I didnt understand this
#propositional calculus
ocean plank
modest yokeBOT
<@&987246841693360200> please have a look, thanks.
Changed the category to Math.
<@&987246746478460948> please have a look, thanks.
ocean plank
its german and i have mo idea what this called in english
worthy galleon
Definition 1.7
Given a set of expressions Φ ⊆ 𝒜. An assignment F is called a model of Φ if F(ϕ) = w for all ϕ ∈ Φ. An expression ψ ∈ 𝒜 is called a consequence of Φ, if for all models F of Φ the statement F(ψ) = w holds. If this is the case, we write Φ ⊨ ψ. □If Φ = {ϕ1, ..., ϕn} is a finite set of expressions, then Φ ⊨ ψ if and only if ϕ1 ∧ ... ∧ ϕn ⇒ ψ holds. This claim shall be proved in Exercise 6.
(English translation)
(ChatGPT)
And here's chatgpt's answer:
Certainly, the text you provided is a formal definition from mathematical logic, specifically dealing with the concepts of models and logical consequences within a formal system. Let me explain each part:
Set of Expressions (Φ ⊆ 𝒜): The text starts by considering a set of expressions, denoted by Φ, which is a subset of some larger set 𝒜. In the context of logic, an expression is typically a statement that can be evaluated as true or false.
Model (F): A model, in this context, is a particular kind of function or assignment (denoted by F) that assigns a truth value to each expression in Φ. If F assigns the value 'true' (denoted by 'w' for "wahr" in German, which means 'true') to every expression in Φ, then F is considered a model of Φ.
Consequence (ψ): An expression ψ (which is also an element of 𝒜) is said to be a consequence of the set of expressions Φ if, in every model that makes all expressions in Φ true, ψ is also true. In other words, ψ is logically entailed by Φ; it follows logically from the expressions in Φ.
Logical Entailment (Φ ⊨ ψ): The symbol "⊨" is used to denote logical entailment. If Φ ⊨ ψ, it means that whenever the expressions in Φ are true, ψ is also true. This is a formal way of stating that ψ is a logical consequence of Φ.
Finite Sets of Expressions: The text then specifies the case for a finite set of expressions. It states that for a finite set Φ = {ϕ1, ..., ϕn}, the set Φ entails ψ if and only if the conjunction (logical 'and') of all expressions in Φ implies ψ. In symbolic terms, if ϕ1 ∧ ϕ2 ∧ ... ∧ ϕn ⇒ ψ holds true, then Φ logically entails ψ.
Proof Requirement: Lastly, the text mentions that the given claim (about the entailment in the context of finite sets of expressions) needs to be proved in an exercise, specifically Exercise 6, which suggests that this text is from a textbook or academic paper with accompanying exercises.
In summary, this definition is outlining the fundamental relationship between a set of logical statements and their consequences within a formal system, and how a model is used to determine the truth of these statements. It’s a foundational concept in the fields of mathematical logic, computer science (especially in logic programming and formal verification), and philosophy.
Sorry if this doesn't help
sand tartan
@foggy crescent please speak English
sand tartan
No that's the rules, please write in english
ocean plank
Sorry if this doesn't help
ok thanks going to look through
muted plover
the basic point here is that you want to consider logical expressions, involving some variables e.g. p and q, for example p ∧ q or p → q e.t.c.
we might want some assumptions / axioms, which would be expressions we assume to be true, and that is what they use Φ for. if for example we're working on the assumption that p ∨ q is true, then Φ = {p ∨ q}.
the "assignment" F is a choice of true / false for each variable, so in this example it would be a choice of whether p is true and whether q is true. for example one assignment F takes p to be true and q to be false, i.e. F(p) = true, F(q) = false, which would then have F(p ∨ q) = true ∨ false = true, and other things like F(p → q) = true → false = false, e.t.c.
such an assignment is a model of Φ if the assumptions in Φ that we want to be true are actually true, i.e. F(φ) = true for every φ in Φ. in the above example with Φ = {p ∨ q} we had F(p ∨ q) = true, so that F is a model for that particular set of assumptions Φ.
then they say an expression is a consequence of Φ if the expression is true in every model of Φ, i.e. it's true regardless of whether the variables in it are true or false, as long as the assumptions from Φ are true
so for example, if we instead take the assumption Φ = {p ∧ q}, then p is a consequence of Φ because all models of Φ must choose p to be true: otherwise they wouldn't make p ∧ q true. (if p is false then p ∧ q is false). this is just a complicated way of saying that if p ∧ q is true, then p must be true.
intuitively, that is what the statement p ∧ q → q is supposed to mean, so maybe p is a consequence of Φ = {p ∧ q} if and only if p ∧ q → p holds
generalizing this idea, maybe ψ is a consequence of Φ = {φ₁, φ₂} if and only if φ₁ ∧ φ₂ → ψ holds, since both should mean "if φ₁ and φ₂ are true, then ψ is true".
(ofc you could have more than 2 assumptions and it's exactly the same, which is what they're saying at the end with the Φ = {φ_1, ..., φ_n} stuff)
in fact that is a theorem in logic, and they say it's shown in exercise 6
ocean plank
hm seems complicated, I understand what you are saying but cant fully connect everything
I think I need to take a step back and need to fully understand what F/weird B/ weird A/V etc is
faint terrace
@foggy crescent thanks for trying to help. u had good intentions only, ur efforts are highly appreciated.
the reasons we don't allow non english are:
- it excludes everyone else from the conversation. like, imagine ur sitting together with two friends and one of them only speaks english. if u then talk german, that friend will be lost and feels excluded and not welcome to join the conversion
- the second reason is moderation. we can't ensure a safe and welcoming environment for everyone if we (the mods) can't read what people write.
in such a case, it's best to provide that sort of help via DM, or translate to english - if possible.
sorry for the inconvenience and thanks again for ur efforts 👍
ocean plank
so everything starts with V
The language of propositional logic contains as components the countable set V of propositional variables [...], the operator symbols [...] and the brackets ( and ).
a propositional variable is something that evaluates to either true or false right?
The set of expressions A is the smallest set that satisfies the following two rules:
[...]
what do they mean with this weird A here?
also what is the difference of p1, p2, p3, ... and this Φ and ψ
faint terrace
its a A written in calligraphic style. its the set of all logical expressions u can formulate
the variables V are p1, p2,...
its like x, y, z
a variable
so now u can formulate an expression like
(not (p2 and p7)) or p3
and all possible expressions u can ever formulate are called A
their text there says exactly that. just in the mathematical precise way
ocean plank
okay and this just says that phi is a variable of A and follows the "rules" of it
faint terrace
yeah. so an expression like the one i just wrote
just means "let phi be an expression"
ocean plank
ok good
ok and here its saying that the statement phi consists of p,q
with p, q defined above
and a statement consists of multiple expressions
faint terrace
yes. it seems they defined V as a "function" that extracts variables from an expression
likely as convenience for writting
so p and q are variables here
ocean plank
here they saying it
faint terrace
yeah. they do that often for convenience
ocean plank
ok thanks, here they introduce B with is simple
and if they saying F: V -> B than they mean that the function takes in some element of V and turn it/map it to a value of B right?
faint terrace
correct
for example f("der hahn kräht") might be false
so that mapping gives u a concrete way to execute/interpret ur expression
faint terrace
A is the entire expression
they say that this second function is valid for a given F: V to B
essentially, they are talking about truth tables here
so, given an expression, u can formulate an entire truth table that tells u the output true/false once u have a concrete mapping
faint terrace
its written weird though
faint terrace
i would have written it as a function taking both, A and V
so F: A x V to B
i guess
ocean plank
ah ok
so a variable like phi here is in the set of V, and smth like phi and psi is in the set of A
right?
faint terrace
correct
ocean plank
ok nice
faint terrace
and that first F is a mapping. for example phi = true and psi = false
a concrete mapping that will, applied on an expression, give u a result
ocean plank
ok thats nice
so I fully understand this
ok so I read through Definition 1.7 again and its a bit clearer
ok I think I understood that
though I didnt understand the use case yet
let me look through the example
hm the example just shows where this applies in an example Φ but still didnt understand what the use case of saying Φ |= p or any variable
but something I will understand later on probably
now I am going to try finish task 6
faint terrace
let's say u have a few expression that say something
perhaps sth with temperature and people visiting the swimming pool
and another with tenperature and snowing
now, ur looking for a concrete mapping for ur list of claims such that they all are true
so perhaps u need snowing = false, going to pool = true and is_hot = true
or sth like that
a concrete setup such that this 2 expressions resolve to true
those 2 expressions together are that big phi subset A
the concrete mapping that u just found, leading to all expressions being true, is called a model of that expression set
now, if u add another expression like
"when it's hot, ill eat icecream"
and it happens that, while using the same concrete mapping, that it also resolves to true, it can be called a Folgerung of that model
so based on ur existing expressions, this new expression could also be formed
it follows out of them
faint terrace
both directions
if psi follows from phi, show that its true that all phi together follow psi
and the other direction
if all phi together follow psi, then psi follows from phi
ocean plank
hm im quite lost
ocean plank
hm ok
faint terrace
if psi follows from phi, what does that mean?
ocean plank
if psi follows from phi, what does that mean?
you mean this
faint terrace
yes. that's the left side of the first direction
now, what's the next step. what does that mean?
that for all models F of phi, F(psi) = w
ocean plank
that means that F(psi) = w
faint terrace
yes, for all models F of phi
ocean plank
yeah yeah
faint terrace
now. what is psi? its an expressions
faint terrace
what are the little phis? they are expressions as well
ocean plank
yeah but how am I supposed to plug and play? I have no expressions, its all so abstract
faint terrace
and u just had that all mappings F will have F(psi) = w
now, what are those mappings again? they are the mappings that lead to all expressions being true
cause phi is model
so logically, when u apply a mapping that leads to all little phis behind true and that also leads do psi being true
then, what u have is "if i can make all little phis true, well, then psi will become true as well"
ocean plank
ok thats understandable
faint terrace
yeah. now u have to write that down mathematically precise and that's where it becomes a bit annoying
and then later u have to do the other direction
ocean plank
and now this means that this is always true for Φ |= ψ because anything -> true will be true
ocean plank
yeah. now u have to write that down mathematically precise and that's where it b...
ok im trying
ocean plank
hm I have smth but I dont even know 
but how does this prove anything
it doesnt probably lmao
going to bed now, thanks for all the help 
faint terrace
btw, u should always add text between the math fragments. otherwise it's unclear what is what and how it goes together
unless it's a single concrete chain
but in this case it's not
u have a bunch of things there and then a blue arrow from one to another and it's more a puzzle then a clear proof chain
regarding what u wrote down, the phis aren't always true. only under the mapping F
but under the mapping F, psi is also true
for a "->" to be a true statement, the right side must be true when the left side is true
when the left side is false, the right side doesn't matter
ur left side is true exactly under the F mappings and no other
and under those mappings the right side is true as well (can also be true under more though)
u have to approach this differently. for the -> to be a true statement, the right side must be true when the left side is true. when is the left side true? when all phis are true. when is that the case? only under mappings F, nothing else
and for mapping Fs, the right side is true as well bc psi follows from phi (whats given in this direction)
true statement. done
other direction
the other direction is similiar
u got a bunch of expressions phi. now take the mappings F that lead them all to true
in this direction, the -> is given. so u follow directly that under those F, psi is true as well
now u got a bunch of F that lead all phi and psi to true
by definition that means u found a model phi and psi follows from it
done
ocean plank
Ok going to looks through this later, I am unsure how to correctly show that I mean „only under mapping of F“
faint terrace
that's how they were defined. cause model
"for all models F of..."
a model has the property that it leads all the expressions to true
so the only way to get all expressions to true is by using one of those F
some random mapping K won't do the trick, only those F lead to all true
ocean plank
so overall I need to show that this is true, right?
(Φ |= ψ) <-> (φ_1 ∧ ... ∧ φ_n => ψ)
ok I think I might have the first part
should probably restructure and rewrite some stuff
but I think its correct for the first direction
im not a fan of the first sentence
faint terrace
its hard to follow for me, since it's still scattered around a lot and u abuse math symbols for non math stuff (the arrows) to combine ur snippets and thoughts. but in a non mathemarical way
start with "to prove:..." writting down the entire thing u have to show
then "we have to show either direction" and then have two sections. one per direction
start again with what to show in this direction
then start. "given..." write down what u have for this direction
then start the chain. combing thoughts with text, not random arrows
ocean plank
ok im trying
faint terrace
u can also write some stuff in a more math way
for example ur "jede belegung" sentence
in math, it would be
modest yokeBOT
faint terrace
sth sth that direction
if u find it hard to read, write some of the binding things as words
like "such that" instead of :
or "for all" instead of the symbol
ocean plank
ok I think I mostly understand though
faint terrace
text or full-math, it should follow the same flow
ocean plank
this okay for the start
or am I doing smth wrong already
im finishing it tomorrow
otherwise I will get depressions
faint terrace
it looks much cleaner now 👍
note there is an idiomatic symbol for "zu zeigen" (to prove), its two Z interlaced
(or TP interlaced in english)
looks like this
this part here at the top is confusing since it has nothing written with it
are those ur definitions?
then perhaps start with "Given"
or "Definitions"
modest yokeBOT
faint terrace
that way its clear what that stuff is supposed to be and why its mentioned
and if u accompany the math with some text, its easier to read and understand
u can do the same with the "to prove" part, adding some text to give a good overview
i.e. after writing ur "to prove: ...", u can add:
"i.e. iff psi follows from phi, that means that all varpsi imply psi"
then, u say "aus .. folgt ...", why does that follow? by whom?
by definition? then add that
perhaps write
modest yokeBOT
faint terrace
then the next bit
modest yokeBOT
faint terrace
now, ur missing the actual crucial part of the proof. so far u only elaborated on the left side, but didnt follow the right side
modest yokeBOT
faint terrace
now, u have control over this expression. u know when its true and when its false
to show that the claim holds, u can either resolve the => into its individual components, i.e. a => b is the same as not a or b
or use a truth table
depends on how u defined it, i guess
so, with that truth table, u have to show that its impossible to get that F case
i.e. where the phis together are T, but the psi is F
modest yokeBOT
ocean plank
ok thanks for this extensive review
faint terrace
btw, i get that it feels sorta trivial and as if everything is crystal clear based on the definitions already.
the task is to get more comfortable with how to express urself in math-accurate terms
and of course to get comfortable with those phi/psi/expression/model terms
and i might be a bit strict, perhaps stricter than ur tutor.
ive had the "luck" to have tutors that give u zero points if u put a => on the left side just bc u wanted to use it in a "intuitive way" to connect stuff
like u did earlier
"0 points, => is a math symbol for implies and u used it incorrectly here"
"0 points, what are those things at the top there? ur definitions? connect and explain ur thoughts"
rough first semesters. but it also helped to improve the quality (the hard way)
ocean plank
ok good to know, I use these symbols in physics alot, shouldnt do for math ig
ocean plank
and i might be a bit strict, perhaps stricter than ur tutor.
ive had the "luck" ...
its a book, I walk through on my own, so no tutor
faint terrace
i remember when we had to prove something like (5 * (3 * 2)) / 6 = 5 but only with axioms
its like "dude, thats clear, just do the math, lol, what am i supposed to show here"
but if u actually do it proper, the proof will span over an entire A4 page
ocean plank
you need like the fundamental stuff like „Körper“, „Ringe“ etc right? which define the fundamental operator, I think I saw smth about that once
faint terrace
yeah. it even starts before that. with the axioms itself. sth simple like -1 * 5 + 5 then turns into 2 lines of transformations
to explain that -1 * 5 turns 5 into -5, which is defined as the inverse element to 5 under operation +, which then results in 0, the neutral element defined under the operation +
and so on
ocean plank
this fine right
ocean plank
is there a mathematically way of writing that F is a model of phi?
faint terrace
since they didnt introduce a symbol for that in their definition, no
u can make one if u want
modest yokeBOT
faint terrace
or u also use the same symbol again. although that might lead to confusion:
modest yokeBOT
ocean plank
ok unlucky
faint terrace
id say almost. its not stressing on the fact that the only way that all phi are true is by using a model F
like, what if u found another mapping K that isnt a model but it also leads to all phi true
ur sentences kinda allow this scenario still
and u couldnt tell how psi behaves under that K
its crucial to stress the fact that all phi becoming true can only be achieved by using a model
ocean plank
like, what if u found another mapping K that isnt a model but it also leads to a...
wouldnt this make K automatically a model of phi?
faint terrace
exactly. thats what u sorta have to stress with ur wording
the way its worded now, kinda leaves that scenario up in the air still
u say "for a model, all phi are true"
but u dont say
"all phi are true if and only if model"
like, u say model -> all phi true
but u need all phi true <-> model
so just slightly reword that sentence and ur gucci
ocean plank
ok thanks
faint terrace
also, minor, but before u use a symbol like F, u should have explained before what that is
"Für eine Belegung F, ..."
ocean plank
ah i forgot
faint terrace
technically, u should also write "wenn alle phi_i individuell ..."
instead of just phi
since u used phi_1, ...
but thats kinda nitpicking now
ocean plank
also, minor, but before u use a symbol like F, u should have explained before wh...
you mean to include this F: A -> B
faint terrace
its enough to just say "Für eine Belgung F, ..."
instead of "Für eine Belgung, ..." and then use F without mentioning what F is
the details of F are in the word "Belegung", kinda
u can write it out fully if u really want, but imo u dont have to
ocean plank
ok ty
ocean plank
im struggling a bit for the second part of the prove
I need to do the same in the other direction
but if I write this:
(φ_1 ∧ ... ∧ φ_n => ψ) => (Φ |= ψ)
the left side can be either true or false
I have no clear starting point
for the first direction I always used Φ |= ψ as the starting point, so like it was the definition of ψ instead of just a statement that can be either true or false
faint terrace
but if I write this:
```
(φ_1 ∧ ... ∧ φ_n => ψ) => (Φ |= ψ)
```
the left side ca...
the claim on the left side is known to be true though. so u can have
true -> true
false -> true
false -> false
but
true -> false
is known to not exist
so what u know is that either the phis aren't all true, in which case u don't know what's going on with psi
or all phis are true, in which case u know that psi is also true
either case, u can take a look at all mappings F that lead all phis to true
u dont know much about those mappings yet, other than that
but by definition it follows that they are then called a model
since thats exactly how a model is defined
and u know bc of the -> that under those F, the psi is also true
so... by definition u got that psi is an extension of Phi
and done
ocean plank
the claim on the left side is known to be true though. so u can have
true -> tru...
Ah so I can always start with left side is true here
faint terrace
u cant assume that that is the case. but ur not interested on those cases
u specifically want to look at the mappings that lead the left side to true
and the A -> B part, which in this direction is given to be a true statement, then tells u that B is true as well under those mappings
like, u start with "lets look at mappings for which the left side is true"
and then "since A -> B, B must then be true as well"
and that's pretty much it, definitions follow that F are mappings and psi follows from phi
modest yokeBOT
@ocean plank
ocean plank
@faint terrace can you pls take a look
faint terrace
<@157994153806921728> can you pls take a look <:peepo_heart:885480929206874132>
the fact that the statement is a true statement doesn't tell u whether psi is true or false
psi is B in ur table
ocean plank
yea but I argued that if A is true then B is also true
faint terrace
if u take "any mapping", u also include the mappings that have
f f
u can't take any mapping. u have to take a mapping that specifically leads to A being true
ocean plank
ah ok
faint terrace
perhaps u even meant that. but the word "any" sounds as if u could take any mapping
ocean plank
yeah lmao
faint terrace
u have to take a mapping that's a model
the first sentence is kinda obsolete. about what happens for false A
its unnecessary for the proof
ocean plank
ok
faint terrace
otherwise it's all good. i would mention that F is a model of Phi
faint terrace
looks good to me
u might want to reword either "für alle belegungen F" or "O.b.d.A. sei F eine Belegung,..."
obda means ohne beschränkung der Allgemeinheit
or u say "sei F eine beliebigige Belegung die Modell von Phi ist..."
just to make clear that this holds for all of them. since that's required for the folgerung
its only a Folgerung if it holds for all models
not just perhaps one model