1. Taking the universe of discourse to be the set {1,2,3}, specify extensions for "F", "G", and "H" that make the following schemata true: (a) (x)(FxGx).-(x)(FxGx).-(x)(GxFx)

(b) (x)(Fx.Gx Hx).-(x)(Fx Gx.Hx).-(x)(Gx Fx.Hx)

2. For each of the following pairs of schemata, show that the first does not imply the second by providing an interpretation with the universe of discourse {1,2,3} that makes the first schema true and the second schema false:(a)(x)Fx.(x)Gx (x)Hx(x)(Fx.Gx Hx)

(b) (x)Fx.(x)Gx(x)(Fx.Gx)(c)(x)Fx (x)-Gx(x)(Fx - Gx)

3. Show that "(x)-Fx(x)Gx" implies "(x)Fx (x)Gx". Do this by arguing about the interpretations that make the relevant schema true or false.

4. In each of the arguments below, the premises monadically imply the conclusion. Show this by monadically schematizing the arguments (assuming a U.D. of persons), and then arguing that any interpretation that makes all of the premise schemata true must also make the conclusion schema true (or by arguing that any interpretation that makes the conclusion schema false must make at least one of the premise schemata false).

(a) PREM1: No faculty member who either teaches logic or plays the harmonica has a bodyguard.

PREM2: Nobody who does not have a bodyguard will survive. PREM3: Every philosopher who does not teach logic plays the harmonica.CONCLUSION: No faculty member who is a philosopher will survive.

(b) PREM1: Everyone involved in the recent crime spree is a professor.

PREM2: If no professors are criminally insane, then none of the inmates are professors. CONCLUSION: If any the inmates are involved in the recent crime spree, then some professors are criminally insane.