Fundamentals of Logic

[Buffy]

I used that same example of street light being either (discreetly) red or green to point that precisely, there isn't such a thing in absolute terms as Red or Green - we don't even see Red the same way, .......... discreetness doesn't exist as a matter of reality - that's just how we perceive reality.

Pbbbbbtttbbt! Cop-out time! The judge in traffic court is more than happy to tell you even if you're color blind, the red one is the one on the top. I'm the first to admit to much fuzzyness (the other half of my post), but to say there are not discrete or absolutes *at all* is not true, something that you seem to admit:

Also, it's no coincidence we perceive reality sufficiently equal so that society can work

But 'cha lose me here:

that means that reality itself isn't different for all of us, that Absolute truths do exist after all, but we just can't perceive exactly as it is and thus absolute truths have no place in our understanding....In the ultimate sense, we are not guided by truth and light, but by faith.

So...they do exist, but because my brain "interprets" the wavelength of red "differently" then there is no truth and everything is based on faith... Uh... To me that sounds like a leap of logic to justify faith. If you wanna do that, that's okay, but to me you just are denying a logical conclusion because it conflicts with your world view, so don't expect me to be convinced....

 

Cheers,

Buffy

[Guest loarevalo]

I'm the first to admit to much fuzzyness (the other half of my post), but to say there are not discrete or absolutes *at all* is not true, something that you seem to admit:

But 'cha lose me here:So...they do exist, but because my brain "interprets" the wavelength of red "differently" then there is no truth and everything is based on faith... Uh... To me that sounds like a leap of logic to justify faith. If you wanna do that, that's okay, but to me you just are denying a logical conclusion because it conflicts with your world view, so don't expect me to be convinced....

I suppose whether you believe the universe to be a perfect continous sea of space (or energy, whichever you prefer), or a discreet grid (as asserted by Quantum Loop theory) is a matter of choice as long as those thories are just philosophy - not proven science. I, personally, would bet on the perfect continuum.

 

As a matter of opinion, I think all of our systems are (ultimately) discreet because that's what we can manage - we catalog, group, stereotype, form general descriptions of things, so that we can recognize them and use them in our mental processes. I could look on something and say "those are two slices of pizza" while another may "it's one slice" while still another may say "it's half a pizza" - we all interpreted reality differently, while reality was undoubtely the same. Frankly, physics wouldn't care to distinguish between one slice, two or whatever - the laws of physics are the same at every point everywhere.

[Qfwfq]

I really don't see the point of discussing the existence or not of Absolute Truth, concerning the fundamentals of logic. Logic is a mathematical topic. It is very useful, when applied appropriately.

[Guest loarevalo]

I really don't see the point of discussing the existence or not of Absolute Truth, concerning the fundamentals of logic. Logic is a mathematical topic. It is very useful, when applied appropriately.

I agree that Logic is useful, but like any other field of mathematics (and philosophy in general) its foundations are subject to debate, because the foundations are only assumptions based on our common sense and experience and intuition, and because some have a more enlightened intuition than others, there are bad and also good systems.

 

 

Sorry for straying a bit off topic. I have trouble distinguishing between the fundamentals of a theory and its foundational aspects. The fundamentals are pretty clear, one can find them in any logic (predicate calculus) textbook. Logic, in regards to mathematics and not philosophy, is a simple mathematical model: binary arithmetic. From that perspective, mathematical logic never says anything about truth, but only about T and F, or 0 and 1. Mathematical logic is simply a system modeling our understanding of truth, and it's not necesarily a correct or best model of it.

 

Mathematicians generally care little about the objectiveness of their systems - how they relate or apply to reality. Interestingly, they have lately cared a great deal about logic and not only about its consistency, but about it's accuracy as a model as if it were a model of physics. I think mathematicians have struggled with logic because logic is what creates the link between their mathematical systems existing in the "platonic realm" and objective reality; thus, the common of notion of "truth" among all intelligent humans is what gives their systems value and meaning. What do you all think?

[Qfwfq]

When Aristotle wrote his book entitled Logic, I'm sure he was aware that it couldn't prove the truth of an assert, not based on the truth of other asserts. The aim of logic has never been that of pulling a rabbit out of a hat.

 

Logic, in regards to mathematics and not philosophy, is a simple mathematical model: binary arithmetic. From that perspective, mathematical logic never says anything about truth, but only about T and F, or 0 and 1. Mathematical logic is simply a system modeling our understanding of truth, and it's not necesarily a correct or best model of it.

It is much more.

Mathematicians generally care little about the objectiveness of their systems - how they relate or apply to reality.

Of course, it isn't the point of mathematics.

Interestingly, they have lately cared a great deal about logic and not only about its consistency, but about it's accuracy as a model as if it were a model of physics.

Logic is very fundamental to mathematics, just like it is to all philosophy based on the Greek system. While it can be formalized as one branch of math, Boole algebra and even as a lattice, it is yet what the whole proceding of math is based on. Unknown to many people, in modern mathematics logic and set theory are more fundamental than numbers.

 

Although I doubt many mathematicians being seriously worried about it, the hypothetical event of someone proving "inaccuracy" of logic would mean demolition of mathematics altogether. Russel's paradox is more of a head scratcher and has caused much work to rebase set theory.

 

I think mathematicians have struggled with logic because logic is what creates the link between their mathematical systems existing in the "platonic realm" and objective reality; thus, the common of notion of "truth" among all intelligent humans is what gives their systems value and meaning. What do you all think?

I see this as a misconception of the role of logic in mathematics. This link isn't the point of math and it isn't logic anyway. The link is simply that a mathematical construct may be applied to anything real that resembles the math closely enough. Note, a physicist or an engineer might tend to see it the other way around.

[Guest loarevalo]

I think mathematicians have struggled with logic because logic is what creates the link between their mathematical systems existing in the "platonic realm" and objective reality; thus, the common of notion of "truth" among all intelligent humans is what gives their systems value and meaning. What do you all think?

 

Thank you for that opinion. What I meant by that "logic creates the link" is this: The notion of truth in mathematics is very diferent from other mathematical objects like sets, or functions, which don't necesarily exist objectively. However, truth needs to be the same both mathematically and in the real world. Where would mathematics be if we couldn't talk about the truthfulness of it? Truthfullnes, in my view, is what makes the theorems "real" in the same of being objectively real, like the desk or a cup of coffee are real. Truth, to me, is that ultimate object in mathematics that is in the "edge" of mathematics. What do you think about this, the fact that we can't go higher than truth:

 

1+ 2 : 3

1+2 = 3 : TRUE

1+2 = 3 <-> TRUE : ?

 

The operation in each statement are +, =, <->. What is on the other side of : is the result or evaluation of the operation. Is there a higher level of objects other than truth? What makes truth so special?

 

Obviously (1+2 = 3 <-> TRUE ) <-> TRUE

 

Are we stuck forever on this level?

Computer programers are probably more familiar with this phenomenon since they have to generalize (standarize) everything and can't settle on ambiguos intuitionistic definitions like mathematicians often do.

[Buffy]

"You’ll find, that the only thing you can do easily is be wrong, and that’s hardly worth the effort." -- The Mathemagician, "The Phantom Tollbooth"

 

I agree with Q, I think you're misinterpreting what logic's role is in math: when we run into fun paradoxes like those of Russell or Goedel, they're just fodder for more thinking, which sometimes takes time. The philosophical issue of mapping mathematical logic onto the real world is an interesting issue, and the folks that specialize in applied mathematics would take issue with your conclusion. Being a computer scientist, I can certainly attest to the interesting things that happen when you get into application, but the issues of implementation of theory rarely end up requiring throwing the baby out with the bath water, unless you're completely clueless...

 

Cheers,

Buffy

[Qfwfq]

However, truth needs to be the same both mathematically and in the real world.

Nooooooooooooooo!

 

Logic and epistemology are two different disciplines.

 

Computer programers are probably more familiar with this phenomenon since they have to generalize (standarize) everything and can't settle on ambiguos intuitionistic definitions like mathematicians often do.

:rant:

I don't see what computer programmers have to do with Tunisian penguins. I'm a computer programmer and I often need to apply logic or even a bit of arithmetic to real problems, in a way appropriate to those problems. A computer programmer, like many others who apply math to something real, isn't a mathematician.

 

Mathematicians don't settle on ambiguos intuitionistic definitions.

[Guest loarevalo]

I never said Logic and Epistemology were the same, actually I remember distinguishing them in a previous post: "mathematical logic says nothing about truth, only about T and F, 0 and 1,..."

However, what I meant was that truth is like a "protocol" by which all fields of study communicate. In that sense, truth is the most elemental object of philosophy. Mathematicians can (and do) create many versions of logic variating in their strict definition of truth, yet there remains always a primitive notion of truth that mathematicians can't tamper with. It's by this notion that mathematicians can propose their "new" logics without running in a vicious cycle of the "new" logic used in proving theorems of the same "new" logic - if they did this, that "new" logic becomes meaningless. The axioms of the "new" logic can't be expected to be judged by the same "new" logic - our primitive notion of truth must intervene do add some sense and interpretation to the system.

 

There always remains that primitive notion of truth, and mathematicians/logicians know it, and that's why they rather say nothing about truth, but about T and F, or whatever. Of course, now and then, mathematicians do have to defend and assert their theorems, and thus dare talk of truth.

 

Maybe the "primitive" logic I am refering is none other than the classical "T disjoint from F" logic of the greeks. What do you think? Could anyone rightfully disagree with the classical logic, and therefore must recur to a "primitive" notion of truth? How could we assert the truthfullness of classical logic (and thus assert that there is no more primitive notion) without at the same time making use the principles of said logic?

[Qfwfq]

I never said Logic and Epistemology were the same, actually I remember distinguishing them in a previous post: "mathematical logic says nothing about truth, only about T and F, 0 and 1,..."

But you did say:

However, truth needs to be the same both mathematically and in the real world.

which implies an expectation that math should be a search of Truth.

 

Logic being a formalization in which the notions of true and false are two values, it cannot ascertain anything about reality. As you're aware of this, I don't see what you're getting at.

 

Although logic is an example of a lattice, I disagree with those that consider other lattices like "other logics" because I find it misleading. Strictly, true and false are boolean values in math but mathematicians also dare use the same words as shorthand for whether a theorem can be proven or disproven. This is only a matter of not being pedantic, it's equivalent to saying whether or not a certain assert is true in a given set of axioms.

 

Maybe the "primitive" logic I am refering is none other than the classical "T disjoint from F" logic of the greeks. What do you think? Could anyone rightfully disagree with the classical logic, and therefore must recur to a "primitive" notion of truth? How could we assert the truthfullness of classical logic (and thus assert that there is no more primitive notion) without at the same time making use the principles of said logic?

Logic is Aristotelian logic. The only "primitive" notion of truth is what we observe in reality, that which epistemology talks of.

[Guest loarevalo]

I agree.

Logic is Aristotelian logic. The only "primitive" notion of truth is what we observe in reality, that which epistemology talks of.

Why do yo consider Aristotelian Logic to necesarily be THE logic?

I guess Godel's incompleteness also applies to system of logic, therefore there is no such system as THE system, either a mathematical formal system like Zermelo-Fraenkel, or a construct of logic like Aristotelian logic, or whatever logic.

Because of the inability to find the ultimate logic system, is why we need recurr to the "primitive" notion of truth.

[EWright]

Logic isn't based on absolutes.

 

Only a Sith speaks in absolutes. Consumed by the dark side he is.

[Guest loarevalo]

STAR WARS episode III summary:

enlightening story,

primitive dialogue,

suicide-inducing acting,

undecidedly a good movie.

 

Jesus said this (I can't remember where so I won't quote):

- If they're not against me, they are for me (talking about followers making miracles, or something).

- If you are not for me, you're against me.

 

STAR WARS ep. III is all about moral ambiguity - somewhat postmodernistic in that sense. :lol:

Sorry for straying off topic.