My Thoughts - 4 - Sociopath Community

RE: My Thoughts

WarmPotato · 2020-12-30T23:18:24.1200000

I'm too dumb to participate here

Continuation

Bob said:
Well, I used the word "mindlessly" more specifically to indicate the negative connotation that "conventional" typically carries, namely that there's implicitly no reason beyond convention to follow said rule. Naturally, no sensible person considers the practical application of Classical Logic to be mindless per se.

Okay, I understand. However, despite what you’re describing being something that occurs (there are militant X who believe Y is superior) in general I don’t see this a whole lot nor does it seem explicit nor implicit within the works of the Logicians that describe alternative mathematical logics (Brouwer may be an exception but reading him makes his strong position seem to be a necessity of the time as opposed to viewing LEM as inherently worthless). Kleene for instance, whom is an intuitionist, writes his major work Mathematical Logic via the classical logic, hence implicitly he views the classical logic as one with merit and perhaps the most useful given how common it is as a mathematical foundation.

What seems to be in question is where does this ‘merit’ come from? What makes a Logic viable? These are the questions the Logicians set out to answer and for them it is merely a system that is consistent in its verification and translatable to any other consistent system. Hence, Classical, Intuitionist, Linear, Modal, etc are not superior to one another by this view but rather systems that meet the conditions of a definition.

It seems the common view is that all logics/languages/systems that meet the definition are the most useful sort, it just so happens that Classical Logic is not the only one.

Bob said:
Now with that being said, I'm not sure how exactly one could investigate the "factuality" of such a language, especially if one's required to use a particular language to even argue its factuality, thereby adhering to certain assumed axioms.(edited)

Yes, this is an issue. In the context of the 20th century and beyond the solution seemed to be emphasis on Tautologies and Explicative processes as a means to verification.

This prompted my first Mathematical crises after reading Russell. Contructivism because of its reliance on language seemed to be on shaky ground and outright superficial. In the end I came to accept that in so far as the system is consistent and holds some correspondence that I can at the least use the system given its consistency.

Bob said:
Well, I was under the assumption that LEM was rejected, so if that isn't the case in practical application, naturally there would be little reason to have the consequences of such a rejection.

In intuitionistic logic LEM is 'rejected' in so far as it does not hold within the confines of the system, however it is a formula within the logical framework. So its existence is not rejected but rather by the assumption the system is constructed upon (this proof theoretical framework) it does not make sense.

Bob said:
Well, here you yourself at the end start distinguishing between ambiguity and reducibility as you seem to admit an inherent ambiguous nature in conceptual variability. The very nature of variability implies ambiguity, even if the ambiguity's range or nature is very accurately described, represented, reduced or limited.

I claim that all expressions when not defined are ambiguous. If a variable is symbolically expressed and is not defined, then it would indeed be ambiguous. However, I can reduce ambiguity and perhaps outright do away with it through definition (explication does seem to achieve its goal). I do not see why ambiguity must be innate to all expressions nor why it must be immutable

Bob said:
In this case, the fact that within colloquial algebraic notation variability is symbolically (and precisely) represented by (or reduced to) a mere "x" does not take away from the innate ambiguity concealed behind the symbol. The reducibility in this case doesn't take away from the conceptual ambiguity, and algebra acknowledges the need for this.

A variable (in the confines of a mathematical logic) is just the symbolic representation of a domain, it is well defined and conceptually not ambiguous imo.

X is a variable, and a variable is a symbolic representation of domain. As such you are saying that domains are ambiguous concepts and just because I hid it behind the symbol X doesn’t reduce the ambiguity of domain. We of course can on like this, start talking about ordered sets (lists), sets, elements, and eventually the very meaning of mathematical object. At such a point I concede and will say okay, at the foundations there may be some ambiguity within the system (assumptions or the like), but that ambiguity is inconsequential as it does not affect the consistency of the system. I still wonder if variable is ambiguous here though, does the ambiguity belong to variable merely because we can linguistically reach ambiguity from the term variable? Essentially, is ambiguity conserved across concepts regardless of an instantiation of definition?

Bob said:
Asking for a solution requires the ability to express an absence of information and said absence of information necessitates ambiguity.

We must be careful here. We do not do not ask for solutions of variables (we ask for their domains which I've addressed) but we do ask for solutions of expressions which constrain domains by mapping them to other domains. I think you mean that there are instances in which we ask for solutions to expressions and that implies a lack of information which implies ambiguity. As it concerns information, and this too would apply to case in which we ask for a domain of a variable, if a solution exists then technically all information necessary for solving should necessarily exist within the representation given the necessity for consistency. There does seem to be a sort of conservation within a symbolic system which is consistent though that is something I am not capable of articulating as of now.