Posted on 10/25/2002 12:14:19 AM PDT by jennyp
Obviously, I disagree vehemently.
Applying logic to "different functions at different times" with the same name is committing the fallacy of the excluded middle.
Different domains of discourse are "forever separate". Trying to apply theorems from plane geometry to the problem of, say, putting your child to sleep is not likely to produce fruitful results. Discrete logic is about things in sets that are related to each other in some well-formed, functionally explicit way. It is not about the universe, as the universe encompasses more than that.
The separation of domains is the result of logic being a simple, finite mathematical tool which does not have limitless application. Take a tiny subset of the problem you propose for yourself: explain the unified logic theory that encompasses both formal plane geometry and peano's axioms of arithmetic in the same formal logic framework.
To finish your assignment, show me a crossover proof that uses lemmas derived from peano's axioms along with lemmas derived from plane geometry to formally demonstrate something valid--anything at all.
I'll save you some time. You can't even start to do it. The domain of discourse of plane geometry is is the ideal continuous plane. The domain of peano's axioms is discrete sets. This is like looking for recipes to bake an airplane.
A question that has perplexed, among others, Descartes, Kant, Berkeley, and Hume.
I like Popper's speculation on this subject: we can only sharpen our apprehension, principally by confirming that our observations are in line with those of others (many of whom, I might add, appear to be smarter than us). This proves nothing, since others are also outside throwing signals in, and could just as easily be evanescent shadows. However, if we chose to have faith in the existence and analytical instincts of others, we make progress in our joint understanding.
It's not something you can prove, it is a matter, in my opinion, of good faith, and good manners to assume that lining up our perceptions with the mainstream of thoughtful others is the right game to be playing to apprehend the world, despite it's subjective pitfalls.
Aristotalian logic was what was under discussion. I know it supports two operators because that's all I use when do traditional sorite(s).
The IMPLYs operator, as I've now demonstrated several times to Tares, applies between the major and minor predicates to produce a new statement. The NOT operator can be applied to a predicate. This is embodied in the various patterns you memorized that make up the valid implications of Aristotalian logic. All these are, is the IMPLYs operator for every combination of of universal and existential implication that's valid between two predicates.
The reason I know it, is that there ain't anything else to Aristotalian logic. Hard as you look, you will not find an AND or OR or NOR operator in use as a machine to derive predicates. You will only find them absorbed inside a predicate, where their logical function is irrelevant. The way to test this, as I have been trying convince tares, is to substitute a meaningless word for anything you think is logic inside a predicate, and see if you can still work the problem.
Hmmm. If I look at Peano's axioms, and I look at the techniques used in plane geometry, I see the former as helping to form the fundamental basis for the validity of the latter. For example, Peano's axiom #3 is essentially a statement of the technique of inductive proofs, which would have some utility in plane geometry.
I'll save you some time. You can't even start to do it. The domain of discourse of plane geometry is is the ideal continuous plane. The domain of peano's axioms is discrete sets.
However, there are discrete entities in plane geometry, for which discrete treatments are possible.
This is like looking for recipes to bake an airplane.
You've no doubt heard of aircraft made from composites -- which are, in fact, baked..... ;-)
Your larger point remains, however. When we limit discussion to areas that can be addressed by logic, we exclude real things that the logic does not, and cannot touch: things like beauty, deliciousness, ugly, boring, interesting (all of which can vary even within the perceptions of a single observer!)
It's also difficult to applly it to something like Tolkein's books -- which are pure fantasy on one level (Tolkein essentially defined a logic in that world); and yet still have significant meaning to us here in the real world.
There is no basis in deductive logic for inductive proofs. No deductive proof exists that induction works. It is something we take on faith, just like we take axioms and predicates on faith. Induction, like the "like" operator, (with which it has much in common), does not have a corresponding logical tautology, nor a proved theorem to it's name. So your example will not help you. I asked you to combine lemmas from the two disciplines to produce a valid proof. When you are talking about lemmas, you are usually engaged in formally serious efforts, and therefore talking about deductive, not inductive proof. Inductive proof is handwaving over the notion that what you have learned to expect is what you expect. It does not provide formal deductive security to a theorem.
The elements of the domain of discourse of plane geometry are points, lines, and planes in a CONTINUOUS field. All of plane geometry is theoretically accessable from the predicates and axioms. This is not true of the domain of discourse of discrete entities in sets. This seems like an unbridgeble difference to me, but if you have an example, by all means point to it.
Nicely put. I'll bow to your explanation, which sounds about right to me. There's indeed an implicit assumption that no counter-examples can be found -- which is a reasonable assumption in certain discrete cases. Inductive proofs are not necessarily true in non-discrete cases -- to my pea brain the Mandelbrot gives a good visual example of this.
(I'm also not interested in going to great lengths to prove my off-the-cuff comment....)
It's interesting to note, BTW, that your thumbnail description of inductive proofs is a fair description of how we go through life.
Those who claim to have derived a perfectly rational and logical ethical system are essentially claiming to do so on an inductive basis (they can only go by what they observe, and cannot have addressed all possible cases). Even the inductive claim is not convincing, given that there are observable counterexamples. The deductive proof is obviously out of the question.
Can the boolean operator IMPLY be derived from the Aristotalian operator IMPLY if I treat the Aristotalian predicates as boolean objects?
The logical axioms are uproved assumptions of both formal systems. To my knowledge there is no derivation step possible, except a silly tautology using the principle of identity.
In theory, you could derive predicate calculus from Aristotalian logic, in that aristotalian logic supports a non-monotonic set of operators. (NOT And IMPLY). Which ought to make the rest of the logical operators derivable as theorems, where your domain of discourse is set theory itself. At least, that was the received wisdom I got as an undergrad. I've not tried it personally.
Chances are, I'd guess, that it's never been done, except maybe as an undergraduate exercise at some university. I'd bet equivalent things have been done in circuit design, without it occuring to anyone to equate the two domains. It's extremely easy to fold all of aristotalian logic into a tidy corner of predicate calculus (three line sorites fold into one line predicate calculus statements). Producing predicate calculus from aristotalian logic would be, by comparison, an onerous and pointless exercise.
Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.