> For example, if it is true that Aunt May believes that Spider-Man fights crime, then even though Spider-Man = Peter Parker, it does not follow that Aunt May believes that Peter Parker fights crime.
If you can't substitute Peter Parker for Spider-Man, then they are by definition not equal...
It may or may not be true that 12345x54321 = 670592745. (I either put in the correct number there, or changed one digit in the middle at random.)
Let's suppose I didn't change it, and that you haven't yet checked. Then those two things are, really truly, equal -- they are two names for the exact same number -- but you don't know that they are equal. So the _meanings_ of "12345x54321" and "670592745" are not exactly the same, even though if "=" belongs anywhere it belongs between those things.
Philosophers call the thing that's the same the "extension" and the thing that's different the "intension" (not to be confused with "intention" which is an ordinary non-technical word). Sometimes you can safely substitute one thing for another whenever the extensions match -- e.g., when doing algebra. Sometimes you can't -- e.g., when talking about someone's beliefs.
I am not a fan of Edward Feser and I think this article is pretty wrongheaded, but there isn't anything specifically wrong with what he says about Peter Parker and Spider-Man.
But that doesn't mean there's anything wrong with logic itself. Whatever model you come up with is still going to be based on classical logic so at some point you still need to assume that that remains valid in order to do anything at all. And I've never heard of a case where this could fail, at least not within the finite worlds one can in practice model.
and then you can also throw in BELIEVES(AUNT-MAY, NOT(NAME-OF(X, "PETER-PARKER"))) if you like.
That is, these two things are different: EXISTS(X): BELIEVES(AUNT-MAY, FIGHTS-CRIME(X)) AND BELIEVES(AUNT-MAY, NAME-OF(X, "SPIDER-MAN")) and BELIEVES(AUNT-MAY, EXISTS(X): FIGHTS-CRIME(X) AND NAME-OF(X, "SPIDER-MAN")), and the second seems to me like a better translation of "Aunt May believes that Spider-Man fights crime", at least for some of the states of affairs you might summarize with that sentence.
Aunt May has not been given the documentation that explains this, which is probably an intern's fault.
Metaphysically Peter Parker, Spider-Man, Aunt May, and the intern are all mocks anyway. So you can't use them for anything real.
(Except maybe entertainment, distraction, and metacognition.)
For materialists, it is not difficult to see that the brain of Aunt May believing only "Spider-Man fights crime" would be in a different state than if she also believed "Peter Parker fights crime", as going from the former to the latter would require some physical change.
This is too reductionist.
Can you substitute Peter Parker(before the lab accident) for Spider-Man ?
It's obvious that you can't substitute Peter Parker (before the accident) for Spider-man.
The Mathematical idea of equality-as-substitution ignores time. Which is why I called it "reductionist"
But in the extreme case you can also construct universes which only speak of inequalities (total order) as a rejection of the identity axiom.
In that universe Peter Parker and Spider Man are different entities, and so what you can and can't say about them is entirely down to judgment.
The most important question is WHY do you want to say anything about anything and who do you want to say it to?
It's difficult to encode any information in language without knowing who is going to be decoding it.
The most obvious example of A != A being true I can think of is the evaluation of this expression in any programming language:
Time.now == Time.nowAny two objects are the same, except for their differences. Any two objects are different, except for their similarities.
"Sameness" and "difference" are abstract assertions. So in saying that you are doing programming; or maybe I am doing fuzzy logic.
From my perspective it seems like you are evaluating this expression:
same?(Logic, Programming) -> False
You can’t switch between symbolic representations to use the meaning of one to make statements about the other because the meaning of the symbols also switch.
Telling me what I can and can't do is an imperative statement...
But if you want to communicate with others you’ll need to agree on your symbology.
In the common mathematical understanding A != A is defined as a false statement. In any example that one can contemplate it would become a true statement then you are replacing one of the ‘A’ symbols with something that is not ‘A’ in your mind.
Which is to say you are using the wrong symbols to represent the idea you intend to communicate.
A != A
If, however, you wish to communicate the evaluation of a function such as "now()", then you would express that as two invocations of that function
now() != now()
or
t != t'
or simply
A != B
So simply using 'A' to mean things not exclusively 'A' is unnecessarily ambiguous. This makes it not only useless for communication between people but also makes it useless for forming any meaningful conclusions.
Failure to be completely expressive about a paradigm is not itself a paradigm. Its just faulty reasoning.
I am telling you that in my model what I am saying is true . Yes I am talking about function invocation. In Ruby:
a = lambda { Time.now }
a.call != a.call
x = “Time.now != Time.now”
eval(x) == true
f() != f()
!( f() == f() )
A != A
Now is not now.
f() != f()
Do note two things:
The notation f.call is the same as f(), i.e. function invocation. You're not comparing functions but invoking them and comparing their results.
I don't know the equivalent syntax in Ruby, but in other languages the following is always true:
f == f
Note that the assertion "two things that are not the same might be different" is completely unsurprising. We're just letting syntax confuse us in this discussion.
It depends on which evaluation strategy you are using. Call by value or call by reference.
def f
return Time.now
end
f == f
f == f
The evaluation strategy must come after we agree whether we're comparing values after invocation or definitions. If it's invocations, the evaluation strategy you pick can change the meaning of the program, so that we must discuss semantics.
In all of these cases,
A == A
A straightforward way can be to simply compare the text of the functions, as written in your programming language of choice, in this case Ruby.
Just take the text of both functions, do a diff, and if it says they are the same, they are the same.
Another comparison could be more abstract, since we know there's more than one way to accomplish the same with a programming language: two functions are the same if, given the same input, they produce the same output. You can either prove this by extension, if the number of possible inputs is finite and small, or you can prove it logically. With some languages which enforce purity using types, you can even show two functions are the same just by looking at their signatures, no need to evaluate anything! (this is one advantage of some statically typed languages, by the way!)
> There are no implicit arguments to the function I have in mind. Passing an argument to the function I have in mind is an error.
Yes, there are implicit arguments. You don't see them because they are implicit! A function that returns the moment in time, "now", of course depends on an implicit argument: the current time! How else is it going to know the moment of "now", otherwise? Think about how such a function is implemented.
Indeed. I pointed this out and provided a link in my first post on this thread.
>You don't see them because they are implicit!
The fact that my function takes no arguments is implicit in the specification of my function. I know because I said the English sentence "The function that I have in mind takes no arguments".
You seem to be suggesting that I don't know what function I am thinking about but you do. This is a very peculiar hypothesis.
If you want to argue otherwise, you must provide this hypothetical implementation that complies with your specs. I'm telling you that your function effectively can't exist as you describe it. In order for your function to work it requires an implicit argument (the "world" if you want to be coarse, though in this case we know it's the time).
If you want to argue that
magical function is not magical function
I don't know how to respond to this. Just observe the thing I am showing you. THAT is what I am talking about. It's right before your eyes - I took it from my head and made it real.
>If you want to argue that
I DON'T want to argue! All I am doing here is expressing myself - you are the one jumping down my throat.
This is me telling you that I am not interested in any social game of figuring out "who is right or who is wrong". If that's the game you are playing - I am happy to terminate the interaction.
What I mean by f() != f() in the abstract is precisely that Ruby program in the concrete. Nothing more - nothing less.
There is no room for "right" and "wrong" here. Those are moral judgments and I've committed no moral violation of any sort.
I don't need to know the implementation of the real function Time.now from Ruby because, without looking at its source code, I can tell you that it has implicit arguments.
How can I tell this? Because it's impossible for your Ruby program to tell the time without some kind of interaction with an external source (be it the computer clock, the internet or whatever); this source is your implicit parameter and it's why two invocations of your function don't return the same value. It is impossible for your function not to have this source of time, either implicitly or explicitly.
If you disagree, then please explain how your function tells time.
I don't need to know the implementation of the real function == from any programming language because, without looking at any source code I can tell you that it has implicit arguments: a tautology.
If you disagree, then please explain how equality functions.
Being a simple bear, so formal logic breaks my head, I agree.
Just a hunch:
These "impedance mismatches" are one root cause for our society's current epistemological crisis. Cases where we can't even agree to a shared base truth.
Further, "now is not now" is a bit like paradox, no? The ability to hold two contradictory thoughts in your head at the same time. A large fraction of people recoil from ambiguity. Also a source of endless strife.
I am being as expressive as my expressivity allows!
for x in all-entities-in-universe: SOMETIMES(x = x)
class A: pass
class B:
def __eq__(self, other): return False
a = A()
b = B()
assert (a = a)
assert not (b = b)
Don't moralise about my reasoning, when you are done implementing fault() we can pass my function as an input to yours. That will settle the matter without ceremony.
A = Time.now
A != Time.now
A == A
[1] pry(main)> eval("a != a")
NameError: undefined local variable or method `a' for main:Object from (pry):1:in `eval'
[2] pry(main)> eval("Time.now != Time.now")
=> true
It's not equivocation because [2] is expressed exactly the same way in English. Now is not now. It is trivially true.
When I use a word, Humpty Dumpty said in rather a scornful tone, it means just what I choose it to mean — neither more nor less
While the syntax is "now", what you're really saying is now(Ti) for some i.
So the sentence, in both natural language and most programming languages becomes:
now(T0) != now(T1)
Now, some "clever" programming language might automatically translate this from
now != now
a := now(T0)
a != a
The important thing is that "now" is not a comstant, but a function.
f() != f()
A != A
So "now" is a trickery to hide the fact you're really saying "now(Ti)", for varying i.
I'm not tripping over syntax, but I think you are:
You think you're saying
A != A
A1 != A2
now(t0) != now(t1)And of course the second form is unsurprising.
The function I am talking about doesn't take any arguments.
[1] pry(main)> Time.now(1)
ArgumentError: wrong number of arguments (given 1, expected 0)The expression A = A can be trivially re-written in another notation as equal(A,A). The reason you don't question the implementation of the equal() function is because it's declared as being true however it's implemented.
So, I declare f() != f() as being true. I also declare that f() takes no arguments. All I am doing is translating (transpiling?) English into Ruby.
"Now is not now" => "Time.now != Time.now"
Talking about function arguments is not even in point. What's the argument to now() in English?
Incorrectly.
> "Now is not now" => "Time.now != Time.now"
Sure, “Now (pointedly looks at watch) is not now (pointedly looks at watch)” is true, and the equivalent of the Ruby. Ruby expressions lack the implicit simultaneity that English sentences wthout some contextual signal of explicit time dependency have, and your mistranslation ignores this.
The English and Ruby terms map 1:1.
Now(Time.now) is not(!=) Now (Time.now)
Why do you call the correct "incorrect"?
Nowhere; that's the contextual signal that defeats the implicit simultaneity the English sentence would have without that body language.
The English sentence "Now is not Now" describes my thought process (1st person view), not my body language (3rd person view).
My thought process occurs over time. Exactly like the Ruby process.
All impure functions have an implicit dependency on (some subset of) the state of the universe when they execute; in mathematics and logic something with this kind of dependency is typically noted by parameterizing it by time when the actual precise nature of the dependency is outside of the scope of the immediate analysis (because a dependency on the state of the universe at the time of execution is equivalent to a dependency on the time of execution with a hidden function mapping time to state of the universe.)
That impure functions methods are in many programming languages syntactically not distinguished from pure ones does not change the logic here.
That Time.now(at time t0) is distinct from Time.now(at time t1) is... unsurprising, and not an example of A != A in logical terms.
Tell us more! I didn't believe in omniscience until now.
Time.now(t1) and Time.now(t2) is meaningless in this context because t1 and t2 are unbound variables. What are you passing to your function? Where is it coming from?
My function says you are overlooking something.
[1] pry(main)> Time.now(t1)
NameError: undefined local variable or method `t1' for main:Object from (pry):1:in `__pry__'
[2] pry(main)> Time.now(1)
ArgumentError: wrong number of arguments (given 1, expected 0) from (pry):2:in `now'As for the rest: please provide the implementation of your Time.now function and I'll tell you its implicit parameters. If you're not talking about an actual function that can be run by a computer, then please tell me so and I'll drop it -- I'm not interested in hypothetical functions tautologically defined not to be equal to themselves.
You are missing the point about extensional and intensional properties of functions.
If you want to look "inside" my Time.now function then maybe you should also look "inside" your equality function and find its implicit parameters?
Here's some Python boiler plate - implement equality and make the assertions pass.
def f():
return 1
def equal(x,y):
# Implement equality
assert not equal(1,2)
assert equal(1,1)
assert equal(f, f)
As far as I can tell you are not talking about such a function either when you speak about equality, but I'll reserve judgment until you produce an implementation.
>I'm not interested in hypothetical functions tautologically defined not to be equal to themselves.
OK, but you are interested in hypothetical functions tautologically defined to be equal to themselves.
That's perfectly fine - we are interested in different tautologies.
You want to say equal(f,f) is true.
I want to say not equal(f,f) is true.
We want to say different things.
You only want to talk about pure functions. I want to talk about ALL functions.
So lets define the function "pure" such that
pure(f) -> True when f is a pure function
pure(g) -> False when g is an impure function
> "All impure functions have an implicit dependency on (some subset of) the state of the universe when they execute"
You wrote:
> "You only want to talk about pure functions. I want to talk about ALL functions."
Find the error.
Still waiting for you to implement equal() as a "pure" function.
We can discuss what you mean by "error" once we have a referent (that's not just in your head) to talk about.
In [1]: class A:
...: def __eq__(self, other):
...: return False
...:
In [2]: a = A()
In [3]: assert a != a
In [4]: assert a == a
AssertionError
Will you now recognise my right to free speech and free thought, or must I fall in line?
Parameters ARE inputs. Results ARE outputs. Renaming them to something else is just obscuring the fact that you are still talking about a function with side-effects.
So using Python as my model of computation this is a pure function, but it does nothing:
def f(): pass
def g(): return True
def h(x): pass
def j(x): return x
def k(): 2+2