3

My question asks me to specify a true and a false interpretation over the domain {1,2,3} of

(∃x¬Fx → ¬∃xFx)

So I have said that the extensions of a,b,c are 1,2,3 Do I just need to find extensions of F that make the statement true or false?

Like for the true statement if I say the extension of F is {1,2,3} then does that mean ∃x¬Fx is False and therefore the conditional is true?

  • 1
    Correct: you have to specify (i) an interpretation of F in the said domain that satisfies (make it true) the formula and (ii) an interpretation of F in the said domain that falsifies (make it false) the formula. And your choice for (i) is good. – Mauro ALLEGRANZA Mar 6 '18 at 7:08
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Since one knows there is a true and false interpretation for ∃x¬Fx → ¬∃xFx one could use a tree proof generator to find an interpretation that makes the statement false. Here is such a result:

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For this countermodel x=b makes the antecedent, ∃x¬Fx, true, but there also exists an element of the domain, a, that makes F true and so the consequent is false. Hence the conditional is false.

The problem has another constraint: Use the domain {1, 2, 3}.

What we need to do is provide two structures using this domain, one leading to a true result and the other to a false result. The Open Logic Project defines a "structure M, for a language L of first-order logic consists of the following elements": (Definition 12.26)

  1. A non-empty domain.
  2. An interpretation of constant symbols.
  3. An interpretation of predicate symbols.
  4. An interpretation of function symbols.

Using the result from the tree proof generator as a guide and following this definition of structure, the following is an interpretation leading to a false result:

  • Domain: {1,2,3}
  • Constant symbols:
    • a: 1
    • b: 2
    • c: 3
  • Predicate symbols:
    • F: {a}
  • Function symbols: (There are none in this exercise.)

For an interpretation leading to a true result, use the above but change the interpretation of the predicate symbol, as the OP suggests, to F:{a,b,c}. Then the antecedent is false since there does not exist an element of the domain that makes F false and so the conditional is true.


Open Logic Project https://openlogicproject.org/

Tree Proof Generator. Retrieved on June 9, 2019 from https://www.umsu.de/logik/trees/

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