Philosophy of Science

Deduction
All swans are white.
A is a swan.
Therefore, A is white.

Induction
I observe many swans, and all the swans that I have seen are white.
A is a swan.
Therefore, A is white.

Abduction or Inference to the best explanation
All swans are white with long necks.
A is white with a long neck.
Therefore, A is a swan.

Falsification
I hypothesize that all swans are white.
A is a black swan.
The hypothesis is false.

Falsifiability
I have a theory that all swans are white.
I could, in theory, find a black swan.
The theory is scientific.

I have a theory that all swans are either white or not white.
I could never find a swan that is neither white or not white.
The theory is not scientific.

Hypothetico-deductive model
Without observing any swans, I hypothesize that all swans are white.
I therefore predict that all swans are white.
I then seek to observe non-white swans and failure to find any would corroborate my hypothesis.

(Bayesian) probability
My degree of belief that all swans are white is 0.95.
In the face of new evidence, I will update my degree of belief.

Problem of induction
The principle of induction worked successfully on occasion x₁.
The principle of induction worked successfully on occasion x₂.
.........................................................................
The principle of induction worked successfully on occasion x_n.
The principle of induction always works.
The above argument — used to justify induction — relies on induction, so is therefore flawed.

New riddle of induction
Let the word "whack" be defined relative to an arbitrary time t as follows: An object satisfies the proposition "x is whack" if and only if it is examined before t and is white, or is examined after t and is black.
The problem is this. Assuming t has yet to pass, then every swan that has been observed is both white and whack. But surely we believe that swans discovered after t will be white, not whack. So it seems that a white and whack swan discovered before t is evidence only for "All swans are white", not evidence for "All swans are whack." The problem is to explain why.

Problems with falsification

• To verify that "all swans are white" requires that one falsifies an infinite number of hypotheses: "all swans are red", "all swans are blue", "all swans are green", etc.
• The existence of a single dirty white swan would erroneously lead us to reject "all swans are white" (there could be a problem with the evidence, not the theory).
• "There are white swans" cannot be falsified.
• "90% of swans are white" cannot be falsified.
• In practice, a theory will need to be augmented by auxiliary assumptions.