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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?

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    Having said that, deductive reasoning means: assuming something as true (axioms, premises) and deriving consequences. May 21 at 7:15
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    @conifold but then we have produced no knowledge, we just explicited the definition of daytime.
    – armand
    May 21 at 8:08
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    @armand By the same logic, mathematics produces no knowledge, only explicates definitions and axioms. It is not a very satisfactory concept of "knowledge".
    – Conifold
    May 21 at 8:15
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    @conifold: well, isn't it exactly the case ? What good is to you a theorem if you don't have a real world fact to make equivalent to another one ?
    – armand
    May 21 at 10:44
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    @armand What good is it to have a fact if you do not know a theorem to relate it to another fact you can use? Knowledge and utility are different things.
    – Conifold
    May 21 at 11:36
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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.

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  • Not really answering the question as far as I can see
    – benbyford
    May 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? May 21 at 19:51
  • Maybe I’m just not understanding you sorry
    – benbyford
    May 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 May 22 at 2:23
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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. May 22 at 2:50
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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.

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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 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. May 24 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. May 24 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.

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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 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. May 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. May 21 at 20:13
  • Ok so you saying that deductive knowledge stems from evolution...?
    – benbyford
    May 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. May 21 at 21:43

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