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The following arguments is always given as a classic example to deductive reasoning:

All men are mortal. (First premise)

Socrates is a man. (Second premise)

Therefore, Socrates is mortal. (Conclusion)

Inductive reasoning however is about observation.

But in the deductive example above isn't there induction as well. I mean all the premises we start in any deductive reasoning should come from some sort of observation. Isn't it? I'm not able to distinguish these two types of reasoning.

For example Bertrand Russell in his book states:

But in connection with mathematics the one-sidedness of the Greek genius appears: it reasoned deductively from what appeared self-evident, not inductively from what had been observed.

Could you clarify what he means here? I cannot draw a line between deduction and induction here. How come any sort of reasoning be independent of induction? Any premise I can think of is due to my senses and observations (?)

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    In reasoning, it makes no difference where premises come from, what matters is how they are handled. In your example logically valid inference rules are used, so it is deductive. In "all swans I've seen are white, therefore, all swans are white" we have an inductive inference instead, it is logically invalid, only plausible. Also, premises like "a+b=b+a" in arithmetic or "acceleration is the derivative of velocity" in mechanics do not come from observations, they are postulated by convention. – Conifold Jul 22 at 0:44
  • The gist of induction is to establish the truth of general laws, like e.g. "All men are mortals", the truth of wich is assumed "as is" by deductive reasoning. – Mauro ALLEGRANZA Jul 22 at 6:32
  • You are likely referring to the analytical propositions and synthetic propositions distinction. Synthetic propositions are propositions about the world and are sense verifiable. Analytic propositions are not required to be sense verified and can be tautological. Analytic propositions also are usually semantic. That is they are governed be rules of grammar or syntax in a given language. All bachelors are unmarried males is a classic example of an analytic proposition. All swans are birds is NOT analytic. All swans are birds is sysnthetic. It's knowledge about the world & sense verifiable. – Logikal Jul 23 at 4:05
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Deduction and induction are not about observation, but certainty in inference.

It may be tempting to define deduction as moving from general to specific claims, and induction vice versa, but this is not entirely accurate. Let's take an example to show the difference between deduction and induction.

DEDUCTION: If a man is in a kitchen, then he is in the house in which the kitchen is located. Bob is in the kitchen, therefore we can conclude Bob is in the house the kitchen is located. Note, that since the kitchen is in the house, it is an inescapable conclusion that Bob, if entirely in the kitchen, is also entirely in the house. This is a deduction that is called a conditional in logic, and has the form p -> q.

INDUCTION: Bob frequently is at home on Monday mornings. Bob was at home on Monday mornings for the last 11 weeks, and given that Bob's car is always present when he is home, observing that today, on the 12th Monday Bob's car is present, it is likely Bob is home though we have not directly observed him.

Simply put, in the first example the conclusion MUST follow from the premises, and that reflects a basic reality about space-time, in this case a volume in space-time follows transitivity. In the second example, while it is likely that the conclusion is true, it is not known with certainty. For instance, Bob may have gone on vacation this week leaving his car home, and the conclusion can only be verified by empirical means (ringing the doorbell might verify he's home if he answers). That is the essential difference between the types of inference. Note there is a third form of inference called abduction which is aligned with heuristics.

As far as the Greek reference to deduction, of course the Greek named Euclid of Alexandria is recognized as the Father of Geometry and a the proponent of the axiomatic method, which is a chain of deductions that allows certainty of conclusion. Euclid's Elements has served as a model of reasoning for mathematicians and philosophers since it was put to paper, and many thinkers have sought to bring the certainty of logic even to disciplines other than mathematics. Scientists, for instance, have long sought the deductive certainty that math provides in their knowledge of the universe (see Netwonian mechanics) and have often been shocked or refused to believe that certainty in physics is not absolute (see Einstein and Bohr).

Another place where the questions of determinism and probability clash is in the philosophy of mathematics itself, where David Hilbert and Kurt Gödel raised and settled questions regarding the nature of truth and certainty in mathematics itself (see Hilbert's Program).

In review, induction and deduction are not about observation, per se, but rather the certainty of conclusions drawn from premises, regardless of the nature of the observations, premises, and conclusions involved.

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First things first... Deduction works by establishing a class of objects and extending the properties of that class to specific instances. Thus in the syllogism you presented, we start with the class 'human being' and establish that it has the property 'is mortal', then we identify Socrates as an instance of that class, and extend the property of the class to him.

Induction, by contrast, starts with a set of instances (observations) and establishes a class for the instances based on common properties. Thus we might look at a whole bunch of objects and induce the class 'human' based on intrinsic similarities, and we might observe further and see that all our instances of this class seem to kick the bucket eventually, and induce that this class has the property 'mortality.'

In practice, this process is cyclical: we create a class based on our observations, then use that class to observe new things, then refine and reconstruct the class based on the new set of observations we generate. Induction and deduction are fairly inseparable in the real world, though usually one or the other comes to the fore at any given moment.

With respect to Russell... Meh! I suppose he's right enough within the context of his own biases, but it is an ungenerous assessment. The Greeks were actually quite brilliant with induction, but they induced things that we (in our age) take for granted and memorize by rote. I mean, nowadays someone teaches us the Pythagorean theorem and we memorize it (or if we're feeling uppity we deduce it ourselves from known principles). But Pythagoras himself had to induce that theorem from long cogitation on the nature of triangles. But Russell's biases are too long of a tangent to get into here.

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