Solutions to Chapter 31 Truth in FOL

A. Consider the following interpretation:

• ‣
  ​

  The domain comprises only Corwin and Benedict

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  ​

  ‘$A(x)$’ is to be true of both Corwin and Benedict

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  ​

  ‘$B(x)$’ is to be true of Benedict only

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  ​

  ‘$N(x)$’ is to be true of no one

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  ‘$c$’ is to refer to Corwin

Determine whether each of the following sentences is true or false in that interpretation:

1. 1.
   ​

   $B(c)$ False

2. 2.
   ​

   $A(c)\leftrightarrow \neg N(c)$ True

3. 3.
   ​

   $N(c)\to (A(c)\vee B(c))$ True

4. 4.
   ​

   $\forall xA(x)$ True

5. 5.
   ​

   $\forall x\neg B(x)$ False

6. 6.
   ​

   $\exists x(A(x)\wedge B(x))$ True

7. 7.
   ​

   $\exists x(A(x)\to N(x))$ False

8. 8.
   ​

   $\forall x(N(x)\vee \neg N(x))$ True

9. 9.
   ​

   $\exists xB(x)\to \forall xA(x)$ True

B. Consider the following interpretation:

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  ​

  The domain comprises only Lemmy, Courtney and Eddy

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  ​

  ‘$G(x)$’ is to be true of Lemmy, Courtney and Eddy.

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  ‘$H(x)$’ is to be true of and only of Courtney

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  ​

  ‘$M(x)$’ is to be true of and only of Lemmy and Eddy

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  ​

  ‘$c$’ is to refer to Courtney

• ‣
  ​

  ‘$e$’ is to refer to Eddy

Determine whether each of the following sentences is true or false in that interpretation:

1. 1.
   ​

   $H(c)$ True

2. 2.
   ​

   $H(e)$ False

3. 3.
   ​

   $M(c)\vee M(e)$ True

4. 4.
   ​

   $G(c)\vee \neg G(c)$ True

5. 5.
   ​

   $M(c)\to G(c)$ True

6. 6.
   ​

   $\exists xH(x)$ True

7. 7.
   ​

   $\forall xH(x)$ False

8. 8.
   ​

   $\exists x\neg M(x)$ True

9. 9.
   ​

   $\exists x(H(x)\wedge G(x))$ True

10. 10.
    ​

    $\exists x(M(x)\wedge G(x))$ True

11. 11.
    ​

    $\forall x(H(x)\vee M(x))$ True

12. 12.
    ​

    $\exists xH(x)\wedge \exists xM(x)$ True

13. 13.
    ​

    $\forall x(H(x)\leftrightarrow \neg M(x))$ True

14. 14.
    ​

    $\exists xG(x)\wedge \exists x\neg G(x)$ False

15. 15.
    ​

    $\forall x\exists y(G(x)\wedge H(y))$ True

C. Following the diagram conventions introduced at the end of §23, consider the following interpretation:

Determine whether each of the following sentences is true or false in that interpretation:

1. 1.
   ​

   $\exists xR(x,x)$ True

2. 2.
   ​

   $\forall xR(x,x)$ False

3. 3.
   ​

   $\exists x\forall yR(x,y)$ True

4. 4.
   ​

   $\exists x\forall yR(y,x)$ False

5. 5.
   ​

   $\forall x\forall y\forall z((R(x,y)\wedge R(y,z))\to R(x,z))$ False

6. 6.
   ​

   $\forall x\forall y\forall z((R(x,y)\wedge R(x,z))\to R(y,z))$ False

7. 7.
   ​

   $\exists x\forall y\neg R(x,y)$ True

8. 8.
   ​

   $\forall x(\exists yR(x,y)\to \exists yR(y,x))$ True

9. 9.
   ​

   $\exists x\exists y(\neg x=y\wedge R(x,y)\wedge R(y,x))$ True

10. 10.
    ​

    $\exists x\forall y(R(x,y)\leftrightarrow x=y)$ True

11. 11.
    ​

    $\exists x\forall y(R(y,x)\leftrightarrow x=y)$ False

12. 12.
    ​

    $\exists x\exists y(\neg x=y\wedge R(x,y)\wedge \forall z(R(z,x)\leftrightarrow y=z))$ True