Predicate Truth Tree Analysis

Perform a predicate truth tree analysis on the following propositions, pairs of propositions, and sequents. Show the sufficiently decompose truth tree and your analysis of the logical property in question.

1: Test for Contingency, Inconsistency, Tautology

(Ex)(Ay)(Dyx -> ~Dxy)
(SOME x)(ANY y)(IF y D x THEN x NOT D y)


Contingency Inconsistency Tautology

2: Test for Logical Consistency

(Ex)(Sx & Wx) (Ay)~(Wy v Sy)
(SOME x)(BOTH x S AND x W) (ANY y) NOT(EITHER y W OR y S)


Logically Consistent Logically Inconsistent

3: Test for Deductive Validity

(Ax)(Ay)(Mxy -> Nxy) ├ (Ax)(Ay)(Mxy -> (Nxy & Nyx))
(ANY x)(ANY y)(IF x M y THEN x N Y) THUS (ANY x)(ANY y)(IF x M y THEN (BOTH x N Y AND y N x))


Deductively Valid Deductively Invalid

4: Test for Deductive Validity

(Ax)(Sx -> (Fx v Gx)), (Ex)(Sx & ~Gx)├ (Ew)Fw
(ANY x)(IF x S THEN (EITHER x F Or x G)), (SOME y)(BOTH y S AND y NOT G) THUS (SOME x)(w F)


Deductively Valid Deductively Invalid