Truth in FOL

Consider the following interpretation:

• The domain comprises only Corwin and Benedict

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

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

• ‘$N(x)$’ is to be true of no one

• ‘$c$’ is to refer to Corwin

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

1. $B(c)$

2. $A(c)\leftrightarrow\neg N(c)$

3. $N(c)\rightarrow(A(c)\vee B(c))$

4. $\forall x\,A(x)$

5. $\forall x\neg B(x)$

6. $\exists x(A(x)\wedge B(x))$

7. $\exists x(A(x)\rightarrow N(x))$

8. $\forall x(N(x)\vee\neg N(x))$

9. $\exists x\,B(x)\rightarrow\forall x\,A(x)$

Consider the following interpretation:

• The domain comprises only Lemmy, Courtney and Eddy

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

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

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

• ‘$c$’ 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. $H(c)$

2. $H(e)$

3. $M(c)\vee M(e)$

4. $G(c)\vee\neg G(c)$

…

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

(atom1) at (0,2) 1; (atom2) at (2,2) 2; (atom4) at (0,0) 3; (atom5) at (2,0) 4; (atom6) at (4,0) 5; [-¿, thick] (atom1)+(-0.15,0.15) arc (-330:-30:.3); [-¿, thick] (atom6)+(0.15,-0.15) arc (-150:150:.3); [-¿, thick] (atom1) – (atom2); [-¿, thick] (atom1) – (atom4); [¡-¿, thick] (atom1) – (atom5);…