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Please, could you help me make sense of/classify types of deductive reasoning?

When studying mathematical logical, I have noticed there is this Hilbert's axiomatic system (Hilbert calculus) with its inference rules and axioms. We call it deduction when deriving theorems, and sequence of those applied rules is called a formal proof.

But there is also other type of "not so rigorous" deduction, which is somehow based on semantical understanding of the meaning of the words. I will give you an example of what I mean by that: "Suppose that any horse can sit in any chair. Also suppose that cars can fly and can carry things of any weight and any size on its seats. ==> Therefore I deduce that any horse can fly." - there could definitely be a better example but I think it illustrates you what I mean by this "type" of "everyday" deduction.

Than there is this next kind of deduction, which is somehow right between those two deductions in terms of their "rigorousness" - deduction used in most of the mathemacal proofs (informal proofs). This type of deduction also uses semantic understanding, but somehow more strictly than in the previous example (maybe?). Maybe there is another type, I am not sure.

Also there is a difference between those types of deductive reasoning in explicitly stating the rules which I use in reasoning (formal proofs) vs. some implicit rules I use (I would call them a "common sense" reasoning rules, but I would appreciate some more detailed descrition of "those rules" we people implicitly use for informal proofs).

Is there any naming convention for those types of deductions? Could you help me make some clear distinction between those "deductions".

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    You may have a look at the mediaeval theory of " consequences" ( formal and material) See : Stanford Encyclopedia of Philosophy.( online) – user37859 Apr 20 at 21:52
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    It is much simpler than this. There are only two different types of inferences, syntactic (based on axioms and inference rules) and semantic (based on models, truth tables in the simplest case). The former is used in formal mathematics almost exclusively. As for levels of rigor used in regular reasoning there is no classifying them, they vary from inference to inference even in the same argument, and most informal arguments are mixed, containing both syntactic and semantic inferences. But in math they are expected to be formalizable. – Conifold Apr 20 at 23:09
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    To expand on Conifold's reply, in mathematics, to highlight this distinction, there are two theories, one syntactic proof theory and semantic model theory. These differences (one might say opposites on the end of a continuum, are a function of differences in artificial and natural languages. – J D Apr 21 at 17:21
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Just an attempt .

*First distinction ( based on the logical form of the conclusion) *

hypothetico-deductive reasoning / categorical deductive reasoning

*Second distinction : based on the nature of the consequence relation *:

formal deductive reasoning / material deductive reasoning

H ence maybe 4 possibilities, the ideal of modern science ( formal axiomatic systems) being certainly : hypothetical reasoning ( axioms = postulates, not " absolute truths") + formal reasoning.

Note : in hypothetico-deductive reasoning , the premises are only hypotheses, and the conclusion has the form : "if [ premises} then [consequence]", but the consequence, by itself, is not asserted as categorically true.

Note : by " material deductive reasoning" ( home made terminology) I mean a reasoning such as

Peter is John's father. John is David's father. Therefore, Peter is David's grand-father.

it is the meaning of the terms " father" and " grand-father" that makes the conclusion necessary , given the premises. It's not a pure logical consequence relation.

Note : "" [in geometry] One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs" ( atributed to Hilbert ) --> purely formal reasoning

A link concerning pure logical consequence relation: https://www.iep.utm.edu/logcon/

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    I would disagree with your distinctions listed. Firstly, they aren't needed.. Secondly if there is a deduction based on something other than logical form such as relationships that would PROVE all if . . . Then structured statements are not logical consequences. The if . . . Then construct in Mathematical logic may use terminology differently than philosophers. They are not identical. Material logic is not home made terminology. It was brought to light by medevial philosophers. This meant logic or deductive reasoning is not purely based on FORM. Definitions or meanings matter in many cases. – Logikal Apr 21 at 14:44

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