Is there an Ultimate Logic?

[watsup]

It occurs to me that maybe I ought to say a little about what logic is. The history of logic began with Aristotle in the Prior Analytics. Logicians in the middle ages worried over some problems that obsessed Aristotle (like how to relate conditional statements--i.e. if...then...statements, to the other logical operators like AND or NOT) and did manage to complete the square of opposition in a manner that would have pleased Aristotle. But dealing with quantified predicates and having a fully worked-out understanding of hypothetical and modus syllogisms remained elusive.

Finally, in 1898, Gottlob Frege published the Begriffschrifft, which came to the attention of Bertrand Russell. Russell realized that Frege had solved the longest-standing problems--again, dealing accurately with quantified predicates and relating conditional statements to the other logical operators, thereby closing the loop on all the recognized forms of syllogism. His system did contain a fatal flaw, which Russell and Whitehead spent a lot of time trying to fix.

What emerged from those early 20th century efforts is this: logic is simply an artificially constructed language with formal definitions of its terms, and those definitions make it easy to understand the truth-relations between propositions. Consider the following three propositions: Socrates is a man, all men are mortal, Socrates is mortal. It seems intuitively clear that if the first two sentences are true, then the third one must also be true. That's a truth-relation between propositions. What logicians had always hoped is to find a way to set the evaluation of natural language arguments on as firm a footing as mathematics, and while that is now known to be impossible, we can check for validity (defined as: if the premises are all true, it is impossible for the conclusion to be false)--that is, we can check whether someone has reasoned correctly or not.

When I first started investigating logic (over 30 years ago now) I thought for sure that it was simply wrong, that it is an overly mathematized approach to the world, a rigid system that tries to apply artificial rules to what is in fact a complex and messy world. I gradually realized that I was wrong. What the logicians of the 20th and 21st century have done is they've modelled human reasoning almost perfectly. There are a few outstanding issues, but they're not ones that usually come up in conversation or debate.

Bertrand Russell and G.E. Moore apparently thought that the then-newly discovered structures of formal logic implied a materialist ontology. Pretty much every logician after them realized, however, that any such implication is downright silly. Logic needs to be as topic-neutral as possible, and the logicians from about 1930 onward went to work to make sure that logic is free from any kind of metaphysical entanglement.

There are a great many misconceptions about logic and what counts as logical. A lot of people think that science is logical and religion illogical or non-logical or some such, but this is just a hold-over from the logical positivists, who thought that we could essentially replace all of philosophy with science plus formal logic and mathematics. They were spectacularly incorrect, but at the time it seemed to be a promising program. Their ultimate failure showed, among other things, the need for logic to float free from metaphysical committments.

Logic these days can handle pretty much any set of ideas, topic, what-have-you. Scientific theories, religious arguments, philosophical arguments, scholarship methods, etc. are not themselves logic. They only use logic--or rather, they are the products of reasoning, and logic is a system we can use to check that that reasoning is valid.

Has logic ever provided solid evidence for the existence of a God? Seems like that could be seen as an “ultimate logic”.

 

[ashurbanipal]

Has logic ever provided solid evidence for the existence of a God? Seems like that could be seen as an “ultimate logic”.

Not that I'm aware of. Lots of people have made various arguments for the existence of God and claimed that those arguments are logically valid, and of course, opponents of those arguments disagree on that point (or that the premises of the arguments are true). I think you'll find logicians generally want logic itself to provide neither criticism nor support for either position.

 

[ataraxia]

Well, that's certainly true. There's no necessary relation between "tree" and the things that are (for example) in my back and front yards. Nor is there such a relation with "baum" and those things, or "dendron" and those things, etc.

Sure. But that's not quite what I meant. There are organisms which challenge our ability to neatly categorize them as a tree or an animal- let alone their categorization as particular species. Everything is evolving and changing all the time. Our vocabularies and categories, while helpful, are just attempts to try to provide some framework through which we can try to work. And when you look at different cultures, you realize that different cultures have attempted to do this with the same phenomena in very different ways- evidence that we are just trying to create categories based on generalizations that do not, and can never, carve nature at the joints that way- there is always just too much grey. But when you really get down to it, things blur and bleed into each other. Our language is a tool we have created to generalize and

Think about, for example, the naming and categorization of something like "humans" as a species. Scientifically, using our most rigorous definitions and categorizations in biology, we define "species" as meaning organisms which can mate and produce viable offspring able to reproduce in turn (so for example, a horse and a donkey are still considered separate species, because even though they can produce an offspring, the mule, it in turn is sterile and unable to produce any further offspring).

So with that categorization, modern humans, Homo Sapiens, are categorized as a particular species of hominids. Traditionally, they were considered distinct from earlier hominids such as Homo neanderthalis and the Donisovans- because genetic analysis showed these other hominids to be quite distinct from sapiens... until, that is, we learned that there are fragments of Neanderthal DNA in almost all modern humans (except in Africa where Homo sapiens originated and the DNA had not mixed with the neanderthals who had left Africa much earlier). Same with the Donisovans.

So which is it? Were these earlier hominids real humans in the modern sense or not? Surprisingly, this is an area of controvery in modern biology and anthropology. It's not clear because there are no such natural landmarks in nature to distinguish where one species end and the other starts. They all, including plants AND animals AND fungi AND protozoa AND bacteria, etc, etc... all originated from simple, single cell organisms and gradually evolved into each other. These categorizations, while pragmatically helpful, do not really exist in nature- and we find that the concept of "species" is much slipperier than we seem to think.

I'm not so sure this is true, or at least, I don't think it's universally true.

All one needs is one counter-example of how this doesn't work. It turns out it's pretty universal. For example, take the word chair: it's something you sit on, right? Well, what if I choose to sit on a tree stump- could I call that my chair? What about my little daughter who calls the little chairs in her doll house "chairs"? What about the chair of the department of philosophy?

Plato would have been interested in discussing the essence of what it is for something to be a "chair". Wittgenstein would have pointed out to him that the concept bleeds out and blurs and interconnects with all sorts of other words and concepts in all sorts of ways. There is no center, just "family resemblances", and there is no final essence to the word except for how it's being used in the particular "language game" being played.

He would tell Plato that he is just chasing a rainbow- something that looks pretty- but that disappears as soon as you think you are approaching it. So you might as well just sit back and enjoy it, not try to catch it.

 

[ataraxia]

Whatever you think of this argument, it seems clear he's not looking for essences. He's got something else in mind.

Platonism is all about trying to grasp, with the mind's eye, the inner essences and "ideal forms" of things. So just like there is an essence and ideal form that gives shape to our understanding of what a chair or a triangle is, there should be such ideal forms for things like "justice", "beauty", etc.... and it was the job of the philosopher, through deep thought and Socratic dialectic (not observation), to grasp and understand these deeper, higher essences.

As you may know, this concept was further developed by neo-Platonists like Plotinus into the concept of "The One"- the ultimate essence and center of everthing, which, although formless and dimensionless, is the essence of everything, including triangles, and chairs, and beauty, and justice, etc... Now when this idea encountered Jewish monotheism, it was just a matter of time before the two were married- a marriage whose blessing was thought to have been finally given by St. Augustine.

Now many of the early Church fathers were not happy with this marriage. Tertullian, for example, famously objected: "What has Jerusalem to do with Athens?" But by that point it was too little too late. The two had already eloped and signed the nuptial agreements.

It was not until the 20th century that a divorce has been called for.

 

[ataraxia]

That doesn't seem likely to be correct. There aren't any natural correlates to the limitations that Godel found for formal systems. Additionally, there are objects within some formal systems that could not have any natural correlate--imaginary numbers, for example. We use imaginary numbers to solve certain problems in engineering and physics, but it's impossible to have 3i oranges, or something like that. The idea that we learn logic from observing the natural world seems to sit rather ill with observations such as that.


I think that seems to be based on a misunderstanding of what logic is. Euclid and Newton both used logic; they don't have their own logics just as such. Euclidean geometry is often used to introduce deductive reasoning to high school students these days, but that's the only connection as far as I can tell. Euclid and Newton built theories using logic; those theories were eventually proven to be wrong (in Newton's case) or less parsimonious than the alternatives (in Euclid's case), but neither theory is dictated by logic or mathematics. Actually, I think that was Lobachevski's point--mathematicians up to about 1830 thought that Euclid's postulates and axioms were mathematical truths--he showed that they are not.

You are correct that Euclid and Newton weren’t creating their own "logics"—they were using the tools of logic (like deduction and inference) to build theories. Their work wasn't logic itself; it was theory-building with classical logic as the framework. So yeah, saying they had their own separate "logics" would definitely be misleading.

I also totally agree with your point about Euclid and Lobachevski. For a long time, people thought Euclid’s geometry wasn’t just a way of thinking about space, but the true way. Then Lobachevski (and later Riemann) showed you could create other geometries that were just as logically consistent, even if they didn’t match our everyday intuitions. That shift—from thinking axioms were obvious truths about the world to realizing they’re just starting points for different systems—was huge, and it really changed how mathematicians thought about the relationship between logic, math, and reality.

As for the "law of identity" and "law of causation" stuff—yeah, you're spot on again. In formal logic, we don’t really bake those big metaphysical ideas in as axioms. Identity in modern logic is just a technical thing ("this term refers to the same object as that term"), and causation isn’t even part of the basic setup. So when people throw around phrases like “A is A” as if that’s just what logic says, they’re mixing formal reasoning with metaphysical baggage that formal logic actually tries to avoid.

 

[ataraxia]

That's not logic, either. There's no observation that serves as an "input" to logic. I'm not quite sure what would be an input for logic--I suppose maybe you could say that arguments serve as inputs when we use logic to test for validity, but usually, observation is relevant to scientific theories.


Sure, but they're not really systems of logic, or mathematics, at least in the way that I understand systems. They seem to be closer to models rather than systems.

My point is that systems of logic are in themselves our latest models of how things seem to work. In the past, when dealing with everyday experiences, there was only one set of laws: the laws of classical logic. But with the rise of other observations, it became clear that other systems of logic, using different postulates from the classical one, were necessary. In the 19th century, mathematicians started playing around with some of these assumptions, just for fun. But later, with the rise of modern science and its further observations both on the cosmological scale as well as the subatomic scale, it became clear that those mind games turned out to be the actual way the universe works.

So if you look at modern logic, there isn't even just one logic anymore! In addition to classical logic, there is intuitionistic logic, modal logic, paraconsistent logic—they each tweak the "laws" depending on what you're trying to model. That flexibility suggests that "laws of logic" aren't absolute commandments from a law-giving god, but useful tools and models humans create depending on observations and context.

Take classical logic versus intuitionistic logic, for example. In classical logic, we assume that every statement is either true or false, even if we have no idea which. But in intuitionistic logic, you don't get to say something is true unless you can actually prove it. It's a whole different vibe, depending on how you think about knowledge—and both are completely valid ways to reason, just depending on the context.

Or look at paraconsistent logic. In classical logic, if you have a contradiction, everything falls apart—you can technically "prove" anything if one contradiction exists. But in paraconsistent logic, you can actually work with contradictions without the whole system blowing up. It’s used in places like legal systems or computer science where messy, conflicting information still has to be handled intelligently.

Take quantum mechanics—at the everyday level, we expect things to follow classical logic: objects are either here or there, cause and effect seem pretty straightforward. But at the quantum level, things can be both here and there at the same time, and particles can seem to affect each other instantly across huge distances. It’s so weird that scientists had to rethink the underlying assumptions, and basically build new models—new "rules"—to make sense of it. Classical logic just didn’t cut it anymore.

Or think about Einsteinian relativity. Before Einstein, everyone assumed space and time were absolute—you know, common sense stuff. But it turns out that space and time are relative to the observer's motion. What counts as simultaneous for one person isn’t simultaneous for another. That totally shook up the way people thought about cause and effect and even what counts as "the same moment." Again, people had to adjust the rules because the old ones didn’t fit the evidence anymore.

In some of these cases, the grey areas between these different realms where different logics apply gets pretty blurry- sometimes it's not even clear which set of logics should apply to a particular scenario. Sometimes it's just a matter of experimentation to see which works.

So when you look at examples like the above, it’s seems hard to argue that such laws of logic are these fixed universal things handed down from some external lawgiver. It looks way more like humans inventing and adjusting models based on what they find works for trying to make sense of reality. It's not even clear there is ONE set of laws of logic for all. Maybe that's evidence of polytheism?

 

[ashurbanipal]

Everything is evolving and changing all the time. Our vocabularies and categories, while helpful, are just attempts to try to provide some framework through which we can try to work. And when you look at different cultures, you realize that different cultures have attempted to do this with the same phenomena in very different ways- evidence that we are just trying to create categories based on generalizations that do not, and can never, carve nature at the joints that way- there is always just too much grey. But when you really get down to it, things blur and bleed into each other.

I don't think this is correct often enough to be much worry for logic just as such, or even for semantics in most instances. That things are changing all the time seems pretty certain, but that alone does not make something difficult to categorize. For instance, as I understand it, the continent of Antarctica is moving toward South America at a rate of about 1 cm per year--but the fact that it's changing position does not mean that we cannot classify it as being in the southern hemisphere.

So with that categorization, modern humans, Homo Sapiens, are categorized as a particular species of hominids. Traditionally, they were considered distinct from earlier hominids such as Homo neanderthalis and the Donisovans- because genetic analysis showed these other hominids to be quite distinct from sapiens... until, that is, we learned that there are fragments of Neanderthal DNA in almost all modern humans (except in Africa where Homo sapiens originated and the DNA had not mixed with the neanderthals who had left Africa much earlier).

So which is it?

I think at best that's a problem for semantics, not logic...though I'm not sure it's really a problem for either. Plato certainly thought that if things are not static, we could never refer to them or reason about them; he rejected Heraclitus' views for that reason, though if you read what he says carefully, this is just another argument for the existence of forms--he said that even though the water in the Cayster was different now than it was a few moments ago, and even though the shore line is changing ever so slightly from one moment to another, we still have no trouble identifying the Cayster. We do so reliably, all the time--and those few times we don't, the error is easily identified (we got the wrong map or we were drunk or something). Since the physical properties of the Cayster are always changing, reasoned Plato, the way that we identify and define the Cayster must depend on something else.

Generally speaking, I don't think many philosophers who study the problem of constitution and identity today think that identity has much to do with constitution. I suppose those who think there are such things as temporal parts might be something of an exception...kinda.

All one needs is one counter-example of how this doesn't work.

That doesn't seem correct either. If there were exactly one example of something that was difficult to categorize, and everything else were quite easy to categorize, I don't think the upshot would be that we should scrap categorization altogether.

It turns out it's pretty universal. For example, take the word chair: it's something you sit on, right?

No, not necessarily. There are chairs that are never meant to be sat in, and should not be.

Well, what if I choose to sit on a tree stump- could I call that my chair?

No, probably not.

What about my little daughter who calls the little chairs in her doll house "chairs"?

Yes, those are chairs.

What about the chair of the department of philosophy?

That's not really the same issue--terms have different meanings according to context. "Page" can mean a leaf in a book, a particular kind of servant, or it can be used as a verb, and so on. That's not problematic either for logic or semantics. It just means that a term is not usually specified outside its context.

Plato would have been interested in discussing the essence of what it is for something to be a "chair".

That was my point--he was exactly not interested in that.

I think you may have in mind Leibniz' pre-monadology metaphysics. He did think there are properties of things by virtue of which they are what they are--that is, essential properties.

 

[ashurbanipal]

Platonism is all about trying to grasp, with the mind's eye, the inner essences and "ideal forms" of things. So just like there is an essence and ideal form that gives shape to our understanding of what a chair or a triangle is, there should be such ideal forms for things like "justice", "beauty", etc.... and it was the job of the philosopher, through deep thought and Socratic dialectic (not observation), to grasp and understand these deeper, higher essences.

As you may know, this concept was further developed by neo-Platonists like Plotinus into the concept of "The One"- the ultimate essence and center of everthing, which, although formless and dimensionless, is the essence of everything, including triangles, and chairs, and beauty, and justice, etc...

I'd say that confusing Platonic forms and essences is exactly that--a confusion. Aristotle wanted to make the forms be essential properties; Plato did not. For Plato, the form {chair} was not a property of chairs. He knew that you could gather together many chairs and see that there is no property common to all of them, other than that we--quite effortlessly--identify them as chairs. This led him to posit the existence of forms (whether those are actual objects in his ontology, or merely metaphysically "thin" abstractions, is another matter). Forms are explicitly not in the things that we encounter--rather, if anything, the things we encounter are "in" the forms. But you should not think of participation in a form as a property of something--you can tear apart any chair you like atom-by-atom and never find its participation in {chair}.

Now when this idea encountered Jewish monotheism, it was just a matter of time before the two were married- a marriage whose blessing was thought to have been finally given by St. Augustine.

Now many of the early Church fathers were not happy with this marriage. Tertullian, for example, famously objected: "What has Jerusalem to do with Athens?" But by that point it was too little too late. The two had already eloped and signed the nuptial agreements.

Yes; I'm pretty familiary with that history.

It was not until the 20th century that a divorce has been called for.

Not everyone is calling for it...

 

[ashurbanipal]

My point is that systems of logic are in themselves our latest models of how things seem to work. In the past, when dealing with everyday experiences, there was only one set of laws: the laws of classical logic. But with the rise of other observations, it became clear that other systems of logic, using different postulates from the classical one, were necessary. In the 19th century, mathematicians started playing around with some of these assumptions, just for fun. But later, with the rise of modern science and its further observations both on the cosmological scale as well as the subatomic scale, it became clear that those mind games turned out to be the actual way the universe works.

So if you look at modern logic, there isn't even just one logic anymore! In addition to classical logic, there is intuitionistic logic, modal logic, paraconsistent logic—they each tweak the "laws" depending on what you're trying to model.

Two points:

1. Yes, there are by now many hundreds of logics. For most purposes, however, 2nd order S5 alethic-modality logic with identity, functions, and iteration works as an almost-perfect model of human reasoning. Usually, when analytic philosophers talk about logic, that's the logic they mean.

2. It sounds like you still may be confusing theories and logic. If you're thinking of logic as exposing anything else about how the universe works, you're not thinking of logic.

Take quantum mechanics—at the everyday level, we expect things to follow classical logic: objects are either here or there, cause and effect seem pretty straightforward. But at the quantum level, things can be both here and there at the same time, and particles can seem to affect each other instantly across huge distances. It’s so weird that scientists had to rethink the underlying assumptions, and basically build new models—new "rules"—to make sense of it. Classical logic just didn’t cut it anymore.

I think maybe instead of confusing theory with logic, you may instead be making a more subtle error--one that is often difficult to grasp. Logic (that is, 2nd order S5 alethic-modal logic with identity, functions, and iteration, as well as most other varieties of logic) is concerned with syntax only. Semantics does not enter into the picture at the most fundamental level. It's true that logicians have sometimes proven that certain semantic models are sound with respect to a given syntactic system (e.g. K-models and S5 alethic-modal logics), but that should not be confused as meaning that the semantics is the logic.

Consider the following two sentences:

1. The Statue of Liberty is uniformly a certain shade of red over its entire surface.
2. The Statue of Liberty is uniformly a certain shade of blue over its entire surface.

Making sure that we don't equivocate, does 1 contradict 2? The surprising answer is no, as far as logic (2nd order S5...) is concerned. This is because logic is not concerned with the meanings of predicates, only the syntax in which they operate. So, as far as logic is concerned, there's no problem with a particle being both here and there at a given time. There's no notion of causality baked in.

Or think about Einsteinian relativity. Before Einstein, everyone assumed space and time were absolute—you know, common sense stuff. But it turns out that space and time are relative to the observer's motion. What counts as simultaneous for one person isn’t simultaneous for another. That totally shook up the way people thought about cause and effect and even what counts as "the same moment." Again, people had to adjust the rules because the old ones didn’t fit the evidence anymore.

Sure...they just didn't have to adjust any rules of logics.

So when you look at examples like the above, it’s seems hard to argue that such laws of logic are these fixed universal things handed down from some external lawgiver. It looks way more like humans inventing and adjusting models based on what they find works for trying to make sense of reality. It's not even clear there is ONE set of laws of logic for all. Maybe that's evidence of polytheism?

The way that usually works is that a philosopher or mathematician will recognize a particular situation in which a human being has reasoned in a seemingly valid manner, but which classical logic would reject as invalid. Logics aren't typically invented because of discoveries in physics, for instance.