In the philosophy of science inductivism exists both in a classical naive version, which has been highly influential, and in various more sophisticated versions. The naive version, which can be traced back to thinkers such as David Hume, says that general statements (theories) have to be based on empirical observations, which are subsequently generalized into statements which can either be regarded as true or probably true.
The classical example goes from a series of observations:
Swan no. 1 was white, Swan no. 2 was white… Swan no. 3 was white… to the general statement: All swans are white.
In support of this view it can be said that we often appear to think in this manner.
In science the proof that there is a law of gravity would then consist in having recorded a large number of observations of things falling, or of bodies attracting one another. Typically classical inductivism will require large and varied amounts of data, which also means that it has difficulties in explaining the importance of singular observations, such as the one in 1918 where light could be observed to be bent around the sun in accordance with Einstein's prediction in his General theory of relativity.
Another problem is that no inductive conclusion can yield certainty, and here David Hume also famously pointed out that it cannot even be shown that inductive conclusions yield probable conclusions as one gets involved in a circular argument in that case, trying to prove the value of induction through induction.
Karl Popper in The Logic of Scientific Discovery emerged as a major critic of classical inductivism, which he saw as an essentially conservative strategy. He replaced induction with falsification. His simplest argument here says that no induction can prove that all swans are white, since this will require an infinite number of observations, but that the observation of a single non-white swan will falsify the statement that all swans are white. The logical rule invoked here is modus tollens and is purely deductive.
A more detailed discussion of induction involves the whole theory of probability.
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