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On this page in the book Probability Theory: the Logic of Science written by E. T. Jaynes, the author says that:

If A implies B then a false proposition implies all propositions, and if we tried to interpret this as logical deducibility, it would follow that every false proposition is logically contradictory. Yet the proposition: ‘Beethoven outlived Berlioz’ is false but hardly logically contradictory(for Beethoven did outlive many people who were the same age as Berlioz).

It means that one proposition can be logically contradictory.

And in this book another author says that:

A pair of statements is logically contradictory if and only if it is not possible for the statements to have the same truth values.

Then a pair of statements can be, intuitively "reasonably", logically contradictory.

I more agree with the latter and am confused by the former. I wonder what the proposition that Beethoven outlived Berlioz is logically contradictory with?

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  • $\begingroup$ A statement with a false truth value always implies a statement with a true truth value. Hence they never have the same truth values and are logically contradictory. Here, "Beethoven outlived Berlioz" would imply any other proposition being true. Hence you get two propositions that must have different truth values, and are then logically contradictory. $\endgroup$
    – Zain Patel
    Commented Jul 9, 2016 at 10:12
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    $\begingroup$ I am not sure whether this is the answer you are looking for but 'Beethoven did not outlive Berlioz' and 'Beethoven outlived Berlioz' are a pair of statements and it is logically contradictory. For every false statement $p$, you can consider $p$ and $\neg p$ as logically contradictory. $\endgroup$
    – Levent
    Commented Jul 9, 2016 at 10:13

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It's a careless author misusing language to confuse you. He should be talking about a single statement being logically self- contradictory.

  • "Beethoven outlived Berlioz" is not logically self-contradictory. It happens not to be a true statement about our particular world. Chesterton's "tigers grow on trees" is similar.

  • "Beethoven outlived Berlioz and died before Berlioz was born" is not logically self-contradictory. It happens not to be a true statement in our world or in any world in which people aren't born after they die.

  • "Beethoven outlived Berlioz and died before him" is logically self-contradictory. It cannot be true in any world at all. This is like Chesterton's "four-sided triangle".

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  • $\begingroup$ So would you please illustrate why it is not logically self-contradictory combined with the author's reason that Beethoven did outlive many people who were the same age as Berlioz? $\endgroup$
    – Lerner Zhang
    Commented Jul 9, 2016 at 14:20
  • $\begingroup$ In normal mathematical discourse, there is no assumption that the members of a pair are distinct: $(1, 1)$ is a perfectly good pair of natural numbers. $\endgroup$
    – Rob Arthan
    Commented Jul 10, 2016 at 22:09

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