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Given the premise that deductive reasoning needs a premise statement to get started, ultimately do these premises come from inductive observation? e.g.

When the sun is out it’s daytime. The sun is out. Therefore it’s daytime.

This implies that some observation has concluded the premise of the sun being out, and having some sort of correlation to some pattern of the sun coming out?

benbyford
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I don't think that axioms like 'A thing cannot have both a quality and its opposite' can be acquired from observation.

Nobody can show us what 'not' means. The opposite of red is not brown, or blue, or something else we can actually encounter. But we believe right off that there are red things, and there are things that are not red. There is a logical jump we have to make for ourselves.

The idea that things have opposites comes naturally, to the extent that we somehow expect things to have opposites that just can't have them. (I have recently been asked, "What is the opposite of a barrister?")

So, not all deduction can be based on observation. Some of it is in the nature of language, at a level that seems inborn, but is at least tacit and involves necessary, predicted generalizations beyond observation that we are all expected to make in the same way.

hide_in_plain_sight
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  • Not really answering the question as far as I can see – benbyford May 21 '21 at 19:34
  • @benbyford We all deduce that negation works. It is impossible to do so via observation. Ergo, not all deduction happens by observation. I don't see what is missing. If you meant to ask a different question, why didn't you? – hide_in_plain_sight May 21 '21 at 19:51
  • Maybe I’m just not understanding you sorry – benbyford May 21 '21 at 21:34
  • Implication or if-then is an important part of (deductive) logic. Negation — the not operator — is another important part of (most deductive) systems. Your question implicitly meta-inducts (I think) from "if-then is key in deductive logic" to "if-then is all there is in logic". This answer artfully points out that "not" doesn't generally/necesarily follow inductively from "if-then". BTW Computer science as usual has more egs of more nuanced versions of this than than classical math/logic – Rushi May 22 '21 at 02:23
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    @Rusi-packing-up But David Hume! If-then is not reachable by observation either. You can verify it by observing, but you can't get the basic rule from empirical observation. So even if if-then was all there was, stringing together if-thens would still be deduction about stuff not derived from observation, even if it is all not very meaningful or helpful without some outside data, it still exists. – hide_in_plain_sight May 22 '21 at 02:50
  • Yes David Hume! I was just deductively(!) conceptifying your eg to the underlying idea — in boolean algebra terms a complete connective set eg {nand} {nor} as singletons work but {implies} doesnt. Hume of course gives (another) answer to the OP's question. BTW nice to see you back . (@benbyford I missed marking you in earlier comment) – Rushi May 22 '21 at 02:56
  • @Rusi-packing-up I must have taken your comment wrong -- that if we restricted the question to implication, it would then be true. But, of course, it still wouldn't. You can have transitivity of implication as a deduction, or something, and then you have made a deduction out of stuff that cannot proceed from observation. – hide_in_plain_sight May 22 '21 at 03:25
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Does all deductive knowledge stem from inductive observation?

Strictly speaking, there is no "deductive knowledge" and there is no "inductive observation". There is only knowledge, observations, and beliefs. However, what most people would call knowledge is actually beliefs. They would say they know they have a tree in their garden when in fact they only believe that. This is also why our theories are revisable. If we knew our theories true, we could never revise them without producing a false theories instead.

Deductive logic is all we need in a world about which we know nothing, not even that there is one. We only need deductive logic because we are capable of living our lives on the sole basis of our beliefs about the world. We trust our senses. People who don't starve to death and don't reproduce and their genes are selected out. People may be dogmatic about things like God and what not, but they are all prepared to revise some of their beliefs whenever their senses make them feel the pain of their mistakes.

To express this in your quirky terminology, inductive knowledge follows from inductive observations, and this is all we need to survive in our world.

What about knowledge itself? Well, clearly, there are things we know. I know I have the impression of seeing a tree in my garden. From this, I will derive the belief that there is a tree in my garden.

So, essentially, we know what we know, and logic never gets to be used to infer any knowledge which is not already apparent in the premise. So, if the premise is that I know I have the impression of seeing a tree in my garden, then all I can infer is that I have the impression of seeing a tree in my garden.

Speakpigeon
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  • I guess I’m conflating logic with knowledge in my questions. You mention “that we know what we know”... does this suppose logic is useless for knowledge accruement? or... truth? And that your knowledge stated above is devised solely from our beliefs in our senses? – benbyford May 23 '21 at 19:18
  • @benbyford "logic is useless for knowledge accruement?" We can a perfectly logical reasoning (or valid argument) together with a false conclusion. Further, if my initial assumptions happen to be true and my reasoning is logical, I will be able to derive a true conclusion. But, how do I know that my assumptions are true to begin with? They may be true but I will not know that they are true. And if I want to prove them, I will have to make some other assumptions I will not know whether they are true. Logic could work with knowledge but we use it with our beliefs. – Speakpigeon May 24 '21 at 10:09
  • @benbyford "knowledge stated above is devised solely from our beliefs in our senses?" I am not idea and nobody does. It is a fact that I know I have the impression of being in pain whenever I have this impression. This is the whole extent of our knowledge. And from this knowledge, we cannot derive anything about the material world, essentially because it is knowledge of and about our subjective experience. – Speakpigeon May 24 '21 at 10:14
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There's no inductive observation, all observations by definition are empirical through senses (instruments can be classified as our extended senses). I think you're asking about whether deductive reasoning necessarily entails inductive reasoning. These two are completely different modes of reasoning. A deductive reasoning system whose axioms/assumptions/premises (most powerful enough deductive systems need) may not be required to be observed actually true, one can always introduce an unverifiable premise like "It's necessary that if a thing exists then necessarily so" as just recently discussed in a previous PSE question here. Or in math most of us implicitly use ZFC set theory as a foundational deductive system, while the famous C part (Axiom of choice) cannot be observed to be actually true from any experience since it's really about choice from infinitely many sets each of which having no choice function and many mathematicians doubt its soundness and try to avoid using it. Also recently in physics one can express physical laws in terms of impossible or counterfactual/possible modes which goes beyond traditional empirical observations of only what's actual, such as David Deutsch's Constructor theory.

Double Knot
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  • This being the case is it not still necessary to have some basis out side of deduction to hang your assumptions? In maths this would be set theory as mentioned, but sets are derived from something no? A set of things, things that might have existed at some point and have been put into a group or set of things? Otherwise all we have is illusion... or put in another famous way—its turtles all the way down. – benbyford May 21 '21 at 19:39
  • @benbyford. Babies know things they cannot yet have made observations about -- like what version of their visual field at different focal lengths is the 'real' one that makes things look right. A lot of people would equate the axioms of math with those forms of genetic knowledge, built into Chomsky's 'language instinct'. (Not set theory, but older forms of math.) Combinations of things you knew before making any observations are deductions not dependent upon observation. You can declare evolution a form of observation, but it is a very different thing than we expect that word to indicate. – hide_in_plain_sight May 21 '21 at 20:07
  • If it were turtles all the way down, how could we start learning? There is a lot of built-in machinery there. Some knowledge really only depends on combinations of stuff you knew at birth. – hide_in_plain_sight May 21 '21 at 20:13
  • Ok so you saying that deductive knowledge stems from evolution...? – benbyford May 21 '21 at 21:33
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    @benbyford ZFC doesn't have any object which is non-set, only hereditary sets exist. The original ZFA and other alternative set theories have objects (urelements, Quine atoms, etc) such as Von Neumann–Bernays–Gödel set theory, Quine's NF, etc. Many intuitionists or constructivists think similarly like you and reject ZFC's applicability to our actual world and some even only embrace arithmetic foundation like PA or ACA, but ZFC still holds advantage and simplify many proofs compared to all the others. So many are doing illusory math, but same illusion holds for non-Euclidean geometry before GR. – Double Knot May 21 '21 at 21:43
  • @benbyford your question is essentially the thousands of years of debate between empiricism and rationalism... Some people strictly allow finite "observable" natural numbers only such as Leopold Kronecker. A famous historical example is he persistently rejected Cantor's transfinite set theory (the forerunner of all modern set theories) using his authority and made Cantor mad and caused Cantor breakdown eventually... A much ancient example is the discovery of irrational numbers. – Double Knot May 21 '21 at 21:55
  • @benbyford Not most of it, but at least some minimal part. We need to make some basic deductions in order to even know how to start observing. Many of those things become the axioms of basic logic. We havet to be able to extract the patterns that allow us to put information together from somewhere. And we can't learn it, because that is putting infomration together.. – hide_in_plain_sight May 22 '21 at 00:54
  • We know things like the "correlates of correlates are correlated to each other" As Hume proves, that is not something one can learn. One has to assume it. We know that "the opposites of opposites are similar", etc. And we put those together very early on, or we could not learn anything from our observations. That means that not all deductive logic is learned from observation. A tiny bit of it is front-loaded. – hide_in_plain_sight May 22 '21 at 01:13