability" into a
>> notion of belief (it can be wrong), the theaetetus idea makes it into a
>> logic of knowledge, and it obeys Brouwer axioms for the "first person
>> knower": it cannot be defined in arithmetic or in the language of the
>> machine, it has a temporal dimension, and it makes possible to build an
>> arithmetical interpretation of intuitionist logic.
>>
>
> Need more on this!
>
>
>
> Let us fix one or some simple machines: RA or PA, or even an unknown M
> talking the same *language* (perhaps unsound, or inconsistent in its belief
> (the axiom and their consequences)
>
> The language is the arithmetical language. That means that RA's or PA's
> assertion are grammatically correct sequences of symbols belonging to two
> sorts of symbol;
>
> - the logical one: v, &, ->, ~, E, A, t, f, plus the variables x, y, z,
> u, v, ... (plus the parentheses)
> - the arithmetical one: +, *, s, 0
>
> Exemple
>
> Ex(s(0) +s(0) = s(0)) is a grammatically correct arithmetical sentence
> (intuitively false, but we should not care about this at this stage). Note
> that the quantifier Ex is dummy, as x is not in the formula, but this is
> allowed.
>
> EEx(s(0) +s(0) = s(0)) is not grammatically correct, because EE is not
> allowed,
>
> Ex(x + x = x) is grammatically correct.
>
> Ey(x+x = x) also, again Ey is dummy, but we allow it just because this
> will simplify our lives.
>
> (x+x=x), likewise, this is grammatically correct
>
> x++x=* is NOT grammatically correct.
>
> To define cautiously what is a grammatical correct sentence would be long,
> and can frighten the beginners, especially that the math involved here
> might used more than what RA can prove, or as much than what PA can prove.
>
> Now, the difference between RA, PA and M, is that they have different
> axioms, and perhaps different inference rules. We can address them later.
>
>
> *Gödel's beweisbar predicate.* It is supposed to be an arithmetical
> formula translating the statement asserting that PA proves F, with F some
> arithmetical formula, *in* the language of PA. beweisbar(F) should mean F
> is provable by PA, written in the language of arithmetic.
>
> It is what I usually denote by "[]A", and it is the one whose logic will
> be axiomatized by G and G*.
>
> At first sight that seems impossible. PA or M seems to talk only about
> numbers, and says think like Ex( s(s(0) + x = 0) or alike.
>
> There is no miracle, nor magic. To explain to a German how to make a
> pizza, you need to express yourself in german. It is the same with PA, RA
> or any machine M which has only the language described above.
>
> *Beweisbar* means provable in german, but PA does not know german, and so
> Gödel was forced to explain what is a proof to PA.
>
> We have defined a proof by a sequence of formula such that the formula is
> either an axiom or derived by modus ponens from axiom or from formula
> previously derived. And what is a formula: it is a grammatically correct
> sequence of symbol. And what is a symbol?
>
> To explain what is a symbol to PA, we will simply proceed like we do with
> human. As PA knows only the term/object 0, s(0), ...We will just say that
> this number and that number will be used for the symbols. The intended
> meaning will be provided by the use of those symbols, later by PA, RA, or M.
>
> So let us denote the logical and arithmetical symbols v, &, ->, ~, E, A,
> t, f, "(", ")" =, 0, s, +, *
> by the first odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25,
> 27, 29,
>
> You have the lexicon
>
> English French Logician Arithmetic
>
> or ou v 1
> and et & 3
> if then si alors -> 5
> not pas ~ 7
> it exists Il existe E 9
> For all pour tout A 11
> ..
> ..
> plus plus + 27
> times times * 29
>
> The number 1, 3, 5 appearing there abbreviates s(0), s(s(s0))),
> s(s(s(s(s(0))))), ... (the arithmetic language does not contain the symbol
> "1", "3", ...
>
> We can denote the countably (but infinite) many variables by the even
> numbers
>
> x is denoted by 2
> y by 4
> z by 6
> x1 by 8,
> x2, by 10
> x3 by 12,
> etc.
>
> So you see, now PA, is able to get what a variable is, it is informally
> captured by the formal predicate even(x), that is Ey(x = s(s(0)) * y).
>
Here I do use the fact that you can understand such formula, and PA, thanks
> to its axiom, can too, but this is for later.
>
So the convention is
just
that actions and entities
are considered to be
"understandable" by PA
if than can be co
ded in its language
. Given its axioms, that is.
>
> You might prefer Variable(x) = Even(x) & (x ≠ 0), that is Ey(~(x = 0) & x
> = s(s(0)) * y), to avoid like above to use 0 as a variable, which might
> lead to some bugs ...
>
> You can define Arithmetical-symbol(x) by: ((x = 23) v (x = 25) v (x =
> 27) v (x = 29))
>
> OK? Humans do not know much. If you ask what is a variable, they will say
> something like "letters at the end of the alphabet, or the same with
> subscript) a long time before digging on possible deeper meaning on such a
> notion. It is good pedagogy to do the same with RA, PA, M.
>
> Now what is a formula? It is a sequence of symbols (obeying some grammar)
>
> So we need to define a sequence of symbols. We cannot just say that
> Ex(x=x) is represented by 9 (the symbol for "E") followed by 2 (the symbol
> for "x", followed by 17 (the symbol for "("), 2 again for x, 21 ("="), 2
> again, and 19 (to close the parenthesis 17).
>
> This does not work, because we don't have defined "follow".
>
> There are many ways, and Gödel will use the fundamental theorem of
> arithmetic for this task. That theorem say that all natural numbers admits
> one and only one decomposition into product of primes (up to the order of
> the multiplication). This suggests representing the formula Ex(x=x) by
>
> (2^9) * (3^2) * (5^17) * (7^2) * (11^21) *(13^2) * (17*17),
>
> At the left of the exponentiation, "^" you see the ascendent prime
> numbers, and at the right of the exponentiation ^, you have the variable,
> in a language that PA, RA and M can "understand".
>
> To define the predicate (adjective, property, or relation) like
> Formula(x), capable of defining, you need to define "grammatically correct"
> in terms of those numbers and using addition, multiplication, and what has
> been just define (variable(x), sequence(x), etc.). I skip this for now.
>
> Note we don't have the symbol "^" in our language. It is "easy" to define
> it "recursively", but, alas, to explain "recursively" to PA we need the
> finite sequence. So it is a bit more tricky to do that. Gödel did use an
> Antic (Sorry John) Chinese insight in modular arithmetic (known as the
> Chinese Remainder Lemma. Hmm... if you want, later. There are many other
> ways, some very beautiful, like by Smullyan, and others.
>
> Well, beweisbar will be the translation of provable in the arithmetical
> language (this does not depends on any axiom or belief by the machines, as
> a matter of translating a definition into a language. Then RA, PA, Me, and
> You, will differ by our basic belief, and have our own beweisbar predicate.
> Human have a big non monotonic layer, which does not obey to any of this,
> but to extract the theology and the physics, that does not matter (for the
> evolutionary biology and psychology, that does matter).
>
>
> Now, a proof made by machine M, or PA, or RA, is a sequence of formula, so
> that each one is an axiom (with respect to each system of axioms) or a
> formula obtained by the preceding one by modus ponens.
>
> But a sequence of formula is just a sequence of sequence of symbols, so a
> proof will be translated by reapplying the idea above.
>
> For example, the proof (where a typical first order logical inference rule
> is used, "known" by RA, PA and M (and some humans, if they remind that I
> use Ax with the intended meaning "for all x".
>
> Ax(x=x)
> (0 = 0)
>
>
>
> is represented/translated in arithmetic by
>
>
> 2 ^ (the number representation of Ax(x=x)) * 3 ^ (the number
> representation of (0 =0))
>
>
> That is
>
>
> 2 ^(2^9) * (3^2) * (5^17) * (7^2) * (11^21) *(13^2) * (17*17) * 3 ^
> <exercise! :) >
>
Sorry, much too quick. I need more explanation here.
David
>
> Sorry have to go,
>
> Bruno
>
>
>
>
>
>
>
>
>
>
>
>
>
>> We will have a theorem (by Solovay) that G proves a modal logic formula
>> if and only if its translations in arithmetic, with []A becoming
>> beweisbar(gödel-number-encoding of A), A being any arithmetical
>> sentence, is provable. And G* will proves all such provability statements
>> which are true, even when non provable by PA or the mechanical extension
>> concerned. G and G* sum up infinities of interview of the machines and
>> their mechanical descendent, as long as they are arithmetically sound.
>>
>
> And this!
>
>
>>
>> There is a beautiful tryptic between G, G* and S4Grz. Let us call []p ->
>> p reflexion (formula):
>>
>> G has Löb and necessitation, but lack reflexion,
>> G¨has Löb and reflexion, but lacks necessitation,
>> S4Grz has reflexion and necessitation, but lacks Löb.
>>
>> No system can have Löb, necessitation and reflexion. Indeed, with those
>> three we would have, with f put for some false formula, like 0 = s(0), that
>> is 0 = 1:
>>
>> 1) []f -> f by reflexion,
>> 2) []([]f -> f) by necessitation,
>> 3) []([]f -> f) -> []f by Löb, with p substituted by f,
>> 4) []f by modus ponens on 2) and 3),
>> 5) f (by modus ponens on 4) and 1).
>>
>> Löb is enforced for all *machine* which are correct, and believe in
>> elementary operation and enough induction axioms. We can come back on this.
>>
>> I let you rest a bit.
>>
>
> Phew!
>
>
>> I have just made a little grand tour. Feel free to ask any question or
>> make any critics. I insist, the main difficulty in the beginning of logic,
>> is to understand that you should not understand the symbols. You need only
>> to be able to verify some formal very elementary mechanical comparison.
>> Then the semantic will also be treated "formally" using powerful
>> mathematics (sets, trees, topological space, Hilbert spaces, etc.).
>>
>
> Good, but I guess I was also asking for some sort of shortcut to the
> intuitive power of all this, because if the only route to this is more of
> the above (however interesting in itself) then the point will go on being
> lost not only on the majority in this forum (unless I'm very much mistaken)
> but a fortiori on any possible wider audience. Which would be a pity.
>
> David
>
>
>> Bruno
>>
>> PS I guess you have missed my early introduction to modal logic on this
>> list, many years ago! No problem.
>>
>>
>>
>>
>>
>>
>> Thanks in advance.
>>
>> David
>>
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>>
>> http://iridia.ulb.ac.be/~marchal/
>>
>>
>>
>>
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>
>
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> http://iridia.ulb.ac.be/~marchal/
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