Deduction versus Induction

[Eric Bright]

For some reason(s), there is a discussion going on about if we need one or the other. Under this topic, we are trying to discover what those reasons might be.

Why might a conversation regarding ‘either this or the other method’ be seen as legitimate or even necessary?

Why should one want to keep one and reject the other? What might motivate one to do so? Aside from personal preferences and opinions?

What are the reasons, if any, and not mere opinions, one might give to keep one and reject the other?

Instead of saying this method is acceptable but the other one is not, or this method is absurd and the other one is not, or this one is necessary and the other one is not, what actual arguments are there, if any, to keep one and reject the other?

Do we have to go for an ‘either...or...’ stance? If yes, then what necessitates it? If not, then what are the reasons?

[Andreas Geisler]

I think historically, Philosophers have aimed for a law of truth, and so have focused on certainty, i.e. deduction.
However, all our endeavors in other areas of inquiry end at induction.
So, in a mental representation built on all sides of induction, is there a use for "pure deduction"? We have no pure input to feed it, so we cannot apply the normal rules of deduction: if we had sound input, we would get True Conclusions from valid arguments. But when we have merely probabilistic input, we can't claim to get True Conclusions, even from valid arguments.
Even the form of "If we assume X, then ..." makes use of abduction to "fix" inductive bases into something we might feed into deduction.

So, can logic be a study primarily of deduction, or should it properly be a study of induction, abduction and deduction?
(Please note, my background is in linguistics, so I do not use the full philosophical technolect. I suspect that is welcome among other amateurs, but I realize that for people inducted into the the more precise technolect, it may seem crude, and for that I apologize).

[Caroline Zhenning Pei]

Apparently logic is defined as "A method of human thought that involves thinking in a linear, step-by-step manner about how a problem can be solved. Logic is the basis of many principles including the scientific method." (Wiktionary). To solve a problem, we need to understand the conditions, which involves induction. Whenever we reach a new conclusion with the conditions and those conditions only, we are necessarily using either induction, deduction or abduction. To define logic as a means of thinking that aims at problem solving is like to define that logic is about deduction, induction and abduction.

[Eric Bright]

Please allow me to elaborate on this issue a bit further. I am going to use language and grammar as an example.

Do we make grammar or do we document it (i.e. discover it in different languages, or unearth it so to speak)? Aside from artificial languages, all natural languages have grammar. However, we do not make those grammars in most cases. At least not consciously. Grammarian and linguists do not invent grammar of, let’s say, English. They document it. In some sense they discover it. Languages do not operate because someone has put some words and grammar up for others to use. People do not follow grammar in a natural sense of it. They just talk.

Talking is possible because of the regularities and the relationships that there are between sets of sounds and noises that a group of people use to communicate. Long after when humans started to communicate via sounds, actually a huge while later, some of them started to actually notice some regularities in these sounds and noises used for communication. And a long while later yet, some others started to document it and called it, in their own tongue, “grammar.”

These relationships existed way before anyone consciously noticed them. Without them, communication was, and is impossible. These might change and the time goes by, but they cannot simply disappear. It is because of them that language is possible.

Now, let me tie it back to your question Andreas. We didn’t invent deduction. Our thinking does not have to follow it. Our thinking is it. Our thinking is possible only because of deduction. What we can do, at best, is to document it, to unearth it, to discover it.

I shall elaborate on this.

All the science is based on deduction and there will be no science without it.
All of our tools, like our computers, work on deductive principles.
All of our theories are evaluated based on deduction principles. That is the only way a hypothesis can be tested.
All of math is based on deduction (so far as I know). Even the mathematical induction is different from the philosophical one. It is more like deduction that anything else.

Here are a few examples (the words in capital letters are logical terms):

E.g. 1: in medicine

(1) IF drug L works THEN my patient must feel better in 10 days
(2) My patient is not feeling better after 15 days of taking it (actually is getting worse)
(3) THEREFORE drug L does not work

E.g. 2: in sciences

(1) IF my hypothesis is true, THEN I must observer E and R AND T must happen in 20 minutes.
(2) I observed E and R
(3) T happened in 20 seconds instead of 20 minutes
(4) It is not true that ((E and R) AND T)
(5) THEREFORE my hypothesis must be wrong (for whatever reason)

E.g. 3: in life

(1) IF all swans are white, THEN every new swan that I see must be white (in the past or future)
(2) Every observed swan is not white (there are black swans too)
(3) THEREFORE all swans are not white

As you can see, to come up with a testable hypothesis, we, as well as all scientists, use observation (induction). To actually test them, we, and scientists, use deduction.

All the statistical tools that science uses work on the principles of deduction amongst others.

E.g.4: in statistics

(1) IF hypothesis H is true, THEN x, y, z must occur 95% of the time
(2) x, y, z, happened in 30% of the time
(3) THEREFORE hypothesis H is not true

Even our biological brains work on the same principles. I use your own example from the discussion in the semantics board (appropriated it indeed). The following is how our brains, or the brains of a rat, notices a likely causal relationship:

E.g. 5: our brains

(1) IF p (i.e. event Z and the vocal sounds (a, b, c, …, n) are co-related), THEN q (i.e. they should co-occur) AND IF q THEN p
(2) Event Z and sounds a, b, c, …, n do not co-occur
(3) THEFREFORE they are not co-related
or
(2)’ Event Z and sounds a, b, c, …, n are not co-related
(3)’ THEREFORE they might not co-occur

And so on and so forth.

As such, we make a hypothesis based on our observation (induction), then we go about testing it (deduction).

E.g. 6: in a factory

(1) Machine G suddenly stopped working
(2) We make a hypothesis that says: based on our observations in the past (induction), IF machine G is broken, THEN it would make a loud noise
(3) Let’s see if it is making a loud noise. Guess what, it is not making a loud noise.
(4) THEREFORE it is not broken
(5) Now, let’s see what else might have gone wrong...
And so on.

All of the above are deductive reasoning.


The role of evidence

If anything, the lack of certainty in our observations would directly affect the validity of our inductions, because it is induction that relies upon observation.

Contrary to that, deduction does not care about observations. It can even work with zeros and ones, ONs and OFFs.

So, indeed what we can do in here is merely noticing the principles that govern the relationships between abstract entities, like in mathematics. Deduction does not need any reality check to operate. It can work, as easily, in the context of a fictional world. But, the principles would be identical to how it is operating in a real situation.

Your question should actually be asked upside down: If we have no certain observation to feed into our induction-driven hypothesis, how is it going to help us form ANY relevant idea about reality whatsoever? Deduction does just fine, with or without reality, because it is not about our physical reality to begin with. But, as Hume first noted, induction collapses under the weight of our scepticism.

[Caroline Zhenning Pei]

E.g. 3: in life

(1) IF all swans are white, THEN every new swan that I see must be white (in the past or future)
(2) Every observed swan is not white (there are black swans too)
(3) THEREFORE all swans are not white

As you can see, to come up with a testable hypothesis, we, as well as all scientists, use observation (induction). To actually test them, we, and scientists, use deduction.

May I suggest that the process from 2 to 3 is actually induction? We cannot deduce about all swans if we only know about part of them.

[Eric Bright]

Okay. Here is the more explicit version. It is indeed a deductive argument with the following format:

(1) IF all Ss are Ws THEN all observed Ss must be Ws (this is a hypothesis based on induction)
(2) There is at least one observed S that is NOT W (this is a single observation)
(3) THEREFORE it is not true that (all Ss are Ws) (this is the refutation of the hypothesis in 1)

Here is the simpler version:

(1) IF p THEN q
(2) NOT q
(3) THEREFORE NOT p

And the explicit example:

(1) IF all swans are white, THEN every observed swan must be white
(2) It is NOT true that (every observed swan must be white) (i.e. there is at least one observed swan that is black)
(3) THEREFORE it is NOT true that (all swans are white)


I hope this makes the example more clear and shows why it is still a deduction.

[Andreas Geisler]

Going formal actually clouds the issue.
First of all, how do we know that deduction works?
How did we discover it?
Because it works. That's induction.

How do we know the kinds of things that may make a machine stop working? Induction and abduction.
How do we know the kinds of things that may serve as operating principles for a machine? Induction.

We cannot claim that deduction is human thinking, when deduction itself, its properties and processes, including the fallacies, are simply an inductively based catalog of things that do or do not tend to produce good results.

In a nutshell: Deduction can be inductively shown to give useful results.

[Eric Bright]

Your point is as interesting as it is relevant to logic. I hope I would be able to investigate it further with you in here.

To begin, let us make a quick clarification:

Today, logic is a branch of mathematics. As a subset of mathematics, everything that applies to mathematics also applies to logic.

Which university departments logic is mostly debated in is another issue that is not relevant to our current topic, so I am not going to dive into that.

What is inductive reasoning and where is its place?

Excellent questions indeed! No one ever tries to kick induction out of logic. Science will stop functioning immediately if that be the case. But, what is the role of inductive reasoning then?

In science, induction is responsible, almost always, for generating hypothesis. And that is awesome on its own. The question is not if induction is necessary. It is necessary and absolutely so. It is the bridge that connects the abstract world of mathematics to the real physical world.

Mathematics, nevertheless, does not rely on induction per se. To discover mathematical relationships, one does not have to observe anything. It can be done, and is done, in abstraction.

Also, mathematics does not have to concern itself with its relationship with reality. That bit is the duty of physics. The fact that mathematics never minds what happens in reality does not diminish the value of physics in any way. As such, all sunsets of mathematics do not concern themselves with reality and that does not discard nor does it reject anything that does concern itself with reality.

Logic, as a subset of mathematics, is the everything else but induction. Induction plays a role when we want to connect an argument with reality. Induction is usually responsible for the first few steps in a realistic, deductive argument. Deduction, itself, is not an argument. It is nothing. It is only the illustration of relationships that concepts, empty place-holder if you wish, can have with one another.

Have a look at this and tell me if it makes any sense:

-2x + 3y = 5
3x- y = -5


Does the equation pair say anything about reality? Is it connected to reality in any way? Not at its current form.

However, it potentially can. It can be about something in real world too.

The connection between the above-mentioned pair of equation and its relationship with reality is made through physics.

Given this, one has to work REALLY hard to convince anyone that mathematics is nonsense and a bag of garbage. By really hard I don’t mean by writing a blog post. Actually, if anyone can demonstrate that all the mathematics that has ever been done has been in vain, they certainly deserve more than one Nobel prise and certainly in several fields and not just in mathematics.

In most examples that I provided in my previous posts, the first statement (i.e. step number (1) in the examples), are usually the work of induction. Without induction, most of those IF...THEN... statements cannot be provided. Once they are provided by induction, the rest of the implications automatically result. How come? Well, the implications do not need anyone’s permission to appear. They just do, and that is because the relationship are necessarily so. Deduction is the study of these necessary relationships.

Induction, in comparison, only provides the first IF...THEN... hypothesis based on previous observations. Once such a hypothesis is proposed, we can test and see what implications that hypothesis might have. That is the job of deduction. Induction does not do so and cannot do so. Induction is a whole different tool.

To conclude:

• Induction is as essential to science as deduction is
• Deduction is a subset of mathematics (not the other way around)
• Everything that applies to mathematics, also applies to logic as its subset
• Deduction and induction are two completely different tools
• Induction is a hypothesis generating machine for science
• Mathematics can be studied without any reference to reality
• Induction can be studied only with reference to reality
• Deduction is what makes the implications of an inductive claim investigate-able
• One does not exclude the other
• To say that deduction is useless is to say that mathematics is useless
• To say that mathematics is useless one needs to prove it (a claim is not enough)
• It is not possible to prove that mathematics is useless unless one uses mathematics one way or another (i.e. by merely counting their objections)
• It is not possible to prove that deduction without induction is useless unless one uses deduction in ever step anyway (i.e. by merely saying just about anything and  then wanting for the words to mean anything)
• And so on...

[Caroline Zhenning Pei]

Response to conclusion 3:
Does pure, abstract mathematics include induction?

[Eric Bright]

Response to conclusion 3:
Does pure, abstract mathematics include induction?

Are you referring to mathematical induction?

[Andreas Geisler]

According to the links first posted, mathematical logic is but one of the things that are called logic, is it not? There is polysemy.
Logic is three things, induction, abduction and deduction. It is the three kinds of reasoning that humans in fact engage in, and of those three, induction is the most crucial to understand.
Cognitive studies support this, as well as the massive amounts of completely useless purely deductive work done through the ages.

And historically, the fight between empiricists and teleologists has been over the inclusion of induction, so there absolutely has been an effort to exclude induction.

Finally, I think it is fairly absurd to say that mathematics is deductive, when we learn it inductively.
We do in fact take two apples, and two oranges, put them into one bowl, and then count the resulting contents, 1, 2, 3, 4.
That is how we learn mathematics, and that is how we established mathematics in the first place, how we learned the rules.
Same with deduction.

And to top that off, the big paradigm shifts, even in mathematics, have come about through observation. We do maths because it works. It is in the application of mathematics to new problems, real problems, that we learn what else works.

Now, induction has little role in the formation of hypothesis, it goes far deeper than that.  Induction goes to the formation of concepts themselves, of identities (which deduction can only handle through tautology) and relations. Hypothesis is primarily about abduction; looking through the concepts and identities and relations, to see which ones might fit into the causal chain.

In order to be meaningful, a study of human reason, which is what logic has traditionally been, has to include all three forms. Otherwise we'll just keep on writing gibberish for ever.

[Alain Van Hout]

Interesting points, all around. I wanted to respond to two specific (and related) parts of a single prior comment:

In science, induction is responsible, almost always, for generating hypothesis.
...
   

• Induction is as essential to science as deduction is

I think the first is generally correct, but conceptually irrelevant, while the second is not correct, though from exactly the opposite viewpoint as the intonation of the sentence implies.

With regard to the first, induction is indeed the most often used method to generate hypotheses, though this fact has no great bearing on the scientific process as such, only on its efficiency: hypotheses could without too much effort be generated entirely randomly, with no actual problems arising for the application of the scientific method. What induction does do is expedite the scientific process by allowing hypothesis generation to anticipate useful results, i.e. we look where we expect to have the highest probability of finding results that we are interested in. That's why I see it as correct, yet irrelevant.

As to induction being as essential to science as is deduction, what I would ask is: where in science does deduction, i.e. by logic forming valid claims about the specific based on certain knowledge about the general, play an important role? In fact, scientists who would employ this method would more likely be scoffed rather than accepted to a journal. This is mainly because they don't actually provide evidence that their claims about the general necessarily carry over to the specific, only a rationale for why it could be expected to do so. So I would say: contrary to deduction, induction is essential to science.

The only exception to the latter would be if we're e.g. talking about using formulae to make the needed calculations for our specific experimental setup. But then we're not actually talking about deduction as a means to generate knowledge.

[Eric Bright]

I would like to know if there is (or are) any reason to say what you said Alain. Opinions and assertions are certainly very interesting, but they eventually must be supported by something rather than a personal preference. So far, there has not been any demonstration of the claims such as:

I would say: contrary to deduction, induction is essential to science.

Which means, deduction is not. This is only a claim and that is fine. What might be the reason why this might be the case?

Here is another one:

[...] induction has little role in the formation of hypothesis [...]

How come? What does that mean to say it “has little role”? How else a hypothesis is made then aside from randomly generating stuff that might or might not make sense?

We are not talking about how a meaningful statement might be generated. If that was the question, pure chance can also generate meaningful statements; infinite number of them indeed. However, that is not what this conversation is all about. Also, science might come across a discovery by chance, but that is not how a hypothesis is made.

So, please explain the following:

(1) In science, how a scientist form a hypothesis?

When that is done, then please explain this:

(2) How any hypothesis, regardless of its origin (by chance, by a random hypothesis generator, by magic, etc.), can be tested?
(3) How a test of a hypothesis can work without deduction?

Please consider giving reasons for your opinions. Try to refute my position with clear counter examples and demonstrations if possible.

On a different note related to what Andreas said, how we “learn” numbers is one thing, and the nature of mathematics is altogether another thing. We become aware of everything through experience and through our intellectual faculties. That is the pragmatic part of it. Abstraction, however, comes later as concepts form.

Concepts, also are comprehended by our brains. That does not make them any more tangible or physical in the sense that a rock is physical. We become aware of the presence of a rock with the same apparatus of cognition that we become aware of the concept of number one. That does not make numbers like rocks and tissue-papers.

Concepts are abstractions. They might or might not have relationships with themselves and with the physical world. In here, we are talking about some concepts that, regardless of they genesis, have a special relationship with one another. Mathematics is a set of such concepts. The genesis of the ideas are certainly through experience. The relationship between the concepts are not. As soon as you define some concepts, and some relationships between them, the rest come necessarily; whether or not we become aware of them, know them, experience them, or be totally oblivious to them.

I suspect that the answers to the following questions might not still be clear enough in this discussion:

(4) What is deduction?
(5) How does deduction work?
(6) Can one give a few examples of a deductive argument? Four different examples? Following four different rules, and not just one syllogism repeated in different scenarios?
(7) What do we mean by “reason” and what do we mean by “logic”? Are they the same? What are the popular conceptions of these two terms?

It seems obvious to me that many perplexities, such as the ones we are discussing in this topic, would certainly arise when answers to these questions and other similar ones are missing somewhere in someone’s understanding of the issues at hand. Because, it seems to be impossible to claim what has been claimed so far, all against deduction, if/when one knows what deductive logic means, how it works, the difference between a deductive argument and the concept of ‘reasoning’ in general, and so on.

So, to take this topic back on track, I would like to ask everyone to pause for few days and perhaps read a few articles that are written on these topics (in addition to the Wikipedia articles), and see if we can push this forward later (as a result, any further comment that does not directly address the above-mentioned questions, or the ones in the OP should be discarded without consideration).


Rusty and dusty, perhaps?

Even if we have been teaching logic in a mathematics department (or somewhere else for that matter), we would certainly benefit from a refreshment and reading of a few more recent and relevant articles or books. So, please let us relax our assumed authorities in logic and talk as if we are laypeople in that regard, and try to understand the issue instead. This applies to me too. So, I would reset my expectations now and assume that I don’t know much about logic, and start reading on the topic. I will, therefore, assume that you also know nothing about logic, unless otherwise is genuinely demonstrated in here.

Therefore, accordingly, you and I know nothing about logic and should say nothing. We only say things that we can demonstrate somehow (certainly repeating what one believes does not make a demonstration of the truth of anything) and let everyone know how we go about demonstrating that.

If there is a confusion as for what articles or books might be suitable for our purpose, please post a comment to that effect and we all will try to find the most relevant ones and post them in here for everyone’s benefit. If anyone prefers, I can reproduce the list of the essential readings posted under the first topic in the logic board in here as well. That might help a bit.

[Andreas Geisler]

How else a hypothesis is made then aside from randomly generating stuff that might or might not make sense?

A) That is not what induction does.
and
B) That is not how hypotheses are made.

Induction creates all the stuff that we know. A cursory analysis of reason will reveal that ONLY INDUCTION can introduce fundamentally new concepts. Deduction can only piece together the pieces we already have.
But hypotheses are not made by piecing together things we already know (deduction), nor by extrapolating relations available in the stream of senses (induction).
Hypotheses are made by assaying what we know, locating missing bits of the puzzle and then abducing the kind of piece that could fit in a hole like that. I.e. it draws on the web of knowledge formed by induction and deduction, but the process itself is abduction.

[Alain Van Hout]

I would like to know if there is (or are) any reason to say what you said Alain. Opinions and assertions are certainly very interesting, but they eventually must be supported by something rather than a personal preference. So far, there has not been any demonstration of the claims such as:

I would say: contrary to deduction, induction is essential to science.

Which means, deduction is not. This is only a claim and that is fine. What might be the reason why this might be the case?

My apologies for not constructing my argument in a clearer way. I do however believe I supported that claim: I made the claim about induction being of little or no value in science because nowhere in the scientific process do I see it making a valuable contribution (although my second-to-last paragraph was styled as a series of questions, it did make that argument -  I'll try to be more specific in the future).

As an aside, regarding the second quoted claim, which was made by Andreas (for clarity), I would like to add the distinction that the claim I made is that induction is not an inherent requirement of the formation of hypotheses, although it is indeed the method by which most hypotheses are created (i.e. probabilistic estimations of what avenues of research are most likely to offer interesting results).

(1) In science, how a scientist form a hypothesis?

In practice, this is done by using existing observations to make a(n inherently probabilistic) statement about the causes for those observations, where the likelihood of different alternative explanations for those observations is weighed in an inductive manner (even here, 'alternative' already implies a prior filtering of all possible explanations, pixies included, to begin the weighing with those that are reasonably probable).

If we're talking about the scientific method, as in the minimal parts that are required for hypothesis forming, then inductive weighing of alternative explanations is however not a part of science, since taking any random possible alternative explanation, pixies included, would not brake the scientific method, even though it would probably slow it to a crawl (as would be the case of choosing topics that are less likely to receive grant money).

(2) How any hypothesis, regardless of its origin (by chance, by a random hypothesis generator, by magic, etc.), can be tested?

(3) How a test of a hypothesis can work without deduction?

A hypothesis involves an explanation of currently available observations that also has implications for observations that have not yet been made. The test of any hypothesis is (creating an experimental setup for) making those missing observations and comparing them to the expectations of the hypothesis.

Does that last step, i.e. the comparison of the observed results with the expected results, or more specifically determining what the expected results would be, qualify as deductive reasoning? If so, then I would agree that deduction plays an essential role in the scientific process.

For the remainder, I agree that a refreshment of all the concepts you mentioned would be a good idea. A such, for now I'll leave the above (since I have already written), and come back after I've gone through some refreshment material. As to assumed authorities versus (talking as) laypeople, be assured that when it comes to logic/philosophy I squarely place myself in the latter category, and will continue to do so (I'm here because I know something about science, and want to learn something about philosophy) .