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Topic: LOGIC & MATHEMATICS
Replies: 96   Last Post: Jun 6, 2013 5:19 AM

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Peter Percival

Posts: 950
Registered: 10/25/10
Re: LOGIC & MATHEMATICS
Posted: May 31, 2013 6:24 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Nam Nguyen wrote:
> On 30/05/2013 4:54 AM, Zuhair wrote:
>> On May 30, 1:15 pm, Zuhair <zaljo...@gmail.com> wrote:
>>> On May 29, 5:29 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>
>>>
>>>
>>>
>>>
>>>
>>>

>>>> On 28/05/2013 6:06 AM, Zuhair wrote:
>>>
>>>>> On May 28, 7:44 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>>>> On 26/05/2013 10:17 PM, zuhair wrote:
>>>
>>>>>>> On May 26, 4:49 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>>>>>> On 26/05/2013 3:52 AM, Zuhair wrote:
>>>
>>>>>>>>> On May 26, 11:03 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>>>>>>>>>> On 26/05/2013 12:52 AM, Zuhair wrote:
>>>
>>>>>>>>>>> Frege wanted to reduce mathematics to Logic by extending
>>>>>>>>>>> predicates by
>>>>>>>>>>> objects in a general manner (i.e. every predicate has an object
>>>>>>>>>>> extending it).

>>>
>>>>>>>>>> [...]
>>>
>>>>>>>>>>> Now the above process will recursively form typed formulas,
>>>>>>>>>>> and typed
>>>>>>>>>>> predicates.

>>>
>>>>>>>>>> Note your "process" and "recursively".
>>>
>>>>>>>>>>> As if we are playing MUSIC with formulas.
>>>
>>>>>>>>>>> Now we stipulate the extensional formation rule:
>>>
>>>>>>>>>>> If Pi is a typed predicate symbol then ePi is a term.
>>>
>>>>>>>>>>> The idea behind extensions is to code formulas into objects
>>>>>>>>>>> and thus
>>>>>>>>>>> reduce the predicate hierarchy into an almost dichotomous
>>>>>>>>>>> one, that of
>>>>>>>>>>> objects and predicates holding of objects, thus enabling Rule 6.

>>>
>>>>>>>>>>> What makes matters enjoying is that the above is a purely
>>>>>>>>>>> logically
>>>>>>>>>>> motivated theory, I don't see any clear mathematical concepts
>>>>>>>>>>> involved
>>>>>>>>>>> here, we are simply forming formulas in a stepwise manner and
>>>>>>>>>>> even the
>>>>>>>>>>> extensional motivation is to ease handling of those formulas.
>>>>>>>>>>> A purely logical talk.

>>>
>>>>>>>>>> Not so. "Recursive process" is a non-logical concept.
>>>
>>>>>>>>>> Certainly far from being "a purely logical talk".
>>>
>>>>>>>>> Recursion is applied in first order logic formation of formulas,
>>>
>>>>>>>> Such application isn't purely logical. Finiteness might be a purely
>>>>>>>> logical concept but recursion isn't: it requires a _non-logical_
>>>>>>>> concept (that of the natural numbers).

>>>
>>>>>>>>> and all agrees that first order logic is about logic,
>>>
>>>>>>>> That doesn't mean much and is an obscured way to differentiate
>>>>>>>> between
>>>>>>>> what is of "purely logical" to what isn't.

>>>
>>>>>>> Yes I do agree that this way is not a principled way of demarcating
>>>>>>> logic. I generally tend to think that logic is necessary for
>>>>>>> analytic
>>>>>>> reasoning, i.e. a group of rules that make possible to have an
>>>>>>> analytic reasoning. Analytic reasoning refers to inferences made
>>>>>>> with
>>>>>>> the least possible respect to content of statements in which they
>>>>>>> are
>>>>>>> carried, thereby rendering them empirically free. However this is
>>>>>>> too
>>>>>>> deep. Here what I was speaking about do not fall into that kind of
>>>>>>> demarcation, so it is vague as you said. I start with something that
>>>>>>> is fairly acceptable as being "LOGIC", I accept first order logic
>>>>>>> (including recursive machinery forming it) as logic, and then I
>>>>>>> expand
>>>>>>> it by concepts that are very similar to the kind of concepts that
>>>>>>> made
>>>>>>> it, for example here in the above system you only see rules of
>>>>>>> formation of formulas derived by concepts of constants, variables,
>>>>>>> quantifying, definition, logical connectivity and equivalents,
>>>>>>> restriction of predicates. All those are definitely logical
>>>>>>> concepts,
>>>>>>> however what is added is 'extension' which is motivated here by
>>>>>>> reduction of the object/predicate/predicate hierarchy, which is a
>>>>>>> purely logical motivation, and also extensions by the axiom stated
>>>>>>> would only be a copy of logic with identity, so they are so
>>>>>>> innocuous
>>>>>>> as to be considered non logical.
>>>>>>> That's why I'm content with that sort of definitional extensional
>>>>>>> second order logic as being LOGIC. I can't say the same of Z, or ZF,
>>>>>>> or the alike since axioms of those do utilize ideas about structures
>>>>>>> present in mathematics, so they are mathematically motivated no
>>>>>>> doubt.
>>>>>>> NF seems to be logically motivated but it use a lot of
>>>>>>> mathematics to
>>>>>>> reach that, also acyclic comprehension uses graphs which is a
>>>>>>> mathematical concept. But here the system is very very close to
>>>>>>> logic
>>>>>>> that I virtually cannot say it is non logical. Seeing that second
>>>>>>> order arithmetic is interpretable in it is a nice result, it does
>>>>>>> impart some flavor of logicism to traditional mathematics, and
>>>>>>> possibly motivates logicism for whole of mathematics. Mathematics
>>>>>>> might after all be just a kind of Symbolic Logic as Russell said.

>>>
>>>>>>> Zuhair
>>>
>>>>>>>>> similarly here
>>>>>>>>> although recursion is used yet still we are speaking about logic,
>>>>>>>>> formation of formulas in the above manner is purely logically
>>>>>>>>> motivated.

>>>
>>>>>>>> "Purely logically motivated" isn't the same as "purely logical".
>>>
>>>>>>> A part from recursion, where is the mathematical concept that you
>>>>>>> isolate with this system?

>>>
>>>>>> I don't remember what you'd mean by "this system", but my point
>>>>>> would be
>>>>>> the following.

>>>
>>>>>> In FOL as a framework of reasoning, any form of infinity (induction,
>>>>>> recursion, infinity) should be considered as _non-logical_ .

>>>
>>>>>> The reason is quite simple: in the language L of FOL (i.e. there's no
>>>>>> non-logical symbol), one can not express infinity: one can express
>>>>>> "All", "There exists one" but one simply can't express infinity.

>>>
>>>>>> Hence _infinity must necessarily be a non-logical concept_ . Hence
>>>>>> the
>>>>>> concept such the "natural numbers" can not be part of logical
>>>>>> reasoning
>>>>>> as Godel and others after him have _wrongly believed_ .

>>>
>>>>>> Because if we do accept infinity as part of a logical reasoning,
>>>>>> we may as well accept _infinite formulas_ and in such case it'd
>>>>>> no longer be a human kind of reasoning.

>>>
>>>>> I see, you maintain the known prejudice that the infinite is non
>>>>> logical? hmmm... anyhow this is just an unbacked statement.

>>>
>>>> I did; you just don't recognize it apparently: my "The reason is quite
>>>> simple:" paragraph.

>>>
>>>>> I don't see any problem between infinity and logic,
>>>
>>>> Well, then, why don't you express infinity with purely logical
>>>> symbols, for us all in the 2 fora to see? Seriously, that would
>>>> be a great achievement!

>>>
>>> Infinity: Exist x (0 E x & (for all y. y E x -> {y} E x))
>>>
>>> where E is defined as in the head post.

>
> FOL doesn't have 'E' (as in your "{y} E x") as a logical symbol.
>

>>>
>>> while 0 and {y} are defined as:
>>>
>>> 0=e(contradictory)

>
> That's a bizarre concoction of symbols as far as FOL logical symbols
> are concerned: on both sides of '=' there are _invalid_ FOL logical
> symbols, namely '0' and 'e'. Iow, '0' and 'e' aren't FOL logical
> symbols.
>

>>> {y}=e{isy}
>>
>> a typo
>> correction: {y}=e(isy)

>>>
>>> Where 'contradictory' is defined as: for all x. contradictory(x) iff
>>> ~x=x
>>> and 'isy' is defined as: for all x. isy(x) iff x=y
>>>
>>> I consider the monadic symbol "e" as a "logical" symbol, also identity
>>> symbol is logical.
>>>
>>> The above infinity is a theorem of this logic.

>
> I did specifically specify "FOL" when I posed the challenge. Right?


Different authors define logical symbols in different ways, Zuhair is at
liberty to define them as he does.

--
I think I am an Elephant,
Behind another Elephant
Behind /another/ Elephant who isn't really there....
A.A. Milne


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5/26/13
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