As will be demonstrated - in some detail, it's relatively trivial to yank-out a flaw from the Logical Problem of Evil - in the form of a knowledge paradox. (N.B. The following essay presupposes a familiarity on the part of the reader with this ancient, theological "puzzle".)
With the LPoE, its exponents (dogmatic atheists) haul it out it - as proof - that God and evil cannot coexist. 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 coexist." inferentially contradicts one of its premises, namely that of God's all-knowingness, then the LPoE cannot be used to rule-out the coexistence of God and evil - as a matter of logic.
The LPoE contains a series of 4 propositions, which - when combined - includes truth-possibility; that is, they are not of the logical form (A & -A), but rather of (A & B & C & D). Since the LPoE does not posses the logical structure of truth-impossibility, any attempt to palm-off the conclusion as invariably 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 deprive this mind of omniscience, i.e. that this all-knowing mind could not isolate a logically possible counter-example to the LPoE - which blocks its conclusion. When questions of logic and knowledge are involved, omniscience is a strength - never 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).
Bear this in mind: Only one logically possible counter-example to the conclusion of the LPoE reconciles God's existence with that of the existence of evil in the world, 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 coexist." makes it logically impossible for the God who knows everything to know something, i.e. a logically possible counter-example to the LPoE - which permits the co-possibility of himself 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 coexist.
17. The conjunction of propositions 1, 2, 3, and 4 is not of the logical form "A & -A". 1, 2, 3 and 4 contain 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 coexist." is true, then God is not omniscient. God does not know something: To wit how the propositions of the LPoE can evade inconsistency - by God locating one counter-example. 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 coexist." from "19" by the "proof by contradiction method".)
20. If God is omniscient, then it is logically possible for God to know a logically possible 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 logically possible 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 logically possible counter-example to the conclusion of the LPoE preventing it from reaching inconsistency, then it is logically possible for God and evil to coexist. (from 2, 17, 18, 19, 20 and 21)
23. It is logically possible for God and evil to coexist. (From 21 and 22)
In the LPoE - once declared omniscient by proposition "2", the God described therein was conferred the final-say over all matters regarding knowledge. But - in "16", God's final-say is wrenched-away by the conclusion. 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. Logically, there is nothing which this God cannot know, and the conjunction of the propositions, comprising the LPoE, is not a formal contradiction. Only the all-knowing God of the LPoE can objectively, decisively and non-paradoxically decide whether his existence and evil are co-possible. Once a super-human being is declared to possess infinite knowledge, finite human knowledge (about the world, logic and the LPoE) becomes - at best - second-rate.
Proposition "2" gives the game, set and match to theists. Since - by definition, humans are vastly less intelligent - than an omniscient mind, then this mind (assumed omniscient for the sake of argument) could hypothetically figure-out how the coexistence of himself and evil are logically possible, blocking these propositions from veering-into inconsistency. Given the paradox - logically - the LPoE cannot be employed to rule-out God's existence and evil. Attributing omniscience to a mind concedes the point that this mind knows everything, including a possible way - within which himself and evil are logically possible. 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 the (self-referential) inconsistency (2 & 16). (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, granted to God, i.e. if it is not logically possible for God to know a logically possible 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. To be redundant: The conclusion of the LPoE makes it logically impossible for a being who knows all logical possibilities to know one 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]
(It is not the case that God and evil can coexist.)
(God knows everything.)
3. E(w) E(x) E(z) -[Gw ^ Ex] ^ Czl
(It is not the case that God and evil can coexist, but there is at least one logically possible counter-example (due to the logical structure of the LPoE) - which blocks the conclusion of the LPoE, i.e. E(w) E(x) -[Gw ^ Ex] from being true.)
4. E(w) E(x) E(z) -[Gw ^ Ex] ^ Czl -> -Kga ]
(If it is not the case that God and evil can coexist but there is a logically possible counter-example to the LPoE, then there is something which an all-knowing God does not know, i.e. the God, as defined in the LPoE as omniscient, is not omniscient.)
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)