What is the role of logic in philosophy, mathematics, and other disciplines?

[Willowz]

I will try to make my OP based on the rules this sub-forum subscribes to, if not please inform me. Sorry.

My question is about logic. How did we acquire it? Was it evolutionary? How is it that Japanese logicians do very much the same work as white American ones do. I am asking because logic seems so fundamental to everything we do.

Logic (from the Greek λογική logikē)[1] is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science. Logic examines general forms which arguments may take, which forms are valid, and which are fallacies. In philosophy, the study of logic figures in most major areas: epistemology, ethics, metaphysics. In mathematics, it is the study of valid inferences within some formal language.[2] Logic is also studied in argumentation theory.

 

[wuliheron]

There is no single type of logic proven to describe everything we observe and the brain is still a huge mystery in many respects so this question prompts endless metaphysical speculation. However, I would suggest that looking for patterns in the world around us promotes survival and it is pretty obvious evolution selects for the ability to use logic.

 

[FlexGunship]

Perhaps it's worthwhile to define logic more broadly (as it is commonly used) as ANY formal system of thought; any system of reasoning which follows a consistent set of rules.

Therefore, you can say that someone has poor logic (a system of reasoning that yields incorrect results) or someone is being illogical (making decisions that would seem to defy a formal system of thought).

So, how did we acquire it? Through judicious application and refinement. The universe seems to be imbued with a sort of implicit logic which can be discovered and mankind seems to be able to formulate its own system of logic which can take known truths and help to reveal new truths.

 

[LaurieAG]

Therefore, you can say that someone has poor logic (a system of reasoning that yields incorrect results) or someone is being illogical (making decisions that would seem to defy a formal system of thought).

Don't forget about the difference between Sophism, to use clever deceit, and Paralogism, an argument violating the principles of valid reasoning.

 

[Upisoft]

Logic is abstraction of the world surrounding us. Or more precisely, it is abstraction of our perception of the world. Just like math. It is not fundamental at all. Only your perception makes you feel it is fundamental.

It is quite possible that some life can exist who can perceive directly the quantum weirdness. They probably would have similar feeling that randomness is fundamental.

 

[FlexGunship]

Logic is abstraction of the world surrounding us. Or more precisely, it is abstraction of our perception of the world. Just like math. It is not fundamental at all. Only your perception makes you feel it is fundamental.

But only valid, working logic can carry you from initial known truths to new, previously unknown, truths.

It is quite possible that some life can exist who can perceive directly the quantum weirdness. They probably would have similar feeling that randomness is fundamental.

What?

 

[Upisoft]

But only valid, working logic can carry you from initial known truths to new, previously unknown, truths.

Please, give an example of that happening.

What?

This question does not give any clue what is your problem with understanding what I've said.

 

[FlexGunship]

Please, give an example of that happening.

Chemistry.

This question does not give any clue what is your problem with understanding what I've said.

I don't understand why you speculated about a lifeform that could experience, first hand, "quantum weirdness." The topic is logic.

You're talking about intuition. The workings of the universe on very small scales are non-intuitive, they're not illogical. I don't even know what it would mean for physics to be illogical. Furthermore, the antithesis of logic would not be randomness.

 

[chroot]

It is possible that logic evolved simply as a way to win arguments, and thus exert power in social groups. Humans are intensely social animals, and it makes good sense to me that the primary purpose of our cognitive tools is to succeed in social interactions. Logical understanding, and the pursuit of hard-edged truth, might be secondary purposes.

http://www.nytimes.com/2011/06/15/arts/people-argue-just-to-win-scholars-assert.html?_r=2&hp

- Warren

 

[FlexGunship]

It is possible that logic evolved simply as a way to win arguments, and thus exert power in social groups. Humans are intensely social animals, and it makes good sense to me that the primary purpose of our cognitive tools is to succeed in social interactions. Logical understanding, and the pursuit of hard-edged truth, might be secondary purposes.

http://www.nytimes.com/2011/06/15/arts/people-argue-just-to-win-scholars-assert.html?_r=2&hp

- Warren

I read that article, actually, a while ago; I remember it came out around my birthday. It seems to me that what developed was "reason" more than "logic" as outlined in the study. Would it be appropriate to say that there's a difference between reason and logic?

Reasoning is the act of providing supporting statements for your argument.
Logic is what determines whether those statements are valid or not.

Reasoning with sound logic:

Guy 1: "We should hunt deer tonight."
Guy 2: "Why? I think we should forage for berries."
Guy 1: "We need both food and clothing; berries cannot provide clothing."
Guy 2: "Good point. Let's go."​

Reasoning with dubious logic:

Guy 1: "We should hunt deer tonight."
Guy 2: "Why? I think we should forage for berries."
Guy 1: "It's cold, so deer are more likely to be moving and will be easier to hunt."
Guy 2: "Good point. Let's go."
(Never mind that deer may or may not actually be more mobile in the cold and that, by being mobile, would be easier to hunt.)​

 

[mathal]

My question is about logic. How did we acquire it? Was it evolutionary? How is it that Japanese logicians do very much the same work as white American ones do. I am asking because logic seems so fundamental to everything we do.

Langauge is the first logical system that we acquire as children. Each language has it's own coherent set of rules that all speakers of the language use, unconciously. The ability to think logicaly likely developed historically alongside the ability to speak.

mathal

 

[disregardthat]

I really don't appreciate the notion of logic as somehow accidentally applying. It is as if you somehow could imagine a logical argument not being valid, but at the same time accept that it is since logic has served you in the past. Or that it accidentally worked, hence our brains evolved to work around this assumption.

It's nothing like that.

Logic isn't arbitrary in the sense that it somehow works, and we don't know why.

Logic is part of the structure of language. For example: that we can infer A from (not (not A)) is because that is how the word "not" is used, it is how the word functions in conjunction with propositions.

That fact is arbitrary in the sense that "not" didn't have to function that way. But if it didn't, then "not" wouldn't have the same meaning as it does now. One could imagine that double negation was to be interpreted as strong denial, such that the proposition (not not A) would be interpreted as (not A). This isn't a challenge to logic, but a alternative sense of the word "not".

The logical rules are simply a reflection of the function our words have in conjunction with propositions. Therefore they are absolute and undeniable, but not in the sense that we couldn't have a different logic. Rather, a different logic would require words with different meanings. But this would give a different sense of "proposition". Furthermore, the reason our logic seem undeniable is not because our words function correctly (as if they could function incorrectly), but because we know how they function. We know how to apply "not". Therefore, denying a logical argument is like denying the meaning of our words. That is, denying the way we know to apply them. That's the source of the firmness of logic.

 

[apeiron]

Logic is part of the structure of language. For example: that we can infer A from (not (not A)) is because that is how the word "not" is used, it is how the word functions in conjunction with propositions.

You seem to be arguing a Wittgenstein "meaning is use" approach here. The atomic elements of logic (the operations and quantifiers like: "not", "or", "and", "there exists", "for all") are true by definition. They are true because that is the system as we invented it.

Yet there is still seems to be a need to justify this invention. We use these syntactic elements not simply as an arbitary custom, but because they seem fundamentally true to us. So what is the basis of that truth?

As Tarski's undefinability theorem argues, this basis is not going to be found within the terms. We can't use the logical elements to prove themselves.

But still, there seems to be a rationalistic support for them (arguments from reasonableness) as well as empirical support (arguments from observation).

The empirical support comes from successful modelling of the world. If we think about things this way, we can see that it is a form of analysis that gives us control over the events described.

And we can trace a history of improvement in such modelling. Animal minds evolved to be pretty successful at controlling their worlds. There is a proto-reasoning in that animals are good at forming effective habits. Then human language lifted reasoning to another level. Every sentence is a statement of cause and effect (subject, verb, object, or "who did what to whom"). Then philosophers, principly Aristotle, refined the business of reasoning into logic - a mathematical strength syntax for doing world modelling. A formal model of causality.

And it was only this last bit that has been consciously rational in its justification. As with the law of the excluded middle, there was a careful and step-by-step development that pared away the arbitrary so as to leave only what could not be denied as correct - correct through self-consistency, symmetry, irreducibility. Or if we look really closely, through the argument of dichotomy. The arrival at metaphysical categories that are complementary pairs, formed by demanding that they are mutually exclusive, jointly exhaustive.

(I know I keep saying the same thing, but it is the very thing that history has seemed to forgotten - for reasons closely to do with what logic has become.)

So for example, the law of the excluded middle itself was the demand that statements be framed in ways that they are either true or false. Ordinary language always has to be about something, even when referring to unicorns. So semantically, truth and falsehood are rather fuzzy judgements (unicorns might exist undiscovered, the idea exists at least, the component parts of horse and horn exist as facts it seems, etc). But Aristotle cut the syntax free of the semantics. He said if statements can be constrained so that all vagueness, indeterminacy, etc, could be eliminated, then would have a purely syntactic status based on the binary dichotomy of true~false.

So there was a rationalistic process of clarification. In real life - based on empirical experience and the inescapable vagueness of all semantic claims - nothing is actually completely false or true in a way that we "definitely can know it" via our experience. But syntax is where we get to make things true by definition. We say, there just exists this general dichotomy of true~false as a global constraint on properly formed (ie: logical) statements. Of course, this leap from experience to true by definition is immediately then justified in pragmatic terms - we find that this further step works to improve our modelling of reality. But a leap still gets taken.

(Again, all this is a re-hash of Pattee's epistemic cut, Rosen's modelling relations, and other modern schools of epistemology - even though these guys are scientists rather than philosophers .)

What then of the atomic elements of logic syntax then - the fundamental vocabulary of "not", "or", "and", "there exists", "for all"? Where are the dichotomies that are the rational basis of these?

The quantifiers of "there exists" and "for all" are simply the standard metaphysical dichotomy of particular~general (or one~many, specific~universal, local~global - the other allied ways of saying the same thing). "There exists" is a statement about particular existence, and "for all" is a statement about general existence.

You then have the three qualifiers of negation, addition and exclusion. In general, they rely on the atomistic hypothesis. This is the belief that reality reduces to atoms in a void. You have located objects with properties whose existence is irreducible. And they then freely do their thing in a void - in an a-causal backdrop that just is, and does not itself influence the goings on happening within it.

So the metaphysical dichotomy is atom~void. And it must be noted that it is not at all realistic in fact. The demand is that all causality is reduced to a collection of parts. The whole exists only in a way that does not actually count. But still, this was a useful point of view. It certainly did not capture the whole of the truth of reality, but it was a stunningly successful partial story because it was so simplified, and allowed so much to be ignored.

A syntax of logic was then developed from this metaphysics. A leap was made that treated atomism as complete truth.

Atomism is based on the idea of local additive construction - material cause + effective cause in the Aristotelean scheme.

So negation is legitimated by the binary of atom or void - there are only two possible choices, that something exists, or that it does not exist.

Addition is legitimated by the atomistic irreducibility of existence and properties. If something exists, then that doesn't change. So two things coming together are the sum of what existed.

Exclusion is legimated again by atomism in that if something definitely exists at a location, then nothing else can exist at the same place. It becomes a definite case of either/or.

So atomism captures a particular mental image of causality (as causal atoms behaving completely freely within an a-causal void - ie: material/effective cause). And then rationalistic argument extracts a syntactical basis for logic from this. If atomism were true, these must be its consequences. The most basic qualifications of a state of affairs will be in terms of negation (does something exist - yes/no?, addition (what exists can only be summed - a conservation principle for atomistic essences), exclusion (if something exists at a spot, then nothing else can exist at the same spot).

Again, this is a syntax that works. But which is also rationally invented at the final step. An epistemic cut has to be made between the semantics and the syntax so as to have a syntax. At some point, we cut the cord. We say atomism seems enough the empirical truth of reality to just make the jump and treat it as actual known truth. And get on with using this invented tool of reasoning.

Of course, as I also keep saying, atomism is itself one half of a metaphysical dichotomy. There is also the holistic or systems view of causality. In ancient Greece, the two views were still being entertained. If you read Aristotle's metaphysics, it is largely an attempt to reconcile the two apparent extremes of causality. This is why he talked about four causes - including formal and final cause in his scheme. The "causes of the void" we might say. The global, top-down causality of constraints on local freedoms.

But the simple logic represented by atomism took off. The other possible half of logic has languished. It has flared in the work of Hegel and Peirce. It is there again in systems science and semiotics and other attempts to frame a more holistic model of causality.

So there is a "mental image" of holism, just as there was of atomism. But it has not been developed into a completely stripped-down syntax. Although see this paper for some playing around with the possibilities.

http://homepages.math.uic.edu/~kauffman/CHK.pdf

This essay explores the Mathematics of Charles Sanders Peirce. We concentrate
on his notational approaches to basic logic and his general ideas about Sign,
Symbol and diagrammatic thought.
In the course of this paper we discuss two notations of Peirce, one of Nicod and
one of Spencer-Brown...

...The reason, I believe, that portmanteau and pivot are so important to find in
looking at formal systems, and in particular symbolic logic, is that the very
attempt to make formal languages is fraught with the desire that each term shall
have a single well assigned meaning. It cannot be! The single well-assigned
meaning is against the nature of language itself. All the formal system can
actually do is choose a line of development that calls some entities elementary
(they are not) and builds other entities from them. Eventually meanings and full
relationships to ordinary language emerge. The pattern of pivot and portmanteau
is the clue to this robust nature of the formal language in relation to human
thought and to the human as a Sign for itself...

...It is important to note that with the primary arithmetic, Spencer-Brown was
able to turn the epistemology around so that one could start with the concept of a
distinction and work outwards to the patterns of first order logic. The importance
of this is that the simplicity of the making (or imagining) of a distinction is always
with us, in ordinary language and in formal systems. Once it is recognized that the
elementary act of discrimination is at the basis of logic and mathematics, many of
the puzzling enigmas of passing back and forth from formal to informal language
are seen to be nothing more than the inevitable steps that occur in linking the
simple and the complex...

 

[Willowz]

But still, there seems to be a rationalistic support for them (arguments from reasonableness) as well as empirical support (arguments from observation).

The empirical support comes from successful modelling of the world. If we think about things this way, we can see that it is a form of analysis that gives us control over the events described.

I'm not sure; but, you may be mistaking mathematics for logic here.

And it was only this last bit that has been consciously rational in its justification. As with the law of the excluded middle, there was a careful and step-by-step development that pared away the arbitrary so as to leave only what could not be denied as correct - correct through self-consistency, symmetry, irreducibility. Or if we look really closely, through the argument of dichotomy. The arrival at metaphysical categories that are complementary pairs, formed by demanding that they are mutually exclusive, jointly exhaustive.

This is why we cannot depend on dichotomies (or what you call metaphysical dichotomies, unbeknownst to me), they are a result of the LEM which is bound with our "internalistic scaffolding of logic". Where you latter criticize logical atomism for its over-simplicity I think it avoids creating strict metaphysical dichotomies which you present as our only way of understanding things (or I have misunderstood your position or lackof).

 

[apeiron]

(or I have misunderstood your position or lackof).

Apparently so. Are you suggesting that atomism does not depend on the notion of the void?

 

[Willowz]

I just don't see how you draw out these dichotomies.

 

[disregardthat]

It is important to distinguish:

inductive reasoning; reasoning concerning causality,

and

logical arguments.

It is not the same thing, and should not be mixed up.

There is no need nor reason to verify a logical argument. When we state a logical proposition, we decide that it is a logical proposition subject to reasoning by logic rules.

Reasoning of causal consequences are not logical arguments, they require radically (categorically) different means of verification.

Let's take the example mentioned some time ago in this forum: consider the situation; 'A man is standing in the doorway of a room.' We could argue that "he is in the room", and that "he is not in the room". But when we consider both of these alternatives as valid in some sense, we are at the same time deciding that we are not stating logical propositions.

"The man is standing in the room" as a logical proposition does not refer to the man, the room, nor the meaning of 'standing in the room'. As a logical proposition it is entirely separate from these things.

We use logical rules as a criteria for determining whether a means of verification of a statement is valid. In other words, if our means of verification for determining whether the man is in the room allowed us to draw both conclusions, namely "he is in the room" and "he is not in the room", we will automatically dismiss it. We require that the meaning of "standing in the room" must adapt itself to this.

Logic is in this sense part of the structure of language, it is our rules governing propositions. As they don't refer to anything outside of language, it need not be verified. A good illustration of this is the fact that we would be perfectly fine operating with a different type of logic.

 

[apeiron]

I just don't see how you draw out these dichotomies.

It's in the definition: mutually exclusive, jointly exhaustive. It gets easy with practice.

 

[apeiron]

Logic is in this sense part of the structure of language, it is our rules governing propositions. As they don't refer to anything outside of language, it need not be verified. A good illustration of this is the fact that we would be perfectly fine operating with a different type of logic.

All you are pointing out here is that logic is pure syntax, and has been scrubbed clean of semantics. Which is what I said.

But the OP was about the development of logic. Which was what I was addressing.

 

[Willowz]

It's in the definition: mutually exclusive, jointly exhaustive. It gets easy with practice.

But, my point is the method. If I understand this properly, in simple terms you are taking the inverse of anything(?)

 

[apeiron]

But, my point is the method. If I understand this properly, in simple terms you are taking the inverse of anything(?)

More like the reciprocal in that there is a breaking that usually involves scale as well. The canonical dichotomy is local~global, so in some sense one half ends up shrinking to be as small (or localised) as possible, the other expands to be as large (or global) as possible.

So take something standard like the dichotomy of discrete~continuous. The discrete ends up being the local pole, and the continuous is the global pole. So you could perhaps say, in any given reality, discrete = 1/continuous. The larger you make one value, the smaller you make the other value. Even if we are talking here about qualities rather than quantities!

This is an important point because inverse operations can be of the same scale. For instance, the plus and minus of electric charge seems to be a dichotomy. But really it is only an anti-symmetry. And unstable as a result.

Positive and negative, left and right. These are breakings of symmetry where the scale factor is unchanged and so it is very easy to flip one back into the other. In ontological terms, the symmetry breaking is trivial.

But if symmetries are broken across scale, then the two qualities being produced are, in effect, a long way away from each other. It is no longer easy to annihilate the difference. You can't flip the discrete into the continuous or vice versa because they have moved so far apart as kinds of state.

And then as regards to taking the reciprocal/inverse of anything, of course, you can't do this with just anything. What is the inverse of cat, or plutonium, or Venus? Metaphysics is all about getting in behind these kinds of particular instances so as to extract the fundamental abstract possibilities.

After several centuries of debate, the ancient Greeks came up with a bunch of these dichotomies that we still use. We have refined them, but I don't think we've actually added any critical new ones to them.

 

[Willowz]

It's very interesting because it seems you are committing yourself to essentialism. Whereas I and possibly disregardthat take the social constructionism (of science, logic, possibly even math) approach.

 

[apeiron]

It's very interesting because it seems you are committing yourself to essentialism. Whereas I and possibly disregardthat take the social constructionism (of science, logic, possibly even math) approach.

In fact I clearly argue both. We model reality. But there is also a reality. That can be doubted, but there ends up not being very much point in actually doubting it.

You say you take the social constructionist view. And I just summarised the history of that construction of the syntax of logic. If it seems an argument from essentialism, well that historically was the approach that worked.

 

[disregardthat]

Justifying logic is like justifying grammar, logical rules are correct in the same way grammatical rules are correct. In this sense they are arbitrary, but needs no justification.

 

[Willowz]

Justifying logic is like justifying grammar, logical rules are correct in the same way grammatical rules are correct. In this sense they are arbitrary, but needs no justification.

But, they can't be arbitrary and consistent, can they?

 

[disregardthat]

But, they can't be arbitrary and consistent, can they?

How can a logic be inconsistent?

 

[FlexGunship]

Justifying logic is like justifying grammar, logical rules are correct in the same way grammatical rules are correct. In this sense they are arbitrary, but needs no justification.

But, they can't be arbitrary and consistent, can they?

How can a logic be inconsistent?

Are you going for an irony-award with this string of posts? Or was it tongue-in-cheek? If you say that logical rules don't need to be justified and can be arbitrarily defined then there's no guarantee that the resultant set of rules is consistent.

 

[disregardthat]

Are you going for an irony-award with this string of posts? Or was it tongue-in-cheek? If you say that logical rules don't need to be justified and can be arbitrarily defined then there's no guarantee that the resultant set of rules is consistent.

Logic can formally be seen as a method of determining a unique truth value of propositional formulas given the truth-values of the atomistic formulas within. I can hardly imagine logic as such being inconsistent. You are probably thinking of axioms of mathematics.

 

[FlexGunship]

Logic can formally be seen as a method of determining the truth of propositional formulas given the truth-values of the atomistic formulas within. I can hardly imagine logic as such being inconsistent. You are probably thinking of axioms of mathematics.

Are you saying that the axiomatic foundations of a formal logic system can be arbitrarily defined, or that the system can be arbitrarily defined?

If you're saying the axiomatic beginnings of a logical system can be arbitrarily define, then I'm okay with what you're saying now, but that's certainly not how you started with your first post and is consistent with my earlier post:

But only valid, working logic can carry you from initial known truths to new, previously unknown, truths.

Grammar has no such obligation to be consistent and, in fact, it's quite evident in English alone.

 

[disregardthat]

To put it like this as I have done before: (not not A) could be said to have the same, or the opposite truth-value of A. The latter would yield a different logic. But the point of the matter is that we do use the word "not" in the way that (not not A) is recognized as A, and for this there needs not be any justification, metaphysical or otherwise.

I don't know what you mean by saying that english grammar is inconsistent. Could you give an example?

 

[FlexGunship]

I don't know what you mean by saying that english grammar is inconsistent. Could you give an example?

Consistent and inconsistent.
Flammable and inflammable.

 

[disregardthat]

Flammable and inflammable.

A = NOT A

This simply shows that the in-prefix is not universally considered a logical "not"-operation, as we don't mean "the opposite of flammable" by inflammable as well as "flammable". But I get your point. This isn't what we mean by logical consistency though.

It is important to note too (before we get into more peculiarities) that it is the intended (understood) meaning of the word or sentence that matters. That we can mean different things by the same word in different situations and contexts doesn't imply any inconsistency.

 

[FlexGunship]

This simply shows that the in-prefix is not universally considered a logical "not"-operation, as we don't mean "the opposite of flammable" by inflammable as well as "flammable". But I get your point. This isn't what we mean by logical consistency though.

It is important to note too (before we get into more peculiarities) that it is the intended (understood) meaning of the word or sentence that matters. That we can mean different things by the same word in different situations and contexts doesn't imply any inconsistency.

If you plan to hand-wave each example as "an exception to the rule" I'm not sure what would satisfy your conditions for exceptions to grammatical rules. Here's a different type of example where a well defined prefix is used when the root is no longer part of the language.

Counterpoint and point.
Countermand and ?.

 

[disregardthat]

If you plan to hand-wave each example as "an exception to the rule" I'm not sure what would satisfy your conditions for exceptions to grammatical rules.

This is ridiculous, you are the only one saying it's a rule. No one insists on "inflammable" meaning both "flammable" and "not flammable".

 

[FlexGunship]

This is ridiculous, you are the only one saying it's a rule. No one insists on "inflammable" meaning both "flammable" and "not flammable".

I elaborated.

EDIT: However, this is off topic. I took issue with your statement the rules of logic are arbitrary in a similar manner to language. And that they lack the need for justification. Common language to formal logic is not a fair comparison.