How fundamental is logic?

Question

I had always perceived logic as something that exists outside mathematics, physics/the physical, human consciousness, and everything. So when someone in my class posed the question whether logic can exist without humans, my first reaction was "of course it can!". I still tend to think it does, however, I have been pondering about this ever since, and couldn't find much about it online. What did the great minds have to say about this?

I used to think logic is used to formulate mathematics, which is used to describe the way our universe works. But could it be the other way around? Could the way our universe works, have defined our logic? Have our minds evolved such that we have a natural sense of logic that could have been different in a universe that behaves differently? Would this have any impact on the mathematics we use? I get so confused thinking about this, also because I'm unable to imagine what different logics would even look like.

Answer 1

There is a traditional view of logic coming from the Port-Royal Logic, and inherited by Frege and Russell, that logic is a set of laws, the laws of thought (Frege), or the most abstract laws of reality itself (Russell).

Wittgenstein proposed a radically different view of logic in which he questioned the nature of the compulsion of logic. He regarded logic not as a set of laws, but a set of rules. These are not made true by reality, human consensus, or by the nature of thought or the universe. Rather, they are grounded in human activities of inferring, contradicting, asserting and denying. These activities are the bedrock from which we may formulate rules which we then call logic. These rules clarify and specify the nature of and rules for doing these activities properly. Different logics just give different rules for inferring, denying etc. which might be useful in different contexts.

Mathematics is not generally reducible to logic, as Russell and Frege had both hoped. According to Wittgenstein, mathematics is founded in empirical human activities such as measuring and calculating. In general, mathematics specifies rules which license the transformation of empirical statements. e.g. because 2 + 2 = 4 is a rule of arithmetic, the inference from "I had two apples and I bought two more, so I have four apples" is licensed, i.e. it is a valid inference because the rules allow the transformation from the premise to the conclusion.

Empirical claims are always contingent. Logic and mathematics are deemed necessarily true, but this is for Wittgenstein just a way of saying they are rules of the games we play with language, rules of making sense. It is an error of philosophy to treat the necessity of logic and mathematics as a kind of super-rigid empirical force, an error upon which most metaphysical theories are based, such as mathematical Platonism. The traditional view is that logic and mathematics are descriptive, while Wittgenstein's conception of them is that they are normative, i.e. conceptually related to rules, not reality.

This is very, very brief summary of the description of Wittgenstein's view of mathematics and logic in Baker and Hacker's volume 2 of their Analytical Commentary on Wittgenstein's text Philosophical Investigations.

Answer 2

According to Lukasiewicz it is not true that logic is the science of the laws of thought. “It is not the object of logic to investigate how we are thinking or how we ought to think. The first task belongs to psychology, the second to a practical art of a similar kind to mnemonics”. Logic is the study of the laws of argument, i.e. which arguments are valid, which forms are valid. The study of it could not exist without humans, because only humans (as far as we know) are capable of study. However, arguments can be valid or invalid whether or not humans exist.

Answer 3

I used to think logic is used to formulate mathematics, which is used to describe the way our universe works.

Logic and mathematics are distinct subjects; it's only relatively recently that logicians began to be excited about the possibility that mathematics could be reduced to logic - this began with the work of Frege, Russell & Wittgenstein amongst many others during the late 19C and early 20C.

The excitement is driven by the philosophical thinking that logic is somehow a more fundamental topic than mathematics; and any reduction in fundamental notions will be valuable - in the usual sense, that is Occam's Razor - but also in many unforeseen ways too.

But could it be the other way around? Could the way our universe works, have defined our logic?

Still, its worth reflecting that mathematics itself is seen as fundamental; its hard to disbelieve simple arithmetic truths such as 1+1=2; there is a school of thought that there is a world of Platonic Forms that human mathematics reveal; this probably comes from the Pythagorean tradition that Plato imbibed;

Have our minds evolved such that we have a natural sense of logic that could have been different in a universe that behaves differently?

Given that Hume & Kant both, in their own ways, thought that there was an element of consciousness (through psychology for Hume, and through the intuition for Kant) in how we made sense of the world; it seems natural to speculate that this is indeed possible.

I get so confused thinking about this, also because I'm unable to imagine what different logics would even look like.

There's no need to imagine: have a look at these truth-tables for three-valued logics!

Answer 4

It depends on whether you mean logic, the academic discipline, or the actual laws of logic, e.g. the principle of non-contradiction. The former is, obviously, not something hard-coded into the structure of the universe. For example, there is no need to for the symbol for logical conjunction to be what it is. (Logic is somewhat unusual in that there is in English the same word for the discipline and the subject matter -- often these are distinct as in 'cosmos' and 'cosmology.')

Whether the laws of logic are eternally objective is actually a metaphysical rather than logical question. Ultimately this question boils down to whether you believe there is an ultimate objective reality (e.g. God), or whether there are simply "wills to power" which are always in flux. As I said this is, strictly speaking, outside the purview of logic itself and lies in the realm of metaphysics or indeed religion. Many thinkers have argued persuasively for either position. However, there does seem to be a troubling contradiction inherent in asserting that the world is always in flux, because then that assertion itself cannot always be true. (I.e. such statements would themselves be subject to the ceaseless change.) But that argument rests on belief in the principle of non-contradiction.

Answer 5

First-order logic is absolutely fundamental. Put briefly, statements in first-order logic can be represented and evaluated using the natural numbers with a technique known as Gödel numbering, which is based on the fundamental theorem of arithmetic. In contrast, statements in higher-order logics can only be evaluated against man-made models. We know this because Tarski's undefinability theorem shows that Gödel numbering only works with syntactic constructs and cannot be used to prove semantic assertions of truth.

For those who might claim that natural numbers are simply another kind of man-made model (à la Wittgenstein), consider that helium atoms have 2 protons - a fact which does not rquire any kind of human measurement or calculation. Natural numbers are an attribute of the universe. Real numbers, on the other hand, such as Pi and the square-root of 2, only exist because people like to measure things like circles and triangles.

According to Kurt Gödel, who is widely considered the greatest logician since Aristotle, first-order logic "is a science prior to all others, which contains the ideas and principles underlying all sciences."

John F. Sowa writes "Among all the varieties of logic, classical first-order logic has a privileged status. It has enough expressive power to define all of mathematics, every digital computer that has ever been built, and the semantics of every version of logic including itself."

Answer 6

It might seem like a bit of a cop-out to recommend a book, but your questions are far larger than you imagine! In G.W.F. Hegel's "The Science of Logic" he takes on the question: With what must science begin? and goes on from there. It is the most brilliant expository on the subject of science, logic, and thought that I have ever read. He starts off with the concept of nothing and ends up at... well, I won't spoil it for you!

Answer 7

I had always perceived logic as something that exists outside mathematics, physics/the physical, human consciousness, and everything. So when someone in my class posed the question whether logic can exist without humans, my first reaction was "of course it can!". I still tend to think it does, however, I have been pondering about this ever since, and couldn't find much about it online. What did the great minds have to say about this?

The Curry-Howard isomorphism seems to give a satisfying point of view about Logic but we don't have a lot of philosophical interpretations yet.

That correspondence tells us that proofs behave as computer programs and formulas as type for programs : it gives a natural account to Logic, Logic share something with computation which seems to be "natural".

I used to think logic is used to formulate mathematics, which is used to describe the way our universe works. But could it be the other way around?

Logicists such as Frege and Russell wanted to reduce mathematics to Logic but it was a failure. It seems that mathematics are more "free" than Logic. Logic seems to give a constraint/format to force things to be "coherent", "well-behaving". That point of view is renforced by the fact that Curry-Howard see formulas as types for programs. Types in programming are used to restrict the programs to the one which behave well according to a concept of what "coherent" is (for instance we can't sum two characters).

Could the way our universe works, have defined our logic? Have our minds evolved such that we have a natural sense of logic that could have been different in a universe that behaves differently? Would this have any impact on the mathematics we use? I get so confused thinking about this, also because I'm unable to imagine what different logics would even look like.

I think it's fair to say that to imagine a world with a different logic is a bit like imaginating a world with different physical laws.

Answer 8

According to the likes of George Lakoff, Raphael Nunez, Mark Johnson et al., logic (mathematics and language in general) is built up of successive layers of metaphor which have an ultimate basis in sensory-motor activity. Hence, theories of embodied cognition would posit basic sensory motor activity as being more fundamental than logic. I would urge you to take a look at the work of Raphael Nunez, he has some interesting theories regarding “where mathematics comes from” that puts paid to any Pythagorean or Platonic conceptions (like Max Tegmark). You might want to check out the work of Penelope Maddy. If I recall correctly she thinks math has its origins in what’s theoretically referred to as “affordances” (vis. J.J. Gibson and ecological psychology).

Answer 9

I would like to focus only on the question, "whether logic can exist without humans"? The answer depends on how "inclusive" we define "human" and how "limited" we define "logic." If we define human as belonging to "homo sapiens," then the answer is yes, simple logic can exist without humans. It exists in plants and animals - at least at the cellular level.

Answer 10

Logic, mathematics, science, musicology, and most systems of government and commerce are all examples of attempts to create rigor and consistency in our thinking to address the flaws of our built-in logic inherited from our mammalian ancestors.

We are born using a form of Bayesian logic or abduction described in the Theory of Sufficient Reasoning. Immortalized in "Occum's Razor" we gravitate to the simplest explanation, drawn from no more than our personal history of experiences. We judge our experiences and form a collection beliefs in a world view that acts like a filter or paradigm through which we judge future events. Each belief carries with it a credence value or disposition regarding how strongly the belief is held. When an event is witnessed displaying evidence in favor of or against the belief, the rational person will then adjust the credences affected accordingly based on their judgement as conditioned by the prior credence values. https://en.wikipedia.org/wiki/Abductive_reasoning

The strength of abductive logic is that we can make quick decisions without perfect or complete knowledge of the facts, a very valuable tool for survival. This survival skill, however, comes with obvious flaws. Quick decisions without sufficient data results in tragic errors. Also, chronic, traumatic or systematic conditioning can set credences so high that one may not be able to recognize new evidence let alone judge it. Lastly, a credence change cannot be made without maintaining consistency in one's world view. If changing one credence causes unacceptable changes in other, the original belief change may be set aside, thereby weakening the authenticity of one's world view.

So, the logic systems found in scientific, mathematical, and philosophical systems do impose rules of engagement to ensure integrity and reliability and to avoid or mitigate the flaws in abduction. Ironically, when scientific or mathematical discoveries are offered, abduction takes full charge in judging its acceptability. Resistance to change that threatens the system underpinning will be vociferous, even though the logic and evidence are compelling.

Logic systems are like scientific theories. If they consistently provide the right answers in their assigned domains, we accept and use them. But, what happens when a contradiction appears?

Your final concern, whether our choice of logic affects our math and other pursuits is a resounding affirmative. Aristotle's logic based on the premise that a statement must be either true or false created the logical foundation for the industrial revolution and remain embedded in every institution even though it is patently and inhumanly wrong. (Any Charles Dickens novel)

Ultimately, you may master these systems and still have doubt about their credibility because your world view objects regardless of the evidence. On the other hand, you may find, as you learn more, that your inherit logic system can serve you well with help from other logical points of view. I suggest that you suspend judgement for a while, and allow your natural curiosity to seek out the answers to your concerns with knowledge.

Answer 11

First, we should be clear about logic and reason: logic is a set of rules that describe the natural behaviour of physical systems. Reason is the mental application of such set of rules against certain model.

Then, logic is a result of the fundamental causality laws of systems. For example, hitting a head against a wall will result in the head being destroyed, not the wall. Action=hit the head against wall, reaction=head gets hurt. Reasoning is applicating such principles on the right way (should I bang my head against the wall because an student says that such principle is false? No! But perhaps he can try verifying!).

This kind of principles are observed by everyone of us since we're children. When we grow, we have an intuitive sense of such laws. Some people was able to express them using the mathematical language, that's formal logics. For one side, they are fundamental on the sense they are the expression of the fundamental causality laws of nature.

But since the development of the relativity (RT) and the quantum theories(QT) , our intuition (and our classical expression of the principles of logic) does seem to be wrong, or the expression of logical laws seems wrong. Following such issue, on the QT, a position could not be unique (x=k is insufficient to describe a position), or an entity could appear and disappear at different points (the limit of x when approaching a position k is not such position), or even, the weight of a solid changes everytime it changes its speed. Then, logic has been completed to solve such gaps (Dirac notation et.al.). Then logic becomes again a description of the fundamental principles of physics. We cannot be sure, naturally, that this time we have the right description of nature reflected on our logical principles.

In such case, the principles of logic cannot be fundamental, since we perceive them within the context of our physical limitations. We can be close, but never be sure about that.