1. Let P be the proposition "God can create a stone so heavy that he cannot lift it."
2. Let Q be the proposition "God is omnipotent."
P: God can create a stone so heavy that he cannot lift it.
Q: God is omnipotent.
The paradox can be expressed using formal logic as follows:
1. ∃x [(Cx ∧ ~Lx) ∧ ∀y (Cy → Ly)] (There exists an x such that x is a stone that God can create and God cannot lift it, and for all y, if y is a stone, then God can lift it.)
2. ∀y (Cy → Ly) (For all y, if y is a stone, then God can lift it.)
3. Ca ∧ ~La (Assuming a particular stone, a, that God can create and cannot lift)
4. Ca (From line 3, conjunction elimination)
5. ~La (From line 3, conjunction elimination)
6. Ca → La (From line 2, universal instantiation)
7. La (From lines 4 and 6, modus ponens)
8. ~La ∧ La (From lines 5 and 7)
9. ~∃x [(Cx ∧ ~Lx) ∧ ∀y (Cy → Ly)] (From lines 1 and 8, contradiction)
10. ~P (From line 9, existential negation)
11. Q (Assuming God is omnipotent)
12. P → Q (From line 11, conditional introduction)
Now we have arrived at a contradiction, where both P and ~P are derived within the system. This paradox challenges the coherence of an omnipotent being.
2. Let Q be the proposition "God is omnipotent."
P: God can create a stone so heavy that he cannot lift it.
Q: God is omnipotent.
The paradox can be expressed using formal logic as follows:
1. ∃x [(Cx ∧ ~Lx) ∧ ∀y (Cy → Ly)] (There exists an x such that x is a stone that God can create and God cannot lift it, and for all y, if y is a stone, then God can lift it.)
2. ∀y (Cy → Ly) (For all y, if y is a stone, then God can lift it.)
3. Ca ∧ ~La (Assuming a particular stone, a, that God can create and cannot lift)
4. Ca (From line 3, conjunction elimination)
5. ~La (From line 3, conjunction elimination)
6. Ca → La (From line 2, universal instantiation)
7. La (From lines 4 and 6, modus ponens)
8. ~La ∧ La (From lines 5 and 7)
9. ~∃x [(Cx ∧ ~Lx) ∧ ∀y (Cy → Ly)] (From lines 1 and 8, contradiction)
10. ~P (From line 9, existential negation)
11. Q (Assuming God is omnipotent)
12. P → Q (From line 11, conditional introduction)
Now we have arrived at a contradiction, where both P and ~P are derived within the system. This paradox challenges the coherence of an omnipotent being.
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