Wayne Delia
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Disproof of the Compatibility of God’s Omniscience and Man’s Free Will
Definition 1 (D1): Omniscience is the perfectly accurate knowledge of all things knowable, including the outcome of future events.
Definition 2 (D2): Free Will is the unrestricted ability to choose among two or more logically available alternatives in a decision event.
Premise 1 (P1): God exists, and is omniscient.
Premise 2 (P2): Man has free will.
Argument 1 (A1): A logically available choice alternative has a non-zero possibility of being chosen. (From D2)
Argument 2 (A2): An omniscient being has zero possibility of being wrong. (From D1)
Argument 3 (A3): A decision event E has n choice alternatives labeled E1, E2, E3, …, with corresponding possibilities of being chosen labeled p1, p2, p3, … . (Stipulated, from D2 and A1)
Argument 4 (A4): The possibilities pn corresponding to free will choice alternatives associated with decision event E are represented in a vector in which all possibility elements are non-zero. (From D2, P2, and A3)
Argument 5 (A5): God knows the outcome of event E will be choice alternative Ek, which has a corresponding possibility of pk = 100%. The possibilities of all other choice alternatives (pn, where n != k) is 0%. (From D1, P1, A2, and A3)
Argument 6 (A6): The possibility of any choice alternative Ej, other than the alternative God knows will be the outcome (Ek), is both 0% and greater than 0%. (From A4 and A5).
Argument 7 (A7): ~A6 (Law of Non Contradiction; there does not exist a proposition P such that P & ~P is true)
Argument 8 (A8): ~(P1 & P2) (Reductio Ad Absurdem due to contradiction in A6)
Conclusion (C): ~P1 | ~P2 (DeMorgan’s Law)
Conclusion: Either God does not exist or is not omniscient (~P1), or free will is not in effect (~P2).
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Detailed Explanation of Disproof of the Compatibility of Omniscience and Free Will
Definition 1 (D1): Omniscience is the perfectly accurate knowledge of all things knowable, including the outcome of future events.
Omniscient beings possess comprehensive knowledge of every fact. Omniscient beings cannot be wrong in what they know. The theological use of omniscience is a claimed authority based on an omniscient God knowing the outcome of future events with perfect accuracy, called “prophecy”.
Definition 2 (D2): Free Will is the unrestricted ability to choose among two or more logically available alternatives in a decision event.
If a decision event has only one logically available option, it is a coercion, and free will is not in effect. If no options are logically available, there is no decision event. A decision event could have a countably infinite number of alternatives, such as “Pick an integer.” If a choice alternative has no possibility of being selected, it is not considered logically available. An extremely unlikely, yet remotely possible choice alternative is considered logically available.
Premise 1 (P1): God exists, and is omniscient.
The assumption of God’s omniscience is common in Judeo/Christian theology. There are several Old Testament books classified as the Prophets, whose claimed authority is based on the assumption that they were inspired by God’s perfectly accurate foreknowledge. The authority of Jesus as the Messiah is based on claimed fulfillment of God-inspired accurate prophecy of future events.
Premise 2 (P2): Man has free will.
In a theological debate, man’s free will is necessary in order to justify blame assigned to people who fail to “mitigate circumstances” by selecting the least harmful or least sinful option among several supposedly logically available options. Free will is also necessary in Judeo/Christian theology to avoid placing blame on God for anything bad happening.
Argument 1 (A1): A logically available choice alternative has a non-zero possibility of being chosen. (From D2)
If a choice alternative has no possibility of being selected, it is not considered to be a logically available choice. This is because under no circumstances can anyone bring about a situation in which the choice alternative is selected.
Argument 2 (A2): An omniscient being has zero possibility of being wrong. (From D1)
The definition of omniscience requires the perfect accuracy of God’s comprehensive knowledge. God has accurate knowledge of which among the alternatives will not be selected, but He does not possess inaccurate knowledge (propositions with a truth value of “false”).
Argument 3 (A3): A decision event E has n choice alternatives labeled E1, E2, E3, …, with corresponding possibilities of being chosen labeled p1, p2, p3, … . (Stipulated, from D2 and A1)
This stipulation identifies and labels a decision event E as having a vector of possible outcomes
{ E1, E2, E3, …, }
which has a corresponding vector of possibilities of the respective outcomes being selected:
{ p1, p2, p3, …, }
Each of the p-values must represent a selection possibility greater than zero, otherwise the corresponding outcome is not considered to be a logically available choice.
The p-values may represent probabilities of being selected, where a 0% probability implies a logically unavailable choice alternative, and any probability greater than 0% represents a choice option that can possibly be selected. A constraint in this view could be that all probabilities must add up to 100%. The concept of “possibilities” could also be a binary value of 0 or 1 for each p-value, corresponding to whether the choice outcome can possibly be selected (p-value = 1) or cannot possibly be selected (p-value = 0).
Argument 4 (A4): The possibilities pn corresponding to choice alternatives associated with decision event E are represented in a vector in which all possibility elements are non-zero. (From D2, P2, and A3)
The definition of free will in effect over a set of two or more logically available alternatives requires that each of the alterntatives have a possibility of being selected. For each alternative Ek, the corresponding possibility of being selected (pk) must be greater than zero; otherwise, the alternative does not qualify as logically available to be selected.
Argument 5 (A5): God knows the outcome of event E will be choice alternative Ek, which has a corresponding possibility of pk = 100%. The possibilities of all other choice alternatives (pn, where n != k) is 0%. (From [D1, P1, A2, and A3)
The definition of omniscience does not allow for any inaccurate knowledge. A non-zero possibility of selecting an option other than the one which God knows will be selected would imply a non-zero possibility that God is wrong in what He knew would be the outcome of the decision event. No such tolerance of possibly being wrong is permitted for an omniscient entity.
Argument 6 (A6): The possibility of any choice alternative Ej, other than the alternative God knows will be the outcome (Ek), is both 0% and greater than 0%. (From A4 and A5).
When free will is in effect, each of the two or more available options must have a p-value possibility of being selected greater than zero. There is at least one option with a p-value greater than zero which will not be selected.
When omniscience is in effect, each of the presented options other than the one known will be selected has a p-value of zero. This represents zero possibility of selecting an option which God knows will not be selected, because God cannot be wrong.
There exists at least one choice option with greater than zero possibility of being selected (to qualify as logically available for free will), while at the same time having zero possibility of being selected (because God cannot be wrong, and He knows it will not be selected).
Argument 7 (A7): ~A6 (Law of Non Contradiction; there does not exist a proposition P such that P & ~P is true)
A choice option for the outcome of a decision event cannot both be possible (p-value greater than zero) and not possible (p-value equal to zero) at the same time.
Argument 8 (A8): ~(P1 & P2) (Reductio Ad Absurdem due to contradiction in A6)
This results in a contradiction obtained through a valid logical progression starting from a pair of premises assumed to be true. Since there is nothing wrong with any of the subsequent argument steps or the operating definitions, it is necessarily the case that both assumed premises are not true at the same time. The Reductio Ad Absurdem argument technique begins with one or more premises assumed to be true, using valid logical reasoning ultimately resulting in a contradiction, which refutes the one or more premises.
Conclusion (C1): ~P1 | ~P2 (DeMorgan’s Law)
DeMorgan’s Law, ~(A & B) => ~A | ~B, is employed here to establish the conclusion in terms that expresses the conclusion in understandable terms which establishes that omniscience and free will are logically incompatible.
Either God does not exist or is not omniscient (~P1), or else man’s free will is not in effect (~P2).
QED.
Definition 1 (D1): Omniscience is the perfectly accurate knowledge of all things knowable, including the outcome of future events.
Definition 2 (D2): Free Will is the unrestricted ability to choose among two or more logically available alternatives in a decision event.
Premise 1 (P1): God exists, and is omniscient.
Premise 2 (P2): Man has free will.
Argument 1 (A1): A logically available choice alternative has a non-zero possibility of being chosen. (From D2)
Argument 2 (A2): An omniscient being has zero possibility of being wrong. (From D1)
Argument 3 (A3): A decision event E has n choice alternatives labeled E1, E2, E3, …, with corresponding possibilities of being chosen labeled p1, p2, p3, … . (Stipulated, from D2 and A1)
Argument 4 (A4): The possibilities pn corresponding to free will choice alternatives associated with decision event E are represented in a vector in which all possibility elements are non-zero. (From D2, P2, and A3)
Argument 5 (A5): God knows the outcome of event E will be choice alternative Ek, which has a corresponding possibility of pk = 100%. The possibilities of all other choice alternatives (pn, where n != k) is 0%. (From D1, P1, A2, and A3)
Argument 6 (A6): The possibility of any choice alternative Ej, other than the alternative God knows will be the outcome (Ek), is both 0% and greater than 0%. (From A4 and A5).
Argument 7 (A7): ~A6 (Law of Non Contradiction; there does not exist a proposition P such that P & ~P is true)
Argument 8 (A8): ~(P1 & P2) (Reductio Ad Absurdem due to contradiction in A6)
Conclusion (C): ~P1 | ~P2 (DeMorgan’s Law)
Conclusion: Either God does not exist or is not omniscient (~P1), or free will is not in effect (~P2).
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Detailed Explanation of Disproof of the Compatibility of Omniscience and Free Will
Definition 1 (D1): Omniscience is the perfectly accurate knowledge of all things knowable, including the outcome of future events.
Omniscient beings possess comprehensive knowledge of every fact. Omniscient beings cannot be wrong in what they know. The theological use of omniscience is a claimed authority based on an omniscient God knowing the outcome of future events with perfect accuracy, called “prophecy”.
Definition 2 (D2): Free Will is the unrestricted ability to choose among two or more logically available alternatives in a decision event.
If a decision event has only one logically available option, it is a coercion, and free will is not in effect. If no options are logically available, there is no decision event. A decision event could have a countably infinite number of alternatives, such as “Pick an integer.” If a choice alternative has no possibility of being selected, it is not considered logically available. An extremely unlikely, yet remotely possible choice alternative is considered logically available.
Premise 1 (P1): God exists, and is omniscient.
The assumption of God’s omniscience is common in Judeo/Christian theology. There are several Old Testament books classified as the Prophets, whose claimed authority is based on the assumption that they were inspired by God’s perfectly accurate foreknowledge. The authority of Jesus as the Messiah is based on claimed fulfillment of God-inspired accurate prophecy of future events.
Premise 2 (P2): Man has free will.
In a theological debate, man’s free will is necessary in order to justify blame assigned to people who fail to “mitigate circumstances” by selecting the least harmful or least sinful option among several supposedly logically available options. Free will is also necessary in Judeo/Christian theology to avoid placing blame on God for anything bad happening.
Argument 1 (A1): A logically available choice alternative has a non-zero possibility of being chosen. (From D2)
If a choice alternative has no possibility of being selected, it is not considered to be a logically available choice. This is because under no circumstances can anyone bring about a situation in which the choice alternative is selected.
Argument 2 (A2): An omniscient being has zero possibility of being wrong. (From D1)
The definition of omniscience requires the perfect accuracy of God’s comprehensive knowledge. God has accurate knowledge of which among the alternatives will not be selected, but He does not possess inaccurate knowledge (propositions with a truth value of “false”).
Argument 3 (A3): A decision event E has n choice alternatives labeled E1, E2, E3, …, with corresponding possibilities of being chosen labeled p1, p2, p3, … . (Stipulated, from D2 and A1)
This stipulation identifies and labels a decision event E as having a vector of possible outcomes
{ E1, E2, E3, …, }
which has a corresponding vector of possibilities of the respective outcomes being selected:
{ p1, p2, p3, …, }
Each of the p-values must represent a selection possibility greater than zero, otherwise the corresponding outcome is not considered to be a logically available choice.
The p-values may represent probabilities of being selected, where a 0% probability implies a logically unavailable choice alternative, and any probability greater than 0% represents a choice option that can possibly be selected. A constraint in this view could be that all probabilities must add up to 100%. The concept of “possibilities” could also be a binary value of 0 or 1 for each p-value, corresponding to whether the choice outcome can possibly be selected (p-value = 1) or cannot possibly be selected (p-value = 0).
Argument 4 (A4): The possibilities pn corresponding to choice alternatives associated with decision event E are represented in a vector in which all possibility elements are non-zero. (From D2, P2, and A3)
The definition of free will in effect over a set of two or more logically available alternatives requires that each of the alterntatives have a possibility of being selected. For each alternative Ek, the corresponding possibility of being selected (pk) must be greater than zero; otherwise, the alternative does not qualify as logically available to be selected.
Argument 5 (A5): God knows the outcome of event E will be choice alternative Ek, which has a corresponding possibility of pk = 100%. The possibilities of all other choice alternatives (pn, where n != k) is 0%. (From [D1, P1, A2, and A3)
The definition of omniscience does not allow for any inaccurate knowledge. A non-zero possibility of selecting an option other than the one which God knows will be selected would imply a non-zero possibility that God is wrong in what He knew would be the outcome of the decision event. No such tolerance of possibly being wrong is permitted for an omniscient entity.
Argument 6 (A6): The possibility of any choice alternative Ej, other than the alternative God knows will be the outcome (Ek), is both 0% and greater than 0%. (From A4 and A5).
When free will is in effect, each of the two or more available options must have a p-value possibility of being selected greater than zero. There is at least one option with a p-value greater than zero which will not be selected.
When omniscience is in effect, each of the presented options other than the one known will be selected has a p-value of zero. This represents zero possibility of selecting an option which God knows will not be selected, because God cannot be wrong.
There exists at least one choice option with greater than zero possibility of being selected (to qualify as logically available for free will), while at the same time having zero possibility of being selected (because God cannot be wrong, and He knows it will not be selected).
Argument 7 (A7): ~A6 (Law of Non Contradiction; there does not exist a proposition P such that P & ~P is true)
A choice option for the outcome of a decision event cannot both be possible (p-value greater than zero) and not possible (p-value equal to zero) at the same time.
Argument 8 (A8): ~(P1 & P2) (Reductio Ad Absurdem due to contradiction in A6)
This results in a contradiction obtained through a valid logical progression starting from a pair of premises assumed to be true. Since there is nothing wrong with any of the subsequent argument steps or the operating definitions, it is necessarily the case that both assumed premises are not true at the same time. The Reductio Ad Absurdem argument technique begins with one or more premises assumed to be true, using valid logical reasoning ultimately resulting in a contradiction, which refutes the one or more premises.
Conclusion (C1): ~P1 | ~P2 (DeMorgan’s Law)
DeMorgan’s Law, ~(A & B) => ~A | ~B, is employed here to establish the conclusion in terms that expresses the conclusion in understandable terms which establishes that omniscience and free will are logically incompatible.
Either God does not exist or is not omniscient (~P1), or else man’s free will is not in effect (~P2).
QED.