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Studying the book Forall X: Calgary Remix: An Introduction to Formal Logic by P.D. Magnus, came across these exercises and wanted opinion.

A. For each of the following: Is it necessarily true, necessarily false, or contingent?

  1. If Caesar crossed the Rubicon, then someone has. Necessarily true

    • My answer was: contingent. How can I interpret “someone has” ? How can I be sure this is always true?

  2. Wood is a light, durable substance useful for building things.

    • Not sure about this one.

  3. Elephants dissolve in water. If you put an elephant in water, it will disintegrate.

    • Same as above.

F. Consider the following sentences. Which of the following are jointly possible?

  • M1 All people are mortal.
  • M2 Socrates is a person.
  • M3 Socrates will never die.
  • M4 Socrates is mortal.

Sentences M2 and M3

  • Can I presuppose a person is mortal? Or perhaps, those two are jointly possible because I do not have that information?
F. Zer
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    Specifically for "If Caesar crossed the Rubicon, then someone has." imagine Q -> A. For any Q it will always be true also that A is true, meaning it is tautology. If you know the truth tables then Conditional is only false (contingent) if Q was True and A was False, but this would never occur in the above case. Therefore, it is not possible for it to be contingent. – bodhihammer May 03 '20 at 08:53
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    Another example of tautology is, for instance:

    If Adam is a male, then he is masculine.

     – bodhihammer May 03 '20 at 08:59

1 Answers1

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Necessary truth (or logical truth) is commonly described as something that is true "in all possible worlds". One might say that we can realize its truth through logic instead of through experience.

If Caesar crossed the Rubicon, then someone has [crossed the Rubicon].

This is necessarily true. It is not logically possible that Ceasar has done something that nobody has done.

Wood is a light, durable substance useful for building things.
Elephants dissolve in water. If you put an elephant in water, it will disintegrate.

Neither of these statements are necessarily true. Even in our world, there is wood that is not durable or useful for building things.

And perhaps there is a world out there where elephants are made out of sugar. That's a bit silly. But my point is that we only know that elephants can survive in water because we have seen it or trust people who claim that they have, not because of logic.

M1 All people are mortal.
M2 Socrates is a person.
M3 Socrates will never die.
M4 Socrates is mortal.

$M3$ and $M4$ are logically inconsistent, so they are not jointly possible. $M1$ and $M2$ together are logically equivalent to $M4$, so $\{M1,M2,M3\}$ are also not jointly possible. So the maximal subsets that are jointly possible are $\{M1, M2, M4\},\ \{M1,M3\},\ \{M2,M3\}$. Again, in logic we are free to imagine a world in which people are not mortal or a world in which Socrates is not a person unless we have taken those claims as hypotheses.

  • Thank you. When you say: “ there is wood that is not durable or useful for building things“, is it necessary to use common sense to solve these kind of exercises. I am asking because we did not take any claims as hypothesis. – F. Zer Nov 30 '19 at 01:01
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    @F.Zer Exactly right. Your notion of "common sense" seems like it's right on track, and so you can ignore any of the "possible world" phrasings I put in this answer if they distract you. –  Nov 30 '19 at 01:11
  • But the truth or falsity of these statements is based on common sense only? What makes me doubt is this: if, by common sense, it’s possible to determine something is true, then it is a necessary truth; otherwise, it is not. Is that right? – F. Zer Nov 30 '19 at 01:19
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    @F.Zer Only logic can establish necessary truth. For instance, "The sun rose this morning" is not necessary truth, but "Either the sun rose this morning or it didn't" is necessary truth. –  Nov 30 '19 at 14:29