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Parts (i) and (iv) make sense as they are obvious.

Part (ii) says "for every C, if that is a country, then, if it borders Ecuador then it is in SouthAmerica." I want to know whether there is a method to convert "and ($\land$)" to implication; this way, parts (i) and (ii) are interchangable.

I know that part (iii) kinda says that if C is a country then it borders Ecuador (which is not necessarily true), that is, "for every C, if that is a country, then it borders Ecuador, then it is in SouthAmerica." I want to know whether this translation is exactly correct or whether it can be made clearer.

ryang
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dikshank
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2 Answers2

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I know that part (iii) kinda says that if C is a country then it borders Ecuador

No, (iii) doesn't assert that ∀c (Country(c) ⟹ Border(c,Ecuador)), since (iii) is (vacuously) true if this assertion is false!

that is, "for every C, if that is a country, then it borders Ecuador, then it is in SouthAmerica."

Your translation is ungrammatical, and ambiguous, since between (A ⟹ B) ⟹ C and A ⟹ (B ⟹ C) and (A ⟹ B) and (B ⟹ C), there is no equivalent pair. A literal translation of (iii) is

  • If a thing borders Ecuador if it's a country, then it's in South America. ✅

This translation is hard to accurately understand, and most people will just misread it as (i)/(ii) , that is, as

  • if a thing is a country that borders Ecuador, then it's in South America ❌
  • if a country borders Ecuador, then it's in South America. ❌

Fortunately, its (logically equivalent) contrapositive is much friendlier to parse:

  • If a thing isn't in South America, then it's a country that doesn't border Ecuador. ✅

Incidentally, do observe that (iii) implies, ludicrously, that every balloon is in South America.

I want to know whether there is a method to convert "and ($\land$)" to implication; this way, parts (i) and (ii) are interchangeable.

A useful equivalence to remember:$$A ⟹ (B ⟹ C) \quad\equiv\quad (A ∧ B) ⟹ C.$$ Convince yourself of it by first considering the left-to-right entailment, then considering the right-to-left entailment.

On a tangential note: your textbook's omission of the outermost parentheses in its four sample logic statements isn't optimal practice, because for example $$\forall x\, Px\Longrightarrow Q\quad\equiv\quad (\forall x\, Px)\Longrightarrow Q\quad\equiv\quad \exists x\,( Px\Longrightarrow Q) \quad\not\equiv\quad \forall x\, \color\red(Px\Longrightarrow Q\color\red).$$

ryang
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  • Just to be clear, your statement where you say that every balloon is in South America and A country that doesn't border Ecuador can also be in SouthAmerica is because False -> True is valid? My question is let's say hypothetically there was a FOL statement that was valid for all countries (meaning if a country didn't border Ecuador then it would return that it is NOT IN SA, and if it did border then it would return that it is IN SA) but IF it didn't care about objects as the balloon example, why do we care about that since it is out of the domain, I mean why non countries matter. – dikshank Oct 05 '24 at 05:42
  • Hmm, where did I say A country that doesn't border Ecuador can also be in South America? $\quad$ 2. If Every balloon is in South America is false, then (iii) must also be false; this means that the latter implies the former, without invoking any vacuous truth. $\quad$ 3. Every true sentence is valid, but not vice versa; in your above comment, "valid" needs to be replaced with "true". $\quad$ 4. The domain isn't countries, otherwise there'd be no need for the predicate Country(c). We don't have to reference balloons; (iii) implies that Edinburgh is in Ecuador.
    • – ryang Oct 05 '24 at 05:58
  • Oh right, you didn't say it, I mixed two things, I wanted to say a "NON country". But why do we care about a NON country because that is out of the scope, As long as the countries are satisfying our statement "ALL COUNTRIES ....". The question never asked anything about non countries. – dikshank Oct 05 '24 at 06:09
  • Typo above! My last sentence should've read, "(iii) implies that Edinburgh is in South America." $\quad$ 2. You mean my talking about balloons and Edinburgh? Mentioning those consequences are just to help you see that (iii) isn't a faithful translation of "All countries that border Ecuador are in South America", since only the former claims that ballons and Edinburgh are in South America.
    • – ryang Oct 05 '24 at 06:30
  • I mean to say that, The statement is True for all countries but it may make assumptions about Non countries, as you said it can mean (iii) implies that Edinburgh is in Ecuador So we do care about the objects that are Not countries? – dikshank Oct 05 '24 at 06:47
  • I've already corrected my typo yet you are duplicating that typo. $\quad$ 2. A translation exercise is about being faithful (truly equivalent), not about being factually correct (so, we would translate a lie without correcting it). So, yes, we do care that a purported translation claims that Jakarta (a member of the discourse domain) is in South America when the original statement makes no such claim.
    • – ryang Oct 05 '24 at 06:53