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    Basic Logic for Popper

    The simple conditional claim, modus ponens (modus ponendo ponens).

    If A happens, then B will happen. A happens, therefore B happens.

    1.   A→B

    2.   A

    ∴ 3.   B   (1,2 modus ponens)

    People commonly use this argument pattern to make simple if-then arguments. However a common mistake is to commit the fallacy of affirming the consequent, whereby it is argued that if B is observed, then A must have happened. At first this seems logical, but it could be the case the A did not happen and B happened for some other reason. Consider this example: if Aristotle drops the ball (A), then the ball will bounce (B). Suppose it is observed that the ball bounced. Does this necessarily entail Aristotle dropped the ball? No. His friend Plato could have taken the ball and dropped it himself. So it is not necessarily true that because B was observed, A happened.

    1.   A→B

    2.   B

    ∴ 3.   A   (1,2 fallacy of affirming the consequent)

     

    There is a similar argument pattern known as modus tollens (modus tollendo tollens).

    If A happens, then B will happen. B does not happen, therefore A could not have happened.

    1.   A→B

    2.   ~B

    ∴ 3.   ~A   (1,2 modus tollens)

    Consider our friends Aristotle and Plato again. If Aristotle drops the ball (A), then the ball will bounce (B). Suppose it is not observed that the ball bounced (~B). Does this necessarily entail Aristotle dropped the ball? No, because it couldn’t! The ball did not bounce, so it is not possible that anyone had dropped it. So Aristotle (or anyone for that matter) did not drop the ball (~A).

    Given the modus tollens argument pattern there is also a fallacy that is often committed. If Aristotle drops the ball (A), then the ball will bounce (B). Suppose it is not observed that Aristotle dropped the ball. Does this necessarily entail the ball did not drop? No. Again, it is possible that while Aristotle did not drop the ball, Plato took it and dropped it himself causing the ball to bounce. This is known as the fallacy of denying the antecedent.

    1.   A→B

    2.   ~A

    ∴ 3.   ~B   (1,2 fallacy of denying the antecedent)

     

    According to Popper conventional science used the fallacy of affirming the consequent to validate it’s hypotheses. It would make some prediction, A, and then if B was observed scientists assumed A must have been true. This is why science now operates within a paradigm that uses the logical formulation of modus tollens, i.e. scientists try to falsify the prediction, thereby trying to disprove a hypothesis. However this also means it is not possible to prove any hypothesis, because A cannot be derived from ~B.