As will be shown in some detail below, it's relatively trivial to yank-out a flaw from the Logical Problem of Evil - in the form of a knowledge paradox. (N.B. this brief essay presupposes a familiarity on the part of the reader with this ancient, theological "puzzle".)
With the LPOE, its exponents (dogmatic atheists) whip it out it as proof that God and evil cannot co-exist. The attributes assigned to God of all-powerfulness, all-knowingness and all-goodness are asserted to conflict with the existence of evil in the world. However - as will be explained, if the conclusion of the LPOE: "It is not logically possible for God and evil to co-exist." inferentially contradicts with one of its premises, namely that of God's all-knowingness, then the LPOE cannot be used to rule-out the existence of God and evil - as a matter of logic.
The LPOE is a series of four propositions - which when combined incorporates truth-possibility; that is, they're not of the logical form (A & -A), but rather (A & B & C & D). Since the LPOE does not contain the logical structure of truth-impossibility, any attempt to palm-off the conclusion as inexorably yielding: "Not both God exists and evil exists" is a fundamental, low-level error.
When fallible (limitedly knowing) humans ascribe omniscience to a mind in the LPOE, they may NOT go on to whine that this mind could not generate a counter-example to its conclusion. When questions of logic and knowledge are involved, omniscience is a strength - not a weakness. By pronouncing God omnisicient, this God (if only stipulated for the purpose of an argument) has been granted the final say over all matters of knowledge - within the context of this argument.
Keep this in mind: only one logically possible counter-example to the conclusion of the LPOE reconciles God's existence with that of evil, and if God could not know (at least) one logically possible counter-example to the LPOE's conclusion, then God is not omniscient.
The knowledge paradox arising from the LPOE is one - in which a consequence of the conclusion: "It is not logically possible for God and evil to co-exist." makes it logically impossible for the God who knows everything to know something, e.g. a logically possible counter-example to the LPOE's conclusion which permits the co-possibility of God and evil. Given the logical form of the LPOE and the omniscience imputed to God within it, a self-contradiction can be easily teased-out.
Take the following series of propositions:
1. God is omnipotent.
2. God is omniscient.
3. God is wholly good.
4. Evil exists.
(Propositions 5 - 15 omitted for the sake of brevity. See Mackie.)
16. It is not logically possible for God and evil to co-exist.
17. The conjunction of propositions 1, 2, 3, and 4 are not of the logical form "A & -A". It has truth-possibility. It is logically possible that there is at least one case in which the propositions are true together. (From 1-4 and the law of non-contradiction.)
18. If it is not logically possible for God to know a logically possible counter-example to "16" (which keeps the proof consistent), then God is not omniscient. (From 17 and 2)
19. If the conclusion "It is not logically possible for God and evil to co-exist." is true, then God is not omniscient. God does not know something, to wit how the propositions of the LPOE evade inconsistency. The conclusion contradicts "2". (2 & -2) (A & -A) Self-contradiction. (From 1-4, 16, 2, 17 and 18) (The disproof of the LPOE can be halted here. However, one could easily derive the negation of "16", i.e. "It is logically possible for God and evil to co-exist." from "19" by the "proof by contradiction method".)
20. If God is omniscient, then it is logically possible for God to know a counter-example to the LPOE preventing it from reaching, inconsistency. (from 2, 17, 18 and 19)
21. It is logically possible for God to know a counter-example to the conclusion of the LPOE preventing it from reaching inconsistency. (From 2 and 20)
22. If it is logically possible for God to know a counter-example to the conclusion of the LPOE preventing it from reaching inconsistency, then it is logically possible for God and evil to co-exist. (from 2, 17, 18, 19, 20 and 21)
23. It is logically possible for God and evil to co-exist. (From 21 and 22)
In the LPOE - once declared omniscient by proposition "2"- the God described therein was granted the final say over all matters regarding knowledge. But in "16", God's final say is snatched away by a conclusion, assumed to be binding on an omniscient being. The proponents of the LPOE do not take omniscience seriously nor do they recognize that the conjunction in the series of propositions contains truth-possibility. The omniscience ceded to God is a universal generalization. There's nothing which this God cannot know, and the conjunction of the propositions comprising the LPOE is not a contradiction. Only the all-knowing God of the LPOE can objectively, decisively and non-paradoxically assert whether he and evil are co-possible. Once a being is declared omniscient, human knowledge (about the world, logic and the LPOE) becomes - at best - second-rate.
Proposition "2" gives the game, set and match to theists - forever. Since by definition, humans are - being charitable - infinitesimally less intelligent than an omniscient mind, then this mind (assumed omniscient for the sake of argument) could hypothetically figure out how the co-existence of itself and evil are logically possible, preventing these propositions from sliding into inconsistency. Given the paradox - logically - the LPOE cannot be used to rule out God's existence with that of evil. Attributing omniscience to a mind concedes the point that this mind knows everything, including a possible way which itself and evil are logically possible in this "argument". There's a pane in a window through which a ray of logical possibility, built on a "knowledge gap" between an all-knowing God and limitedly-knowing humans - conjoined with the truth-possible, logical structure inherent in propositions "1-4" - forever shines.
A contradiction between the conclusion and a premise of the LPOE is what generates a (self-referential) inconsistency (2 & 16) - not the attributes assigned to God and the evil prevading the world. (Intuitively, the truth-conditions of the conclusion bend back to one of the LPOE's premises, colliding with it.) A consequence of the conclusion contradicts one of the traits assigned to God: if it is not logically possible for God to know a counter-example to the conclusion of the LPOE, then God is not omniscient. Omniscience means - among other things - knowing all logical possibilities; hence, since God knows all logical possibilities, the conclusion: "It is not logically possible for God and evil to coexist." implicitly conflicts with a premise. The conclusion of the LPOE makes it logically impossible for a being who knows all logical possibilities to know a single logical possibility.
In first-order predicate calculus, the knowledge paradox of the LPOE is more glaringly obvious than the above:
1. E(w) E(x) -[Gw ^ Ex]
(God and evil cannot coexist.)
(God knows everything.)
3. E(w) E(x) E(z) [ -[Gw ^ Ex] ^ [ Czl ^ Bzi ]
(God and evil cannot coexist but there is at least one counter-example (due to the logical form of the LPOE) - which blocks the conclusion of the LPOE, i.e. E(w) E(x) -[Gw ^ Ex] from hitting inconsistency.)
4. E(w) E(x) E(z) [ -[Gw ^ Ex] ^ [ Czl ^ Bzi ]] -> -Kga ]
(If God and evil cannot coexist but there's a counter-example to the LPOE, then there is something which an all-knowing God does not know.)
5. -Kga MP 3, 4
6. E(y)-Kgy EG 5
7. -(y)Kgy CQ 6
8. (y)Kgy ^ -(y)Kgy ADD 2, 7 (self-contradiction)