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I was doing practice exercises of chapter-3 of the textbook forall x: Calgary An Introduction to Formal Logic.

There are some questions confusing me (there answers are not given in the solution booklet):

B. For each of the following: Is it a necessary truth, a necessary falsehood, or contingent?

(3) If wood were a good building material, it would be useful for building things.
(5) If gerbils were mammals they would nurse their young.

D. Which of the following pairs of sentences are necessarily equivalent?

(2) Thelonious Monk played gigs with John Coltrane. John Coltrane played gigs with Thelonious Monk.

Now, for the questions, I think (5) is a necessary truth because of the meaning of 'mammals' (but not 100% sure).

But, I am not able to decide for (3), which feels too ambiguous (e.g., good building material for what? Can there be good building material for something which is not counted in our notion of 'things').

Also, the pair in (2) should be equivalent unless there can be a primary and a secondary role related to 'playing gigs', which I am not sure of.

So, what are the right answers to these questions? Please explain.

ryang
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Navneet
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    I agree with the above comment... (2) is not "necessarily true" i.e. it is not provable on the base of logic alone. We have to rely on our empirical knowledge that "playing gigs with" is symmetric. Consider the similar "Thelonious Monk is older than John Coltrane." It is not equiv to "John Coltrane is older than Thelonious Monk. – Mauro ALLEGRANZA May 23 '21 at 14:07

1 Answers1

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pg.18

For the rest of this section, we’ll take cases in the sense of conceivable scenario, i.e., in the sense in which we used them to define conceptual validity.... if we use a different idea of what counts as a “case” we will get different notions. And as logicians we will, eventually, consider a more permissive definition of case than we do here.

pg. 21

  • An argument is valid if there is no case where the premises are all true and the conclusion is not; it is invalid otherwise.
  • A necessary truth is a sentence that is true in every case.
  • A necessary falsehood is a sentence that is false in every case.
  • A contingent sentence is neither a necessary truth nor a necessary falsehood; a sentence that is true in some case and false in some other case.
  • Two sentences are necessarily equivalent if, in every case, they are both true or both false.

From the above excerpts, it appears that the author means this: “necessary truth” means logical truth, “necessary falsehood” means logical falsehood, “necessary equivalence” means logical equivalence, and “valid argument” means an argument whose conclusion is a logical consequence of its premises.

As such, I'd say that

  • (3) $\;[P\rightarrow P]\;$ is a tautological, therefore necessary, truth;
  • (5) $\;[P\rightarrow Q]\;$ is a (probably true) contingent sentence;
  • (2) $\;[G(x,y)\leftrightarrow G(y,x)]\;$ is a true contingent sentence. (Whether the predicate/relation $G$ is symmetric depends on interpretation.) So, the pair of sentences is not “necessarily equivalent”.
ryang
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