3. There seems to be a lack of symmetry in the Theorem on page 9 of...

 

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3. There seems to be a lack of symmetry in the Theorem on page 9 of the Week 6 Slides. Could part (1) of the theorem be improved to say: "If TFy, then IU{v} is consistent."? Justify your answer. Consistent Sets of Formulas Theorem Let I be a set of formulas, and let p be a formula. (1) If T is consistent and I Ey, then IU{4} is consistent. (2) If FKy, then PU{¬9} is consistent. Proof of (1): Suppose that I is consistent and PHy. Assume for a contradiction that PU{4} is inconsistent. Let be arbitrary. Since IU{y} is inconsistent, PU{4} Fp. So by the Deduction Theorem, IF (y → ). But since IF p, TF by MP. As y was arbitrary, this shows that I is inconsistent, a contradiction. Proof of (2): Suppose IF p. Assume for a contradiction that PU{¬e} is inconsistent. In particular, IU{¬y} Fy. By the Deduction Theorem, ITE(¬9→4). We've also seen that ((-y →4) → 4). So by MP, I Ey, a contradiction. 3. There seems to be a lack of symmetry in the Theorem on page 9 of the Week 6 Slides. Could part (1) of the theorem be improved to say: "If TFy, then IU{v} is consistent."? Justify your answer. Consistent Sets of Formulas Theorem Let I be a set of formulas, and let p be a formula. (1) If T is consistent and I Ey, then IU{4} is consistent. (2) If FKy, then PU{¬9} is consistent. Proof of (1): Suppose that I is consistent and PHy. Assume for a contradiction that PU{4} is inconsistent. Let be arbitrary. Since IU{y} is inconsistent, PU{4} Fp. So by the Deduction Theorem, IF (y → ). But since IF p, TF by MP. As y was arbitrary, this shows that I is inconsistent, a contradiction. Proof of (2): Suppose IF p. Assume for a contradiction that PU{¬e} is inconsistent. In particular, IU{¬y} Fy. By the Deduction Theorem, ITE(¬9→4). We've also seen that ((-y →4) → 4). So by MP, I Ey, a contradiction.