Where I disagree with Banno is that it is appropriate to submit inductive reasoning to the criterion of logical validity; which belongs to only to deductive reasoning. I also may disagree with him in thinking that all inductive reasoning can be reframed in deductive form and that it then does become subject to what you would call the "narrow" notion of validity. — Janus

If you define validity the way he does, as truth-preservation, then yes, he's right, induction is invalid because its conclusion can be true and its premises, defined restrictively, false. There is no arguing with this. The question is: why is that relevant? The answer is probably that it isn't relevant. The "problem" with induction is that there is no problem with induction but with deduction. The problem is our understanding of logical necessity. It's a very deceptive concept. We say that a deductive argument is valid if it is impossible for the premises to be true and for the conclusion to be false. This suggests that there is such athing as absolute certainty. And this is caused by the fact that we are confusing two different types of premises: low-level premises (such as observations) and high-level premises (such as general statements.) Put simply, we are confusing observations with assumptions. That's a problem. Induction operates on low-level premises (observations) whereas deduction operates on both low-level premises (observations) and high-level premises (generalizations.) Observations cannot contradict other observations. What observations can do is they can contradict our generalizations or assumptions. This is because generalizations are derived from observations. So if the base of observations from which a generalization is derived changes, it's very possible for the generalization to change as well. Take a look at the following argument:

1. All men are white
2. Socrates is a man
3. Therefore, Socrates is white

Insofar the second premise is a raw observation it cannot change if the conclusion turns out to be false. However, the first premise can because it is a general statement. Such a general statement is derived inductively from previous observations. It's an open system. If it is possible for the set of observations on which it is based to change, for example by making a new observation, then it is possible for the statement to change as well. If the conclusion turns out to be false then we'll have a new observation in our set of observations and this observation, telling us that there exists a man who is not white, would require that we change our conclusion. On the other hand, exceptions do not disprove the rule, they merely make it weaker. So if a thousand men are white and a single man is black then we can preserve our general statement. But if we reach a point where a thousand men are white and a ten thousand of men are black, then we'd have to change it. This is, of course, a simplification of what's going on in reality. Our conclusions need not be this simplistic in practice. We can let exceptions influence our actions. But describing in words how this process works is a chore.

Both deductive and inductive arguments are adaptive it's just that deduction adapts through contradiction whereas induction adapts through observation. The second might also be true of deduction but it's usually not the case because most people see deduction as distinctively negative (or eliminative) process. In fact, they think of it so negatively that they think that every contradiction requires a change in one of the premises. Which is sort of true but is also sort of insane. As Einstein said and Popper thought, "No amount of experimentation can ever prove me right; however a single experiment can prove me wrong." The question is, of course, how precise you want to be. Most people are fine with ignoring exceptions. Within that sort of mindset, falsification/negation is not so different from verification/affirmation. Many deductionists also think that deduction can only tell you what's wrong and never what's right. Think of Socrates, think of his dialectic, think of his famous statement "the only thing I know is that I know nothing."

OK, I think I see where you are coming from now. It may be consistent with "some Ps are Qs" that all Ps are Qs, but not that no Ps are Qs. So, you are thinking of logical consequence, not in the sense of logical entailment, but of semantic consistency. — Janus

I am not sure we are on the same page. I don't think that the argument is valid because the two sentences are consistent. I don't even know what that means. I am saying that the argument is valid because it does not violate the rules of that particular type of reasoning. In induction, the rule is "if most Ps are Qs then you must conclude that all Ps are Qs." The rule is implicit in the narrow definition of induction making inductive reasoning necessarily valid (i.e. it cannot be invalid.) I covered this in one of my previous posts.

Here's a general form of probabilistic argument:

1. X out of Y Ps is Q
2. Therefore, this P is R

I intentionally define probabilistic reasoning to be of this general form in order to make it possible for it to be invalid.

Here's a valid probabilistic argument:

1. 3 out of 5 men are alcoholics
2. Therefore, this man is an alcoholic

Here's an invalid probabilistic argument:

1. 3 out of 5 men are alcoholics
2. Therefore, this man is not an alcoholic

Why is this argument invalid? Because it violates the rules of probabilistic reasoning. The main rule of probabilistic reasoning is that if most Ps are Qs then you must conclude that a particular P is also a Q. If you don't, the argument is invalid.

You are focusing too much on specifics. Sort of like Banno. This can be dangerously deceptive. Banno says that induction is invalid which suggests that there is something wrong with it. You try to counter this by saying that induction is neither valid nor invalid. But does Banno really care? Of course not. He's focusing on his extremely narrow definitions. Nothing can change his mind because what he says is true by definition. Both of you are being too formal. Both of you ignore there's much debate about what logical consequence is. Both of you restrict yourself to Wikipedia and high-school textbooks. I am certainly not the first to speak of inductive validity but is that really important?

An argument is valid in the general sense of the word if it does not violate the rules of reasoning.

This is valid:

1. Some Ps are Qs
2. All Ps are Qs

This is invalid:

1. Some Ps are Qs
2. No P is Q

This is also invalid:

1. Some Ps are Qs
2. Half of Ps are Qs, half of Ps are not Qs

It's all relative to the rules of reasoning.

You seem to think that if the logical entailment of "Some P's are Q's" is not "No P's are Q's" then it must be "All P's are Q's". — Janus

That's exactly what logical consequence is in the broad sense of the wrong.

This is simply mistaken; the only logical consequence of "Some P's are Q's" is that some P's are Q's.

And here you're defining the concept of logical consequence narrowly.

I can say the same about you. In fact, I'd probably be more right than you are. A lot of people think they understand what they are talking about; and they do, but very superficially. If I asked you to define logical consequence, I'm pretty sure you'd struggle. Either that or you would define it the same way that I do just unnecessarily narrowly.

The logical consequence of 2 + 2 is 4. That's what you get when you follow the rules of addition. In the same way, the logical consequence of "Some Ps are Qs" is "All Ps are Qs". That's what you get when you follow the rules of induction. Very simple.

1. Some men are bald

Do you seriously believe that this logically entails that all men are bald?

Wow, man, if you really believe that then I'm not sure there is any point conversing with you further. — Janus

Maybe you should start paying attention to what other people are saying.

I've already said that every conclusion is empirically (or semantically) uncertain. Even if you knew everything there is to know about the past, your predictions can still turn out to be wrong. The future is under no obligation to mimic the past.

The problem is that you do not understand what logical consequence is.

The concept of logical necessity has been mystified. In order to demystify it we must first understand that a logical argument is nothing more than a mathematical function; or more generally, a relation between two sets. You have premises, which are analogous to inputs, and a conclusion, which is analogous to outputs. It's pretty straightforward to determine whether an equation is true or false. If f(a, b) = a + b then f(2, 2) = 5 is not true. The conclusion (5) does not follow from the premises (2 and 2.) That would be analogous to a logically invalid argument. I think this is pretty clear to everyone. What's not clear is that this is exactly what it means for an argument to be logically invalid. Most people think in terms of truth-preservation. But truth-preservation and logical validity are two different things. In the example above, we're dealing with tautalogical (or unconditional) truths. 2, 2 and 5 are all logically true. It is thus impossible for the premises to be true and the conclusion to be false. And yet, the argument is not valid. Truth-preservation is just a common symptom of validity. Nothing more. To equate the two is to say that 2 + 2 = 5 is valid which is counter-intuitive. And to say that induction is invalid merely because it is not explicitly truth-preserving is deceptive since it indicates there is something wrong with it.

Truth-preservation is a very deceptive concept because it suggests that if you know the past that you can predict the future with certainty. That's not true. A much more careful definition is required in order to avoid that but that is a non-trivial task.

Every logical conclusion is empirically uncertain. This means that even if you know all of the past with perfect accuracy, your predictions can still turn out to be wrong. Thus, it makes no sense to emphasize that inductive conclusions are probable since deductive conclusions are no less probable.

I'm actually talking about knowledge acquisition. I believe this is what Hume was also interested in.

Intuition is not only poorly defined, it is impossible to define because it pops out of the experience of living. It just happens. However, the more one practices observation and pattern recognition, the more one is likely to have moments of inspiration because it is sharp observation skills that is the mother of inspiration. With such a process, one is merely traveling on the same path of knowledge forever. I guess one can rely on the inspiration of others — Rich

It makes sense to me to say that objects that are defined to be infinite in scope are impossible to be described in entirety. It makes sense because it is true by definition. You defined the object under your scrutiny to be infinite in size, so it's impossible to fully exhaust it. If you define a human body to be infinite in complexity, you cannot hope to describe it completely. No matter how much of it you include in your description, there is always something about it you have yet to describe. Though you cannot describe such phenomena in entirely, you can nonetheless describe it in part. Your descriptions can be more or less exhaustive. The same thing about intuition. Regardless of whether you define it to be finite or infinite in complexity, you can always describe it to a higher or lower degree.

What I am trying to understand is why do you think that intuition is impossible to describe. I can describe my intuition no problem. All I have to do is make an effort to do so. Again, intuition is merely defined as a process of decision making that is outside of our awareness.

↪Magnus Anderson If we are agreeing, then great. However, I don't know where intuition/inspiration falls within the rules of logic. Also, pattern recognition is not simply generalization. Frequently it is a process of observing differences and similarities among many patterns and then intuitively combining these intuitively conceived newer patterns into an entirely new greater pattern that allows one to acquire an entirely new way of understanding something. Most breakthroughs in science happen this way. — Rich

I think that inspiration is irrelevant to understanding how reasoning works. Intuition, on the other hand, is a poorly defined term that for the most part means nothing other than "knowing something without being able to explain how". That's not very useful.

What is important is how the set of known particulars (i.e. experience, past observations, etc) is mapped to the set of unknown particulars (i.e. predictions and retrodictions.) It might not be appropriate to call this process "generalization" since high-level concepts (i.e. laws, rules, patterns, models, universals, etc) are ignored. Instead, we map directly from known particulars to unknown particulars. We go directly from observations to predictions. In some cases, we go straight to actions.

Logic is the study of this mapping. This mapping is initially intuitive in the sense that we cannot explain how it works. But through time, with careful introspection, we gradually become aware of its inner workings. The intuition, in part or in whole, becomes formalized. It acquires a memetic existence which allows us to employ it mechanically i.e. by simply following its written instructions.

Observations together with pattern recognition combined with inspiration/intuition that gives new meaning to these recognized patterns. — Rich

That's my point: observations together with the process of generalization which you call "pattern recognition combined with inspiration/intuition". That process is not arbitrary. It unfolds according to some set of rules. Logic studies the rules of that process.

If an argument is such that its conclusion follows necessarily from its premises then it is an deductive argument, end of story. That is how a deductive argument is defined. — Janus

Not true.

Here's an inductive argument:

1. Some Ps are Qs
2. Therefore, all Ps are Qs

The conclusion necessarily follows from the premise. You cannot conclude something like "Therefore, no P is Q". It is necessary that you conclude "Therefore, all Ps are Qs". Note that we're talking about logical necessity and not objective necessity.

Whether or not a certain syllogism is "valid" is only relevant on graded tests.

What is relevant is knowledge is acquired by a combination of personal observations, group consensus, and periodic moments of intuition and inspiration. Such knowledge can be used in a formal manner using some symbolic logic, but the root of knowledge is in observational pattern recognition of v various sorts. — Rich

Reasoning isn't merely about making observations. A mass of observations will mean nothing to you if you cannot generalize from them. Logic is the study of this process of generalization. This process of generalization can take any of form but there is one form that we consider "rational" or "valid" and numerous other forms that we consider "irrational" or "invalid". Logic is the effort to discriminate between the two. More generally, it is the effort to discriminate between different patterns of reasoning and analyze their consequences, their pros and cons, under different circumstances.

then induction can be valid.
— Magnus Anderson

Only if it is framed in deductive form — Janus

Not really. If I define logical validity broadly to mean that an argument is valid if and only if it logically follows from the premises (i.e. if it does not violate the rules of reasoning) then induction is by definition valid. Note that I corrected myself? I initially said that it can be valid but then I realized that induction is defined so narrowly that it cannot be other than valid.

1. Some Ps are Qs
2. Therefore, all Ps are Qs

If there is a conditional rule between the premise (the independent variable) and the conclusion (the dependent variable) which states that the P and Q in the conclusion must be the same P and Q in the premise, then it is impossible for this sort of argument to be anything other than logically valid. It is not necessary for the argument to be truth preserving to be considered valid i.e. if its conclusion is false it's not necessary that some of its premises be false.

The problem is that logical validity is poorly defined. Look at this:

In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. — Wikipedia

This suggests that logical validity and truth preservation are one and the same thing. They are not. Truth preservation is merely a symptom of logical validity and even then not always.

Here's an argument that is truth preserving but nonetheless invalid:

1. Donald Trump has an orange hair
2. Melania Trump is Donald Trump's wife
2. Therefore, 2+2=4

Truth preserving but not valid. The conclusion does not follow from the premises.

The reverse is also true. If an argument is not truth preserving that does not mean it's not valid. The following is perfectly valid:

1. All observed swans are white.
2. Therefore, all swans are white.

Logical validity simply means that the conclusion logically follows from the premises. It means that it does not violate the rules of reasoning. It means that the structure of the argument, the combination of the premises and conclusion, is legal.

The problem is that two distinct concepts have been confused: one broad (logical validity) and one narrow (truth preservation.) I believe the mistake stems from the initial observation that logical validity is always coupled with truth preservation. People will always make such mistakes so as long they focus on narrow concepts first and broad concepts second.

And you still haven't addressed my point about the analogical nature of inductive and abductive thinking and the inappropriateness of applying the criterion of deductive validity where it doesn't belong, in order to dismiss the importance of its role in human inquiry into the nature of the world. — Janus

It's a matter of how he defines words. If he defines validity narrowly to mean truth-preserving validity, i.e. that an argument is valid if and only if its conclusion being false means some of its premises are false, then yes, induction is invalid because it can have a true premise and a false conclusion. But if you're like me and you define the concept of validity broadly to mean that an argument is valid if and only if it logically follows from the premises (i.e. if it does not violate the rules of reasoning) then induction is by definition valid.

Really, what he's saying is that induction, in the narrow way that he defines it, is not truth preserving. I think it's strange to say it's invalid because it suggests there is something wrong with it. If an argument is not trurh preserving that does not mean it's wrong. Sometimes, with some kinds of arguments, it does. But that does not apply to induction

Why is it wrong? Because one of its premises is "every swan in the future must be of the same color every swan in the past was"? That's based on the fundamental premise of all reasoning (which is that the future will mimic the past.)

↪Magnus Anderson SO you have re-framed induction as a deduction with a false premise. — Banno

What I did is I made induction more explicit.

↪Magnus Anderson Do I have to point out that the second premise is false? — Banno

The point is that in the following argument it is impossible for the premises to be true and the conclusion to be false:

1. All swans in the past were white.
2. Every swan in the future will be the same color that every swan in the past was.
3. Therefore, every swan in the future will be white.

If both premises are true then every swan in the future must be white.

The fact that you ignore unspoken premises.

My response was this:

1. All swans in the past were white.
2. Every swan in the future will be the same color that every swan in the past was.
3. Therefore, every swan in the future will be white. — Magnus Anderson

And yours?

Magnus, I've shown repeatedly that in an induction the conclusion does not follow from true premises. It's not just my say so. — Banno

What you did is called backpedalling.

You were wrong in this post of yours:

Here's a funny thing. In deductive logic, if the premise is true, and the argument valid, then the conclusion will be true. That's what it is to be valid.

So, in your deductive example:

1. all men have blonde hair,
2. Socrates is a man.
3. Therefore, Socrates has a blonde hair.

If one and two are true, three must also be true. As it turns out, (1) is false, adn hence so is the conclusion.

But in your inductive example

1. All swans in the past were white.
2. Every swan in the future will be white.

(1) is true, and yet (2) is false. That is, the premise is true, the conclusion false - the very opposite of validity. — Banno

You never addressed that.

Sorry - are you saying induction must be valid? — Banno

No, I am saying that induction must be invalid . . . because you say so.

And what is actually going on is far more human, humane and downright interesting than following a mere algorithm. — Banno

Now you sound like a mystic.

It must be.

No.

It is not the case that because every swan seen by white fellas was white, they will never see a black swan in the future. — Banno

Again, logical conclusions, whether they are deductive or inductive, are empirically uncertain which means they can turn out to be wrong regardless of the premises they are derived from.

For example:

1. All men have blonde hair.
2. Barack Obama is a man.
3. Therefore, Barack Obama has a blonde hair.

Although logically certain, it is not empirically certain that Barack Obama has a blonde hair. In fact, he does not. So the premises can be accepted as true and the conclusion can turn out to be empirically false.

Here's a funny thing. In deductive logic, if the premise is true, and the argument valid, then the conclusion will be true. That's what it is to be valid.

So, in your deductive example:

1. all men have blonde hair,
2. Socrates is a man.
3. Therefore, Socrates has a blonde hair.

If one and two are true, three must also be true. As it turns out, (1) is false, adn hence so is the conclusion.

But in your inductive example

1. All swans in the past were white.
2. Every swan in the future will be white.

(1) is true, and yet (2) is false. That is, the premise is true, the conclusion false - the very opposite of validity. — Banno

That's true. And that applies to induction too. It applies to any kind of algorithm. If you want an algorithm to be able to map its inputs to its outputs in a way that it previously didn't, you must modify it. That's what "back-propagation" does to neural networks. Hardly ground-breaking.

The problem is that the manner in which we describe inductive reasoning is not exhaustive. We leave out many premises from our description. They thus become hidden premises.

Here's an example:

1. All swans in the past were white.
2. Every swan in the future will be the same color that every swan in the past was.
3. Therefore, every swan in the future will be white.

This is a gross over-simplification of the process of inductive reasoning but it is nonetheless less of a simplification than the usual description.

Is that enough to prove my point?

Inductive reasoning operates according to some set of rules. This is obvious from the fact that some conclusions are permitted (e.g. that every swan in the future will be white given that all swans in the past were white) and that some conclusions are not permitted (e.g. that every swan in the future will be black given that all swans in the past were white.) When there are no rules, every premises-conclusion combination is fine.

Things exist whether we are aware of them or not. This means that rules exist whether we are aware of them or not. If we are not aware of the underlying rules of reasoning that does not mean that there are no such rules. Does that make sense? When we reason inductively we follow a set of step-by-step instructions on how to do so. But when we describe how we reason inductively we usually leave out a lot of detail. Our descriptions are not exhaustive and detailed but simplistic and shallow. We do not list EVERY SINGLE PREMISE that leads to our conclusions. So if you do not see how premises are bound to a given conclusion, that does not mean that they aren't bound.

If it is valid, show us it's form. — Banno

Do I have to show you its form, i.e. describe the process in detail, in order for you to accept that inductive reasoning is a rule-bound process? Is that necessary? Shouldn't that be obvious from the fact that some premises-conclusion combinations are legal and others aren't?

And if it is valid, why are the swans around here black? — Banno

If an argument is valid that does not mean that its conclusion is empirically true.

Here's a valid logical argument:

1. All men have blonde hair.
2. Socrates is a man.
3. Therefore, Socrates has a blonde hair.

Does that mean it is empirically true that Socrates has a blonde hair? No.

Sure, it is not valid. Because you say so. We'll leave it at that.

An argument is logically valid if it does not violate the rules of reasoning. Both deduction and induction must follow some rules of reasoning.

This is logically valid:

1. All swans in the past were white.
2. Every swan in the future will be white.

This is logically invalid:

1. All swans in the past were white.
2. Every swan in the future will be black.

This is not merely a metaphor and certainly not a deception.

That's what I thought. I merely wanted to confirm it. Yes, Popper hated induction.

A lot of people think there's a problem with induction. They think that the fact that inductive conclusions can turn out to be wrong is an argument against it. As if deductive conclusions aren't equally uncertain. As if thinking that "Because all swans are white and this is a swan, this swan is also white" means that the conclusion that this swan is white cannot turn out to be wrong. Yet we all know that it can. The problem is caused by equivocation. It appears strange to say that deductive arguments, and also inductive arguments, can be both certain and uncertain. That's because we're confusing logical certainty (or properly speaking validity) with empirical certainty. Deductive arguments can be logically certain/valid but they can never be empirically certain; they are always empirically uncertain. The same applies to inductive arguments. They can be logically certain/valid but they can never be empirically certain. Nietzsche said that "absolute certainty is a contradictio in adjecto" and he's right. Assumptions cannot be empirically certain because they are by definition empirically uncertain. Reasoning is the process of forming assumptions regarding events that we haven't observed. Because we didn't observe them, this means that what we think they are can always turn out to be wrong.

Popper's solution to the imaginary problem of induction was his theory of reasoning where we reason by making conjectures (i.e. by making shit up) and then by seeking refutations. He thought that relying on induction is dogmatic i.e. it makes us unwilling to adapt our models of reality in the face of new evidence. What he was blind to is the fact that adaptability is not enough. Models of reality must adapt in a specific manner. In statistics, there is something called "overfitting". It's the situation in which our models fit the data perfectly but do not go beyond it in the way that we expect it. According to Popper, this would be fine. So you have a sequence of observations such as 1 2 3 and you want to model it. Popper says you do so by coming up with any kind of model that fits the data. Okay, so I come up with a model. Now Popper says you use deduction to make predictions i.e. what kind of observations you assume will come next in the sequence. I use deduction and I get something like 1 2 3 0 0 0 0 0. I then set out to test my theory by testing my predictions against reality. I then realize that the next observation in the sequence, quite suprisingly, is number 4. I ask Popper what to do now and he says "adapt!" And I do so by changing my model so that it now makes the following predictions: 1 2 3 4 0 0 0 0. Pretty neat, huh? Well, not so if we ask our intuition. Our intuition wants to build a different model, a model that makes the following predictions: 1 2 3 4 5 6 7 8. Of course, Popper was aware that his theory is stupid, so in order to save it, he added additional rules such as Occam's rule of razor otherwise known as the principle of economy. But he never bothered to define these rules. Is it because he was afraid of doing so? what if he realized that thinking is in fact a rule-bound, deterministic, process? that thinking is fundamentally inductive?

He was a staunch opponent of induction. The purpose of thinking is to generalize. A theory (what he calls a hypothesis) can fit the data very well but be pretty bad at going beyond it. The question is: how do we generalize? "Conjectures and refutations" does not answer that. We do not assume the unknows randomly.

DEDUCTION

1. If it rains, the grass gets wet.
2. It rained.
3. Therefore, the grass is wet.

ABDUCTION

1. The grass is wet.
2. If it rains, the grass gets wet.
3. Therefore, it rained.

INDUCTION

1. It rained.
2. The grass is wet.
3. Therefore, if it rains, the grass gets wet.

Both deduction and induction rely on rules, which is to say higher level concepts, in order to arrive at their conclusions. This suggests to me that they are superficial forms of reasoning in relation to induction which operates on observations, which is to say lower level concepts. You cannot "promote" inductive argument to a deductive or an abductive argument. You can only "demote" it. This means you can turn it into a deductive or an abductive argument only by simplifying it. However, you can "promote" a deductive or an abductive argument to an inductive argument which means you make such arguments complex, that you enrich them with detail, when you turn them into an inductive argument. What this suggests to me is quite simply that . . . induction is fundamental.

Within the context of this post, the three different types of reasoning are defined narrowly to mean different ways of describing the process of arriving at our conclusions. Abduction and deduction are different from induction in that they rely on higher level concepts such as rules. This is what makes them simplistic in relation to induction. By relying on high level concepts, they get rid of a lot of detail. High level concepts = abstractions. Abstractions = simplifications of reality.

They are not equal in rank. Thus, they do not complement each other. Induction can do everything deduction and abduction can do and then some more but deduction and abduction can only do a subset of what induction can do.

So deduction and abduction are simpler than induction. There is one more thing. They are equally simplistic. Deduction is no more simplistic than abduction. This is if we take the definition of abduction to be what it is said to be. There is absolutely nothing complex about the following argument:

1. The grass is wet.
2. If it rains, the grass gets wet.
3. Therefore, it rained.

So what is the algorithm of abduction? You choose some observation (some "surprising fact") and you do so arbitrarily. You can choose any. It does not matter whether it is imaginary or real. Then you choose a conditional that is related to that observation. The conditional must be in the form "if something occurs, then the observation that you previously chose occurs". This something can be any observation of your choice (just like the first time.) Finally, in order for the conclusion to follow from the premises, and for the argument to be logically valid, the conclusion must be the unconditional part from the conditional in the premise #2 stated in the past tense. That's all. These are the rules of abduction. Very simple, right? Anyone can learn these rules and follow them.

I am sure people will say this is not abduction proper but merely a simplification of the process. Sure. What is missing? Oh, that the premises must be true? We need to add that rule? Okay, we'll add it. But wait a second, what exactly does that mean? It's rather vague, isn't it? What does it mean for a premise to be true? You have to define that rule first. And if you want, we can do it right now. Let's say we somehow manage to do it. Would that be enough? It would? Well, Peirce would disagree. If I remember correctly, Peirce revised the definition of abductive reasoning several times. Apparently, some additional rules were missing. Sure, we can add these rules too but would that be enough? Did Peirce ever arrive at a definition of abductive reasoning that he was satisfied with? No? Is abductive reasoning something that escapes definition? Is it like mystical phenomena? Simply impossible to define? Or is it merely so complex that it is difficult to define? Either way, this means we don't actually know what abductive reasoning is. We know some of its features but not all of them. And this means that these features could be anything. So what are we going to do?

This would be the argument that abduction is more complex than deduction it's just that our description of it cannot capture it. The problem with this argument is that we can say the same thing about deduction. If it looks simple, it's only because our description of it is simplistic. It is possible that deduction is infinitely complex and that it only appears to be simple because when we describe it we do so without taking into account all of its features (which would be impossible to do if it is infinitely complex.)

So there is nothing inherent to deduction that makes it simpler than abduction and nothing inherent to abduction that makes it more complex than deduction. According to the manner in which these concepts are defined in this post, they are of equal complexity (which, no matter how great, is always lower than that of induction.)

Another belief related to the concept of deduction that I have to assassinate is that deduction merely extracts knowledge that is contained within the premises. If that's true then it is also true about induction and any other patterned (i.e. rule-based) method of reasoning. Induction employs the same mechanism as deduction. There is fundamentally no difference between the two. But it's not true; it's not true that deduction merely extracts knowledge from the premises. I can easily come up with a complex mathematical expression the result of which was never known to me. Calculating the result of a mathematical expression does not simply mean recalling its result. It means inventing a result. We invent a result and then check to see if its relation to the past is appropriate (i.e. that it does not violate the rules.) Chess is deductive but how many of us actually understand its possibility space in its entirety? The myth that deduction is uncreative is perpetuated by Eastern thinkers (who no doubt suffer from inferiority complex.) Anyone who buys into it is being brainwashed.

One last thing I want to say about deduction is that it does not necessarily proceed from a general statement towards a specific statement. But then this depends on what is meant by "general statement". A general statement, I assume, has the form "all Xs are Ys". Such a statement is equivalent to a genral conditional "if P is X then P is Y". I say general conditional to mean that P is a varible which means it does not refer to anything in particular. An example of a general statement would thus be something like "all men are mortal" and an example of a general conditional would be something like "if P is man then P is mortal". Let's take a look at some examples.

The standard deductive argument:

1. All men are mortal
2. Socrates is a man
3. Therefore, Socrates is mortal

A variation of that argument:

1. If P is a man then P is mortal
2. Socrates is a man
3. Therefore, Socrates is mortal

A specific version of that argument:

1. If Socrates is a man then Socrates is mortal
2. Socrates is a man
3. Therefore, Socrates is mortal

The last argument has no general statements. It also has no general conditionals. Instead, it has a specific conditional.

But I asked you for clarification about this "relevance" of yours. For me, there is a background metaphysics that explains the specific relevance. For you, there must be likewise some background metaphysics - given that it seems you must have some good reason to reject my metaphysics as a relevant grounding.

So what is this metaphysics exactly? Put it on the table. — apokrisis

I would more than like to. The problem is I have no clue what metaphysics is. What is it? I've heard stories about it but they never made any sense and in those rare cases when they did it was a combination of epistemology, logic and conceptual analysis. Maybe ask a specific question? or describe to me what metaphysics is so that I can give you an answer? I might be too demanding, I know.

Or we could discuss some of the things we have already touched upon such as how thinking works. We can discuss "the thinking algorithm" if you want. What rules do we follow when we make assumptions? How do we proceed from a set of particular knowns (i.e. past observations) to a set of ranked candidates for particular unknowns (i.e. predictions and retrodictions)? I already gave you the basic idea behind my approach but you rejected it on the ground that it was a textbook of something you call "deductive thought". I have no idea what "deductive thought" is. You appear to be fond of ANN's but you find something wrong with my approach? I have no idea what's the problem.

It's already difficult to agree on what deductive and inductive reasoning are. Most people have trouble ADMITTING that the key difference between the two is NOT that one is certain and the other is uncertain. Every conclusion is uncertain in the sense that it can turn out to be wrong. The idea of a conclusion that is not uncertan -- that is certain -- in the sense that it cannot possible turn out to be wrong is NON-SENSICAL. It is possible that there are conclusions that WILL NEVER turn out to be wrong but it makes no sense to say that there are conclusions that cannot POSSIBLY turn out to be wrong. A subtle but crucial distinction. Thus, both induction and deduction produce conclusions that are UNCERTAIN. I am not going to say PROBABLE even though I can. It should be a given that every truth-claim is probable since for a claim to be probable simply means that it has a probability value assigned to it. And every truth-claim has a probability value assigned to it. True/false is a measure of probability. And a measure of probability is simply how we subjectively rank possibilities in terms of their likelihood. True/false is two-valued probability measure. It makes no sense to call it something other than probability simply because it is not one-hundred-valued (0% to 100%) or infinite-valued (such as 0.000~ to 1.000~.) Furthermore, both induction and deduction can either be logically valid (or syntactically certain) or logically invalid (or syntactically uncertain.) I gave an example of logically invalid induction in one of my previous posts.

Fine. And yet you kept asking anyway. And I kept explaining why I do find it relevant. And so far you haven't rebutted my reasons for finding it relevant. And importantly so. Yet you want to keep telling me you don't find it relevant - despite offering no supporting reasons. — apokrisis

Can you prove a negative? If so, how do you do it? By showing that there is no evidence supporting the claim, right?

Negatives are problematic. They could be caused by personal deficiency (subjective lack such as ignorance) or they might be real (objective lack.) You can never be sure.

The best I can do in this case is to ask further questions for the purpose of clarification. Maybe my post was deficient in this regard? Could be.

I'm not following you. I've talked about all those things. You seem to want to make some campaign against abduction as a concept. And I am interested in how abduction fits into a holistic and naturalistic scheme of reasoning. — apokrisis

I have nothing against the concept. I am just trying to understand why you place so much emphasis on it. I don't see why such a concept is relevant. That's all. And if what I say appears to be an attack then it's merely due to the possibility that some of the things you say are no more than smokes and mirrors. I have to entertain such a possibility.

You will have to explain why I should be concerned by your problems with seeing a relevance in abduction. I've already explained why it would be relevant to a metaphysics that is irreducibly triadic (rather than dyadic or monadic). — apokrisis

You don't have to if you don't want to. I take it to be a matter of good will on your part.

I'm not getting too hung up on the divisions. There is the more familiar dichotomy of deductive vs inductive argument - necessary inferences vs probable inferences. That kind of works in the sense that deduction proceeds from the general to the particular with syntactic certainty while induction does the reverse of going from the particular to the general with provisional hopefulness. — apokrisis

Syntactic certainty, or logical validity, isn't unique to deduction. Here's an example of inductive argument that is logically invalid:

1. Every swan in the past has been white.
2. Every swan in the future will be black.

That's invalid because it violates the rules of inductive reasoning according to which the future must mimic the past.

Similarly, probability isn't unique to induction. Deductive conclusions aren't certain. They can turn out to be wrong.

The special thing about deduction is merely the fact that if its conclusion turns out to be wrong then some or all of its premises will also turn out to be wrong. This is not the case with induction.

But then a triadic view - one where a dichotomistic separation resolves itself into a hierarchical structure - is the special twist that Peirce brings to everything. It is the next step which completes the metaphysics. — apokrisis

I have yet to see the relevance of placing so much emphasis on the concept of trinity.

But is it?

The classical deductive syllogism is:
Major premise (or the general rule: All M are P.
Minor premise (or the particular case): All S are M.
Conclusion (or result): All S are P.

Abduction then rearranges the order so that the argument is: All Ms are Ps (rule); all Ss are Ps (result); therefore, all Ss are Ms (case).

So you would have to say something like:
- Rain makes things wet.
- This grass is wet.
- Therefore, the grass was (probably) left out in the rain last night. — apokrisis

What's the difference? You changed the order of the premises. You put the rule in front of the observation. Charles Sanders Peirce, according to Wikipedia (1902 and after), had no problem with placing the observation before the rule.

1. The surprising fact, C, is observed. (The grass is wet.)
2. But if A were true, C would be a matter of course. (If it rains, the grass gets wet.)
3. Hence, there is reason to suspect that A is true. (Therefore, it rained.)

You also emphasize that the conclusion is merely probable. I think that such an emphasis is unnecessary since it is a given. All conclusions can turn out to be wrong. No conclusion is certain in the sense that it cannot turn out to be wrong.

I'm not sure why it seems a problem that abduction is retroductive - that the past is being assumed to hold the key to the future. — apokrisis

It is not a problem. The problem is that abduction is a narrowly defined concept. Specific concepts have specific needs. What are these specific needs? This is what I am trying to understand. I want to understand its relevance.

Abduction seeks out the constraints that must underly any observable regularity in the world. — apokrisis

But that's what induction does. Its job is to identify regularities in data. But instead of talking about induction, or more generally intelligence or thinking, you talk about abduction. And instead of speaking in terms of regularities or patterns, you speak in terms of constraints. I don't understand why. There must be a reason. What kind of reason is it? Is it a good or a bad one?

I'm sure the rule you abduce is the simper one - 1+1=2. And so on, ad infinitum. — apokrisis

What does that mean?

So, you can talk about the limitations of inductive reasoning, but the limitations of deductive reasoning are more severe, as it tell us nothing at all other than whether we've correctly solved our Sudoku puzzle. — Hanover

True. I prefer to think in terms of Zebra puzzle. When someone says "deduction" I imagine think of this puzzle (that noone can solve except for genuises.)

My own thinking is informed by modern science and its efforts to build pattern recognising machines, as well as the efforts to understand the same in human brains. — apokrisis

I've read your posts and I am currently trying to make sense out of them. In the mean time, I want to ask you a very simple question in order to make sure that we are on the same page. The question is: do you agree that abductive reasoning is a specific type of inductive reasoning? Ultimately, I understand that this comes down to how we define the concept of inductive reasoning. I believe that if the concept of induction is defined sufficiently narrowly that the answer would be "no". I do not, however, define it that way and I believe that others do not either.

Here's an example of abductive reasoning:

1. The grass is wet.
2. If it rains, the grass gets wet.
3. Therefore, it rained.

It is quite apparent to me that abductive reasoning is a very narrow form of reasoning. By definition, it only forms conclusions regarding events that took place in the past. This means that abductive reasoning is restricted to making "predictions" about the past. In other words, it can only be used to create retrodictions. This is unlike induction which can be used to form beliefs of any kind. This suggests to me the possibility of you defining the concept of induction narrowly as pertaining to making assumptions about the future.

Now let's take a look at a simple example of inductive reasoning. We have a sequence of numbers such as 1 2 3 4. Inductive reasoning can be defined as the process of identifying the pattern that best matches some given data and then using that pattern to form beliefs regarding data that lies outside of this data. Note that there are several ways, perhaps infinitely many different ways, that data can be outside of data. For example, you can ask "what comes after the number 4?" We can all agree it is 5. And we do so intuitively without being aware of the underlying process. We are often unware of answers to questions such as 1) how do we identify the right pattern based on the data that we're given?, and 2) how do we use that pattern to calculate the best guess regarding the unknown we are interested in? (The question #2 less so than the question #1.) We can formalize the question "what comes after number 4?" as 1 2 3 4 ? where question mark denotes the unknown we are interested in. But we can also ask "what comes before the number 1?" This would be analogous to retrodiction. We can formalize this question as ? 1 2 3 4 and we can also immediately answer it by saying that the number that comes before 1 is 0. But we can go further than that and we can ask questions such as "what comes between the number 2 and 3?" We can formalize this question as 1 - 2 ? 3 - 4 where hyphen represents an unknown we are not interested in. The answer is, of course, 2.5. An example of an inductive question that is most similar to what is called abductive reasoning would be a question formalized like 1 2 3 ? 5. This would be analogous to an abductive argument such as:

1. Number 5 is observed.
2. Every number is equal to the sum of the number that precedes it and number 1.
3. Therefore, number 5 was preceded by number 4.

Note that the abductive argument does not specify how we arrived at the rule that constitutes the premise #2. We did so using induction on a number of observations the majority of which have been ommited from the argument (i.e. the sequence of values 1 2 3 ?.) What this indicates is that abductive arguments, like deductive arguments and pretty much all other arguments, are simplifications of reality. They are simplistic. Inductive arguments are also simplistic but they are LESS simplistic than these two types of arguments. Unless, of course, you define induction narrowly.

The interesting question then is how induction, or whatever you want to call it, works. The question is how do we find the pattern that best fits some given data. In fact, my thinking is that the very concept of pattern is unnecessary. We do not need to be aware of any patterns. When I guess that the next value in the sequence 1 2 3 4 is number 5 I do not necessarily do so because I am aware of the underlying pattern. Rather, in most cases, I do so because I know that the set {1, 2, 3, 4, 5} has the highest degree of similarity to the set {1, 2, 3, 4} among the sets that have the form {1, 2, 3, 4, *}. Thinking is fundamentally associative. So the interesting question then becomes the formalization of fluid concept of equality (a.k.a. similarity) between mathematical objects such as sets. The question becomes: how do we measure the degree of similarity between symbols?

Regarding AGI research, most of the research has been dedicated to modelling how the world works rather than to modelling how thinking works. I think that's the problem. Rather than having a programmer create a model of reality, an ontology, for the computer to think within, it is better for a programmer to create a model of thinking which will allow machines to create models of reality -- ontologies -- on their own based on the data that is given to them. This would make machine thinking much more adaptable.

If we are going to use B to justify A, then we ought be more confident in B than in A. It would be odd to attempt a justification with evidence that was weaker than what is being justified. — Banno

Side-question. I do not come from a philosophical background. I might be interested in philosophical problems and I might be willing to try to solve them on my own but I did not study and I generally do not read philosophy. So I do not understand many of the things that a lot of people on this board take for granted. For example, I don't really know what justification is. Everyone is talking about, and everyone is asking how, to justify this belief or to justify that belief; everyone has certain kinds of philosophical problems that they want others to solve for them or that they themselves want to solve or have solved. But most of these philosophical problems are alien to me. I do not understand what exactly is problematic. So I have to ask, unfortunately, and I believe that by doing so I will remain within the boundaries of the topic, what exactly is justification? I can make my own guesses. For example, it is perfectly sensible, based on how we use the word otherwise, to assume that to justify your beliefs means to try to convince someone else to accept them. And in order to do so, you have to do whatever is necessary for the other person to accept your beliefs. That's a very simple, very general, understanding of what it means to justify a belief. But then, some guy named Plato comes along and says that knowledge is "justified true belief" without ever explaining to whom a belief should be justified. To ourselves? But what does it mean to justify our beliefs to ourselves? Very vague term. In reality, we don't justify our beliefs to ourselves, we simply accept them and when we decide to do so we revise them. What is justification in this context? Making sure that we feel that our beliefs are good enough in order to act upon them?

I guess it all depends what you think is a central question of epistemology. Is it how can we hope to have certain knowledge? Or is it how is it that we can reason in optimal fashion? — apokrisis

I think that epistemology, or more simply logic, should be the study of thinking (which I define to be the process of forming beliefs or assumptions.) It should study different patterns of thinking, both actual and possible, and their pros and cons or what consequences they produce under different circumstances. This should answer the second question, which is "how can we reason in optimal fashion?", but the first question, which is "how can we hope to have certain knowledge?", will remain unanswered. I think the first question makes no sense at all. Either it does not or I simply don't understand what it means. Which one is it?

So abduction, if I understand you correctly, is a pattern of no-pattern of thinking. You say that it is the least formalisable pattern of thinking which suggests to me that it lacks pattern to a considerable degree. Or it could be that the pattern is complex and thus difficult to understand and formalise? Which one of the two is the case? I am inclined to think the former but I like to keep my options open.

So if abduction is a process of thinking that has very little pattern within itself, this means that abduction is mostly a random process. It's basically random guessing.

While I believe that there are decisions that are random, I think that a lot of abduction that occurs in real life, at least as I understand the concept, is quite ordered. If you see a disembodied head lying on the floor you are not going to assume "someone clapped his heads and this head popped out of nowhere" you are going to assume something like "someone's head has been cut off". That betrays order. Not necessarily in reality but in thought.

Abduction has the following form:

1. Event B occurs.
2. When event A occurs event B follows.
3. Therefore, event A occured before B.

That's the general version of abduction. It looks sort of like deduction. Its premises are either observed (e.g. there is a disembodied head on the floor), randomly assumed (e.g. when someone claps his hands a disembodied head pops out of nowhere) or inductively inferred (e.g. when you cut someone's head off you end up with a disembodied head.)

I am struggling to see something interest here.

My past experience has proven, that in certain circumstances (ex. the laws of nature) the future is like the past.
— Agustino

You're muddling the tenses. "My past experience has proven, that in certain circumstances (ex. the laws of nature) the future has been like the past." And this says exactly nothing about what will be.

You have to rely on the assumption that the future will be like the past in order for past evidence to be relevant to the future. Which is assuming the conclusion. I don't object to you doing it, I just object to your claim that is is reasoned. — unenlightened

I agree with you. I think this is the case of not understanding the question. The question is: what causes us to think in a particular manner where "particular manner" in the context of this thread refers to thinking inductively. This is an empirical question. We are asking what variable is correlated with the variable that is manner of thinking. Or if you don't want to think in terms of "manner of thinking" we can switch to thinking in terms of the belief that the future will be similar to the past. Our manner of thinking, that of induction, is based on that belief. So the question now becomes what variable is correlated with the variable that is "the future will be related to the past in manner X". There may or may not be such a variable. We're thus asking 1) is there such a variable, and 2) if there is such a variable, what is it? Then there is the infinite regress problem; of if you don't like to call it a "problem" you can call it something like the fact (or merely the possibility) that the past is infinite. If the past is infinite that means there are no "first causes". Instead, it is possible that every effect has a cause which has a cause which has a cause and so on. So explanations can only describe a fraction of a process. We go back only a number of steps back into the past. How many steps? As many as we need for our purposes.

Who wants to answer the question of the evolution of reasoning in humans? I am personally not so interested. But if you asked me to give you my best guess, this would be it: originally, organisms chose their beliefs at random. So some organisms chose to believe the future will be similar to the past and others didn't, others chose something else. Through time, those that chose to believe that the future will be similar to the past continued to exist -- for one reason or another, again, they might have survived for no reason at all but my guess would be that there were reasons for their survival and that these reasons were tied to their choice to believe that the future mimics the past -- and those that didn't, they simply disappeared. According to this story, the original choice is made at random (i.e. it's an arbitrary or an irrational choice) and the subsequent persistence to believe what they decided to believe (i.e. the unwillingness to revise the original decision) is due to the lack of survival pressure to do so. Of course, this is not necessarily true, but it's what I am inclined to think -- more or less. So basically, my explanation is an evolutionary explanation.

But if we are talking about a modern scientific view of reasoning, then introducing the third thing of abduction, or an axiomatic leap of the imagination, makes a big difference. It says knowledge works quite differently from how the traditional conflation of deductive logic and rationality might want to represent it. — apokrisis

I have yet to see the relevance of introducing the third type of reasoning that is abductive (or retroductive) reasoning. It does more to obscure than to clarify the manner in which reasoning functions. Personally, I think you're paying way too much of your attention to a single philosopher that is Charles Sanders Peirce.

There is very little difference between induction, deduction and abduction.

Thought without language is most certainly possible. Artists, for example, think in terms in imagery, symbols. Language is a device for sharing thoughts.

It's not language that forms thoughts but rather it is experiences which are reflected in languages. First and foremost, it is always experiences (memories). — Rich

My preference is to define the concept of thinking so broadly that it refers to a kind of phenomenon that does not have to be accompanied by brain let alone the ability to use language. Most people, however, prefer to think in specifics, so they are inclined to define thinking narrowly as a conscious process that takes place in brain and is intertwined with language.