Is logic the apex of human intellect and understanding?

[Singular Intellect]

Essentially what the thread title asks...can one effectively argue logic as the top of the pyramid for human understanding and reason?

The more I interact with people, the more I realize how the majoity of people have very little grasp of basic logic or the difference between educated opinions and uneducated ones coupled with personal beliefs. Or how easily they are can be swayed by irrelevent bullshit or strings of words without any actual meaning whatsoever. Particularily things like descriptions of spirituality, ie: "we're all different colored lights of thought trying to find the great truth of <insert specific bullshit belief here>", etc, etc.

At times I've found it quite irritating when dealing with individuals, who when crushed under the weight of actual logic in any arguement, ultimately resort to useless sayings like "logic isn't the end all or be all", or "logic doesn't always apply", etc. I like to reply along the lines of "Well then, I'm right because there's dirt on that rock over there." The blank stare or confused questioning I get after I always like to reply with "Hery, you just said logic doesn't apply, right? I'm playing by your rules..."

Personally I think it's just the chickenshit way of dealing with someone who is more knowlegable/logical than yourself.

Thoughts? Can one proudly assert logic as the apex of human intellect, or must one grovel to some religion/spirituality as some 'superior' understanding/feeling about any particular issue?

[Darth Wong]

Imagine if someone had a compelling argument that made perfect sense except for one flaw: it assumes that 2+2=10. No matter what the rest of the argument looked like, one could simply not accept it as long as it incorporates that ridiculous assumption.

Logic is like math. It does not constitute a full understanding by itself, but incorrect logic invalidates any argument.

[Zac Naloen]

The arguement of "Logic can be used to prove anything" is the one that gets my goat.


Really?

So logic can be used to prove the earth is flat?

[Dooey Jo]

If most people didn't have a basic grasp of logic, I doubt Monty Python and similar types of comedy would be as popular as they are. Instead of laughing, they'd go:
"Ooh, they are slapping each other with herrings. That makes sense."

So logic can be used to prove the earth is flat?

As much as it can be used to prove that the Earth is spherical. The premises any such argument would rest on would need to be empirical and wouldn't constitute much of a proof.

[Darth Wong]

If most people didn't have a basic grasp of logic, I doubt Monty Python and similar types of comedy would be as popular as they are. Instead of laughing, they'd go:
"Ooh, they are slapping each other with herrings. That makes sense."

What the fuck kind of argument is that? People must have sound logical thinking skills because they get Monty Python? Are you trying to be sarcastic or something?

[Singular Intellect]

Imagine if someone had a compelling argument that made perfect sense except for one flaw: it assumes that 2+2=10. No matter what the rest of the argument looked like, one could simply not accept it as long as it incorporates that ridiculous assumption.

Logic is like math. It does not constitute a full understanding by itself, but incorrect logic invalidates any argument.

I take it you're just basically saying logic is only as valid as the premise of any particular arguement.

Would it be fair to say that any specific premise is also dependent upon sound logic? For example the shape of the Earth, based upon observations and tests, it's logical to conclude it's roughly a sphere as opposed to declaring some illusion is going on.

[Metatwaddle]

If most people didn't have a basic grasp of logic, I doubt Monty Python and similar types of comedy would be as popular as they are. Instead of laughing, they'd go:
"Ooh, they are slapping each other with herrings. That makes sense."

What the fuck kind of argument is that? People must have sound logical thinking skills because they get Monty Python? Are you trying to be sarcastic or something?

He's saying Monty Python is funny because it's illogical to most people (it's basically a string of total non sequiturs), and to have a sense of what's illogical, we must have a sense of what is logical.

[Dooey Jo]

If most people didn't have a basic grasp of logic, I doubt Monty Python and similar types of comedy would be as popular as they are. Instead of laughing, they'd go:
"Ooh, they are slapping each other with herrings. That makes sense."

What the fuck kind of argument is that? People must have sound logical thinking skills because they get Monty Python? Are you trying to be sarcastic or something?

He's saying Monty Python is funny because it's illogical to most people (it's basically a string of total non sequiturs), and to have a sense of what's illogical, we must have a sense of what is logical.

Yes, that was what I meant. I'm not saying it necessarily means they have a good grasp of it, but I think people generally lack rationality rather than logic. Might be interesting though, to ask the most illogical of people what they think of Monty Python...

[Terralthra]

Imagine if someone had a compelling argument that made perfect sense except for one flaw: it assumes that 2+2=10. No matter what the rest of the argument looked like, one could simply not accept it as long as it incorporates that ridiculous assumption.

Logic is like math. It does not constitute a full understanding by itself, but incorrect logic invalidates any argument.

I take it you're just basically saying logic is only as valid as the premise of any particular arguement.

Would it be fair to say that any specific premise is also dependent upon sound logic? For example the shape of the Earth, based upon observations and tests, it's logical to conclude it's roughly a sphere as opposed to declaring some illusion is going on.

Given that logic is a set of rules governing generating conclusions from premises, it necessarily relies on accurate premises.

Premises are dependent on whether or not they match reality. In effect, the conclusion of a logical argument depends on a) truthful premises and b) valid logic. The logic does not make the premises true; true premises do not make an argument logical. The two together, make a valid conclusion.

[Darth Wong]

If most people didn't have a basic grasp of logic, I doubt Monty Python and similar types of comedy would be as popular as they are. Instead of laughing, they'd go:
"Ooh, they are slapping each other with herrings. That makes sense."

What the fuck kind of argument is that? People must have sound logical thinking skills because they get Monty Python? Are you trying to be sarcastic or something?

He's saying Monty Python is funny because it's illogical to most people (it's basically a string of total non sequiturs), and to have a sense of what's illogical, we must have a sense of what is logical.

So if someone can laugh at other people being embarrassed (ie- humour based on someone's lack of social skills), this must mean that he himself has good social skills? This doesn't follow at all (ironic, for a thread about logic).

[Molyneux]

The arguement of "Logic can be used to prove anything" is the one that gets my goat.


Really?

So logic can be used to prove the earth is flat?

You're working from the wrong definition of "to prove". Like "the exception that proves the rule", what you're looking for here is "to prove" as synonymous with "to test" - you can't use logic to prove a false proposition is true; however, if something is a proposition, then logic can be used with empirical evidence to find its truth or falsity.

[Starglider]

All logic is essentially set theory, something I've become intimately familiar with over many years of implementing various AI logic systems. The core assumption that makes logic work is simply that binary classifications exist; 'an entity cannot simultaneously be part of a set and part of its complement'. The various forms of logic are formal symbol systems based on that assumption (where the inference rules are pre-derrived from it for convenience). If the assumption holds in the system you're trying to reason about (and as far as we know, it always does in reality) and you correctly encode true facts about the target system into your logical premises, then correct use of logic will always generate models of other facts true in reality (your conclusions).

Of course in practice there's plenty of scope for error in the encode/formalise, decode/apply and logical inference stages. Thus the existence of 'inconsistency tolerant' logics that attempt to derrive useful results even when the input data can be proven to be inconsistent (broken). Probability essentially comes from expanding the model to represent the observer's beliefs about the target system seperately from the system itself, then derriving inference rules (the axioms of probabilistic logic) that will always (provably) produce the best calibrated (not quite the same thing as 'most accurate') beliefs possible. A subset of this is 'where do the priors come from', an active debate; IMHO it's possible to objectively demonstrate that Kolmogorov complexity (essentially formalised Occam's razor) is the optimal prior but a) I can't personally do this formally and I have yet to see a truly convincing formal proof from anyone else and b) AFAIK this does require some additional (although sensible) assumptions about the nature of reality. Of course there is also the fact that it seems to work best in practice, hardly formal but rather compelling.

The next complication after that is the fact that full formal analysis rapidly becomes impossible on real systems due to computational intractability. This leads to things like defeasible reasoning and heuristic approximation if you just grope around for solutions. The best solution (in AI design, not human reasoning) is to include the inference system itself in the model, along with an expected utility model for how to allocate computational effort, so that the system effectively generates the best possible beliefs given limited knowledge /and/ computing power. The problem with /that/ is reflective regress; reasoning about how to apply computing power itself consumes computing power, and so on in a reflective regress. You have to make this converge to zero for the system to do anything at all, which is a non-trivial design challenge.

[Starglider]

If most people didn't have a basic grasp of logic, I doubt Monty Python and similar types of comedy would be as popular as they are. Instead of laughing, they'd go:
"Ooh, they are slapping each other with herrings. That makes sense."

Most comedy comes from a violation of expectations, not logic as such. Those expectations can be logically derrived, but more often they're intuitive or even arbitrary social norms.

Oh, and

Thoughts? Can one proudly assert logic as the apex of human intellect,

Hell yes. Unfortunately humans are fairly awful at it (at least relative to the raw computational power of our brains), both because it's hard to implement in wetware and because there has been very little selection pressure to develop formal reasoning skills over our evolutionary history (as opposed to say good motor skills or intuitive social skills). Fortunately technology has gotten steadily better at providing mental prosthetics, a trend that looks set to continue (and indeed accelerate).

or must one grovel to some religion/spirituality as some 'superior' understanding/feeling about any particular issue?

That would be a 'fuck no'.

[Surlethe]

Logic is not the apex of human intellect and understanding, but the foundation. Because it is necessary for any argument to hold, all of mathematics and the sciences incorporate it as an assumption.

[Darth Wong]

All logic is essentially set theory, something I've become intimately familiar with over many years of implementing various AI logic systems. The core assumption that makes logic work is simply that binary classifications exist; 'an entity cannot simultaneously be part of a set and part of its complement'.

That's binary/boolean logic. There is also probabilistic logic.

[Starglider]

All logic is essentially set theory

That's binary/boolean logic. There is also probabilistic logic.

I covered that in my original post. Probabilistic logic is also based on set theory; it has to be, since it's an extension of propositional logic to cope with uncertainty (partial knowledge). It works by formally modelling 'beliefs' about the state of the target system (the 'external referant') rather than just having a one-to-one correspondence between logical entities and external entities. Probability distributions are essentially a set of subsets of the set of all possible outcomes. The 'probabilities' are the relative cardinalities. This is easy to envision for discrete events, such as dice rolling. Roll two six sided dice and there are 36 possible outcomes, which we can group into 11 sets (with cardinalities 1:2:3:4:5:6:5:4:3:2:1) to get the probability distribution over the total value. For continuous probability distributions it's a bit more complex. We don't know whether infinite sets actually exist in real life (e.g. is fundamental physics fully quantised or not), but for normal reasoning purposes it doesn't matter; we normalise the cardinalities to sum to 1.0, such that the absolute cardinalities no longer matter. Similarly it doesn't really matter whether we simply don't know what happened in the target system (i.e. the frequencies exist in the model only) or if it is a many-worlds type setup with objective frequencies. PL works regardless.

Thus all the theoretical derrivation of probabilistic logic is expressed in boolean logic and ultimately set theory; unsurprising since set theory is the most common basic grounding for maths in general. This is at the core of the AI system I am currently working on; during 'cold startup' it bootstraps PL (and various other things) starting with a basic propositional logic system and some minimal axioms.

[Gil Hamilton]

So if someone can laugh at other people being embarrassed (ie- humour based on someone's lack of social skills), this must mean that he himself has good social skills? This doesn't follow at all (ironic, for a thread about logic).

I think the point they are trying to make is that alot of comedy, including alot of Monty Python skits, are based on an inherent and well crafted disconnect between what is expected and what is performed. There is a real world logic being violated in an unexpected way, that's why it's funny.

However, it's not really formal logic in the same sense you are talking about. with arguments and Ps and Qs. I think the word is being used a bit broadly in their example to mean "the way things are supposed to happen".

[Wyrm]

All logic is essentially set theory, something I've become intimately familiar with over many years of implementing various AI logic systems. The core assumption that makes logic work is simply that binary classifications exist; 'an entity cannot simultaneously be part of a set and part of its complement'. The various forms of logic are formal symbol systems based on that assumption (where the inference rules are pre-derrived from it for convenience). If the assumption holds in the system you're trying to reason about (and as far as we know, it always does in reality) and you correctly encode true facts about the target system into your logical premises, then correct use of logic will always generate models of other facts true in reality (your conclusions).

Starglider, you confuse "can be stated in terms of" with "is founded upon". The rules of logic stand on their own. The law of negation, which you state as 'an entity cannot simultaneously be part of a set and part of its complement,' can be stated without referring to sets: 'for any proposition P, P ∨ ¬P is true' — a statement that encodes a lot of information. (And your statement is incomplete anyway.) Just because you can cast the law of negation into set-theoretic terms does not mean that it is the only way to do so.

On the other hand, the operators of set theory (union, intersection, and complementation) are usually defined in terms of propositional logic. If logic is founded upon set theory, then it would be improper to use logic to define its most basic operations.

Finally, you can't get very far in set theory beyond defining terms without logic. Without logic, you can't even say 'Suppose A = {0, 1, 2} and B = {0}. Therefore, B is a subset of A' because that is a logical deduction. It is logic that allows you take two true propositions (the givens of A and B) and the definition of subset and discover a new true proposition.

All logic is essentially set theory

That's binary/boolean logic. There is also probabilistic logic.

I covered that in my original post. Probabilistic logic is also based on set theory; it has to be, since it's an extension of propositional logic to cope with uncertainty (partial knowledge).

You're right here, for the wrong reasons. Probabilistic logic assumes sigma algebras, which is an extension of boolean algebras. Set theory doesn't come in through propositional logic.

[Starglider]

Starglider, you confuse "can be stated in terms of" with "is founded upon".

No, but I am using a wider context than you.

The rules of logic stand on their own.

You can't even state the rules of a logic without a context capable of expressing them. To actually evaluate boolean logic, set theory or any other calculus you need a production rule system. The inference rules of the logic are the productions, the axioms and derrivations for any particular problem are the statements. Strictly a production system does not generate statements, it categorieses them, so really any such implementation of ;boolean logic ;is an implementation of propositional calculus, possibly with some extra restrictions. Classic boolean logic starts with a small set of true statements and every other expressible statement implicitly undecided, and progressively moves statements from the 'undecided' category to the 'true' category until there are no executable productions left. More sophisticated versions implement 'false', 'provably undecideable' and (in inconsistency-tolerant logics) 'undecideable due to axiom inconsistency'.

The law of negation, which you state as 'an entity cannot simultaneously be part of a set and part of its complement,' can be stated without referring to sets:

This is not the law of negation, because set membership is the primitive; the statement that an element is a member of a set and the statement that an element is a member of another set that happens to be the first set's complement are distinct. We are saying that E (element of) S <=> ! E (element of) S'.

Just because you can cast the law of negation into set-theoretic terms does not mean that it is the only way to do so.

That is the only way to do so in a genuinely independent fashion. You can express set theory, including the entire system required to evaluate set theory, in terms of set theory (including the additional information about casuality needed to produce a working system, though clearly to actually produce a working system from the spec you must already have a working evaluator).

On the other hand, the operators of set theory (union, intersection, and complementation) are usually defined in terms of propositional logic.

You can define set theory in propositional logic only in the context of an external evaluation system that requires at minimum predicate logic to express. Predicate logic is effectively a comination of boolean logic and set theory and in fact it is exactly what you get if you load both kinds of production rule into a system. AFAIK you can't get set theory from propositional logic alone even given an external evaluation system, but you can get propositional logic from set theory alone given such a system and some extra axioms regarding external reference.

I admitt that the notion of derriving boolean logic from set theory is somewhat academic in that to actually make it work, you must have that evaluator that implicitly or explicitly uses first order logic anyway. But then you seem keen to ignore the implicit mechanisms humans use to evaluate boolean logic anyway, so I don't see why you shouldn't ignore that too.

Boolean logic systems have much less relation to reality than set-theoretic ones and no innate generality. In applying boolean logic to a real world problem you are entirely dependent on a human to formulate the axioms and translate the results to and from reality - or a fixed translation mechanism (i.e. binary sensors and acutators) build by humans. Boolean logic can't capture the external reference semantics themselves; it can't reflectively model the desired model / reality correlation. Set theory can, and forms the basis for a system that can generalise automatically.

If logic is founded upon set theory, then it would be improper to use logic to define its most basic operations.

Actually no it wouldn't. Reflection is a key requirement of many applications of logic, so eventually you're going to have to create a circular definition. Such a regress must always be grounded by a preexisting external system capable of evaluating the logic without reflection.

Finally, you can't get very far in set theory beyond defining terms without logic. Without logic, you can't even say 'Suppose A = {0, 1, 2} and B = {0}. Therefore, B is a subset of A' because that is a logical deduction.

No, it is not. It's a production rule firing. You're calling it a logical deduction because you usually think of it that way, but you can (and I have) built systems that evaluate set theoretic propositions and come to derrived conclusions without having boolean logic explicitly defined.

It is logic that allows you take two true propositions (the givens of A and B) and the definition of subset and discover a new true proposition.

No, that's production rule firing again. In practice, that is actually a kind of causality, but that's getting into a whole temporal can of worms which you don't need to consider at the basic logic level. Boolean logic is of cour

Probabilistic logic is also based on set theory; it has to be, since it's an extension of propositional logic to cope with uncertainty (partial knowledge).

You're right here, for the wrong reasons. Probabilistic logic assumes sigma algebras, which is an extension of boolean algebras. Set theory doesn't come in through propositional logic.

??? Sigma algebra is defined in terms of set operations.

[Wyrm]

The rules of logic stand on their own.

You can't even state the rules of a logic without a context capable of expressing them.

Same goes for set theory.

To actually evaluate boolean logic, set theory or any other calculus you need a production rule system. The inference rules of the logic are the productions, the axioms and derrivations for any particular problem are the statements. Strictly a production system does not generate statements, it categorieses them, so really any such implementation of ;boolean logic ;is an implementation of propositional calculus, possibly with some extra restrictions.

Boolean logic ≠ propositional calculus/logic. A calculus is a rule system for manipulating statements of symbols. (Compare with lambda calculus.) That has nothing to do with set theory.

Classic boolean logic starts with a small set of true statements and every other expressible statement implicitly undecided, and progressively moves statements from the 'undecided' category to the 'true' category until there are no executable productions left. More sophisticated versions implement 'false', 'provably undecideable' and (in inconsistency-tolerant logics) 'undecideable due to axiom inconsistency'.

Categories ≠ sets. Also, in propositional logic, we don't even really care whether or not an idividual atom is true or false, just on the kind of logical deductions we can form from them.

This is not the law of negation, because set membership is the primitive; the statement that an element is a member of a set and the statement that an element is a member of another set that happens to be the first set's complement are distinct.

Or, put another way, an individual proposition may be either true or false, but not both or any other truth value. No set theory there.

I'm still not seeing the "based on set theory" you're claiming.

We are saying that E (element of) S <=> ! E (element of) S'.

Your statement is false. x∈S≢¬(x∈S) by definition.

That is the only way to do so in a genuinely independent fashion. You can express set theory, including the entire system required to evaluate set theory, in terms of set theory (including the additional information about casuality needed to produce a working system, though clearly to actually produce a working system from the spec you must already have a working evaluator).

And I can state the propositional calculus entirely in terms of the propositional calculus. Your point?

You can define set theory in propositional logic only in the context of an external evaluation system that requires at minimum predicate logic to express. Predicate logic is effectively a comination of boolean logic and set theory and in fact it is exactly what you get if you load both kinds of production rule into a system.

Again, boolean logic ≠ propositional calculus.

AFAIK you can't get set theory from propositional logic alone even given an external evaluation system, but you can get propositional logic from set theory alone given such a system and some extra axioms regarding external reference.

Again, confusing "can be stated in terms of" with "is based on". The propositional calculus works on propositions. If we regard "x∈S" as atomic propositions for each x and S, and "S⊆T", "S∩T", and "S∪T" defined in terms of those atomic propositions, then propositional logic provides a completely sound foundation for set theory.

I admitt that the notion of derriving boolean logic from set theory is somewhat academic in that to actually make it work, you must have that evaluator that implicitly or explicitly uses first order logic anyway. But then you seem keen to ignore the implicit mechanisms humans use to evaluate boolean logic anyway, so I don't see why you shouldn't ignore that too.

Boolean logic systems have much less relation to reality than set-theoretic ones and no innate generality. In applying boolean logic to a real world problem you are entirely dependent on a human to formulate the axioms and translate the results to and from reality - or a fixed translation mechanism (i.e. binary sensors and acutators) build by humans. Boolean logic can't capture the external reference semantics themselves; it can't reflectively model the desired model / reality correlation. Set theory can, and forms the basis for a system that can generalise automatically.

Once again, boolean logic ≠ propositional calculus.

If logic is founded upon set theory, then it would be improper to use logic to define its most basic operations.

Actually no it wouldn't. Reflection is a key requirement of many applications of logic, so eventually you're going to have to create a circular definition. Such a regress must always be grounded by a preexisting external system capable of evaluating the logic without reflection.

Except using propositional logic in the way I described, to define set theoretic operations, is not an example of reflection, but a violation of the knowledge hierarchy. To say "'S∩T' means, for all elements x, x∈S∩T implies x∈S and x∈T", means you know what it means to 'imply' and what 'and' means, as well as for the statement to apply to 'all elements x'.

No, it is not. It's a production rule firing. You're calling it a logical deduction because you usually think of it that way, but you can (and I have) built systems that evaluate set theoretic propositions and come to derrived conclusions without having boolean logic explicitly defined.

One more time, boolean logic ≠ propositional calculus.

Here's a challenge: Prove to me that if x∈B, then x∈A∪B without using propositional logic. I'll be very impressed if you can do it.

No, that's production rule firing again.

Shut up about production rules and answer my challenge.

??? Sigma algebra is defined in terms of set operations.

Which is where the set theory of probabalistic logic comes from, idiot. I just denied that the set theory came in through propositional logic.

[Darth Wong]

Starglider, would you say that all of mathematics is also based on set theory? Because I don't see how the argument for that proposition would be any different from the argument you're making now about all of logic being based on set theory.

[Darth Wong]

So if someone can laugh at other people being embarrassed (ie- humour based on someone's lack of social skills), this must mean that he himself has good social skills? This doesn't follow at all (ironic, for a thread about logic).

I think the point they are trying to make is that alot of comedy, including alot of Monty Python skits, are based on an inherent and well crafted disconnect between what is expected and what is performed. There is a real world logic being violated in an unexpected way, that's why it's funny.

However, it's not really formal logic in the same sense you are talking about. with arguments and Ps and Qs. I think the word is being used a bit broadly in their example to mean "the way things are supposed to happen".

It's not "logic" at all. Monty Python humour is generally based upon violating storytelling conventions and social conventions, not logic. If you want examples of humour based on a truly irrational universe, try Looney Tunes. Especially situations like the Coyote painting a picture of a road on a roackand the Road Runner simply running into the fake road (and of course, when Coyote tries it, he slams face-first into the painted rock). That's a good example of humour based on a truly irrational universe.

Mind you, the fact that someone gets that kind of humour still doesn't mean he knows anything about logic. The fact that someone can recognize something which is really irrational doesn't mean he won't subscribe to all manner of lesser irrationalities. I'm reminded of religious people who believe that they are rational because they reject the beliefs of even more irrational religious people.

[Kuroneko]

Normally this would be splitting hairs, but Starglider is conflating working with a logic system with studying the properties of that logic system. It doesn't take set theory to state the primitives and axioms of a formal logic system and start proving particular theorems, but as a practical matter it does take set theory to study the general properties of the kinds of propositions that can or cannot be proved, as well as other properties like relative consistency, etc. For example, to prove that a proposition is not a theorem in the system requires some sort of metatheory that characterizes provability, which in turn requires something like set theory.

I say "as a practical matter" and "something like" in the above because although it is true that there is a general trend in mathematics to ground everything in set theory, that is simply because it's easiest, not because it's irreplaceably fundamental. For example, instead of developing graph theory in terms of set theory, one could treat graph theory axiomatically and develop set theory from it instead (the chain of development would be graphs→accessible pointed graphs→well-founded apgs = sets of standard ZF set theory), turning the usual approach on its head.