STUDIES IN LOGIC, GRAMMAR AND RHETORIC 1 (14)

Kazimierz Trzȩsicki

OMNISCIENCE, OMNIPOTENCE AND RELATED NOTIONS

The theological notions of omniscience and omnipotence play an important role in philosophy. They are connected with the notion of freewill. The main question concerns the compatibility of freewill with existence of omniscient and omnipotent being. Here the notion of time has a crucial function – the answer to this question depends on the structure of time. The main aim of this discussion is to show possible logical interrelations between notions of omniscience, omnipotence and time using the language of formal logic, in particular, tense logic. The subject has a long history and is a matter of interest of contemporary philosophical logic.

1. Omniscience

The notions of omniscience and omnipotence could be defined as attributes of an atemporal being^{1} or – as it will be done here – something that is in time, in other words, as attributes of a temporal being.

A Cantorian argument against a set of all truths is raised to show that there is not possible an omniscient being, as a being that knows all and

1 Divine atemporality was offered by Boethius in the sixth century as solution to the problem of theological fatalism. There is no question of God foreknowing human actions because God’s knowing cannot be located at any point in time. God is ‘outside’ time, nowhere on the line of time, but with exactly the same epistemic access to each moment of time. Unchanging ‘presence’ which on this view all things have to God, is in some way less like our own present than our past. The view is held by many very reputable philosophers, e.g. St. Thomas Aquinas.

God knows those truths, if any, which are themselves timeless. God’s knowledge is in some way right outside of time, in which case presumably the verb ‘knows’ in translation would have to be thought of as tenseless. The idea of omniscient atemporal being is questioned by Nicholas Wolterstorff (God Everlasting [in:] Contemporary Philosophy of Religion, edited by David Shatz and Steven M. Cahn. New York: Oxford University Press, 1982). Eternal being could know only tensed statement. Tenseless statements are not translatable into tensed ones. If God knows every true statement then he cannot be timeless.

Kazimierz Trzȩsicki

only true propositions.^{2} This and other problems depend on the notion of omniscient being. Following A. N. Prior an omniscient being G (for “God”) could be defined as follows:^{3} (Os) G is omniscient iff it is, always has been, and always will be the case

that for all α (∈ L): if it is true that α, then G knows that α is true, where L is a language (a set of sentences).

Let us add that: (I) G is infallible iff it is, always has been, and always will be the case

that for all α (∈ L): if G knows that α is true, then it is true that α.

If we suppose that:

1. If it is true that α, then it has always been true that it would be the case that α,^{4} then by the definition of omniscience (Os) we obtain:

2. If it is true that α, then G has always known (as true) that it would be the case that α. G possesses an infallible knowledge of man’s future actions. How is this

provision possible, if man’s future acts are not necessary?

According to the Arystotelian principle that what is true is necessary. In consequence of that (2) expresses the theological fatalism. The rejection of (1) is a necessary condition avoiding of (2). This is possible if there are some future contingencies, i.e.: (FC) for some α (∈ L) and t (∈ T in the future, after the present moment)

neither at present it is true that α at the moment t nor at present it is true that not-α at the moment t.

Only in such a world in which FC holds, there are alternative futures between which choice is possible. This is a necessary ontological condition of possibility of free deeds. The contingent, considered as future (ut futurum), cannot be the object of any sort of knowledge of a temporal being – with the exception of almighty and omniscient G – which cannot fall into falsehood. G is omniscient, thus G knows all the contingent deeds. Because G is omnipotent, G is able to decide about any of such deeds. The infallible

2 See Grim Patrick, Logic and Limits of Knowledge and Truth, “Philosophical Studies”, 22 (1988), 341–367; Plantinga A. and Grim P., Truth, Omniscience, and Cantorian Arguments: An Exchange, “Philosophical Studies”, 71 (1993), 267–306; Mar G., Why “Cantorian” Arguments Against the Existence of God Don’t Work, http:////w.w.w.sunysb.edu.//philosophy//faculty//gmar//cantor.txt

3 Prior A. N., The Formalities of Omniscience [in:] Papers on Time and Tense Oxford 1968.

4 Id quod est verum in praesenti, semper fuit verum esse futurum.

Omniscience, omnipotence and related notions

knowledge of G has the source in its taken in advance irrevocable decisions concerning all the contingent deeds.

The idea of ‘closed’ past: quod fuit, non potest non fuisse – what has been, cannot now not have been – has the main stock source in the “Nicomachean Ethics” (vi, 1139b).^{5} Agaton says that even God is not able to change what has been done (1139b5-10; 2, T.4). For C. S. Peirce the past is the region of ‘brute fact’. Some writers to support the idea of ‘open’ future maintain that also the past is a subject of some kind of change, namely some past facts are falling into abyss. For Karneades even Apollo is not able to know past facts if there is no trace of they, thus, moreover, future facts that are not decided yet could not be known by him. In antiquity, the same view was maintained by Cicero. /Lukasiewicz to avoid fatalistic consequences of the principle of bivalence – already discussed by Arystoteles in the famous see-fight passage of “De Interpretatione” – invented many- -valued logics which principles would have to govern our thinking not only about the future but also about the past. He writes:

We should not treat the past differently from the future. If the only part of the future that is now real is causally determined by the present instant, and if causal chains commencing in the future belong to the realm of possibility, then only those parts of the past are at present real which still continue to act by their effect today. Facts whose effects have disappeared altogether, and which even an omniscient mind could not infer from those now occurring, belong to the realm of possibility. One cannot say about them that they took place, but only that they were possible. It is well that it should be so. There are hard moments of suffering and still harder ones of guilt in everyone’s life. We should be glad to be able to erase them not only from our memory but also from existence. We may believe that when all effects of those fateful moments are exhausted, even should that happen after our death, then their causes too will be effaced from the world of actuality and pass into the realm of possibility. Time calms our cares and brings us forgiveness (/Lukasiewicz “On Determinism”, pp. 127-128).

This does not mean that for Karneades, Cicero or /Lukasiewicz the past is ‘open’. On the places of ‘forgotten’ facts no new ‘facts’ occur. The places remain ‘empty’.

The ‘closed’ future seems to be a consequence of ‘closed’ past. The past truths belong to the realm of necessity thus each sentence in past tense form, if true, is necessarily true. But some sentences are essentially future in sense though past in form. The rule that true past sentences are necessarily true has to be restricted to sentences past in sense – this is Occamists’ answer^{6} –

5 See also De Caelo i, 283b13.

6 See Prior, The Formalities of Omniscience [in:] Papers on Time and Tense, Oxford 1968, pp. 26–44.

or – such is Perceian answer – the future tense operator has to be conceived as symmetrical to the past tense operator, i.e. it has to mean “necessarily it will be that”.

2. Omnipotence

We can consider the notion of an omnipotent atemporal or – what will be done here – a temporal being. Questions like “is an omnipotent being able to change something in the past?”, or “will an omnipotent being able to do that-and-that?” have sense in the case of temporal being and are senseless in the case of atemporal being.

What does it mean that a temporal being is omnipotent? It is clear that even an omnipotent being is not able to create an absolutely immovable boulder that the being can move. Generally, the omnipotent being is not able to do something that is not possible for logical reasons. This answer bears new questions. First of all: “what does it mean ‘possible’?” and “what logic we are talking about?”. Moreover, we can ask if the omnipotent being is able to change the logic.

The possibility should be conceived in such a way that only these deeds are possible that are not necessary and that are not impossible, i.e. these deeds which are contingent. Such a notion was searched for by /Lukasiewicz.^{7} If we assume that what is true is necessary, then we have to admit other logical value than truth and falsehood, namely possibility.

Omnipotence could be defined as follows: (Om) G is omnipotent iff for each α (∈ L): if Cα, then G is able to do α, where �C” stands for “contingent”.

Let us remark that if it is true that not-α, then α is not contingent. Thus: Cα iff Cnot-α.

G is omniscient, thus its knowledge is complete: for any α (∈ L), G knows whether α (it is true that α) or α is contingent (it is true that Cα) or α is false (it is true that not-α). G is omnipotent, thus for any contingent α (Cα) G is able to decide if α or not-α. But in a world in that all possible deeds are done by G, another free agent – if there is any – has no possibility to choose anything; there is no contingent α left for it. G’s omnipotent providence exercises a complete and perfect control over all events that happen, or will happen, in the universe. How is this secured

7 We mean here his three- and four-valued modal logics.

without infringement of man’s freedom? Our answer is: G is not obliged to decide about any possible α. G is free to leave it to other free agents.

3. Logic of free world

Let us consider the possibility of a world in which there are some free deeds, i.e. a free world. In the case of “closed” world there is no place for free deeds of a free agent. Even if there are some free agents there is no possibility to act for them. The existence of a free world does not prove the existence of some free agents.

Openess of the future is expressed by FC. In a slightly modified form it says: (oF) The future is open iff there are α (∈ L) and t (∈ T) such that neither

it is true that it will be the case at the moment t that α nor it is true that it will be the case at the moment t that not-α.

Instead of oF we can consider a stronger condition: (OF) The future is open iff there is α (∈ L) such that neither it is true

that it will be the case that α nor it is true that it will be the case that not-α.

The condition oF or OF does not imply or contradicts to: (OA) There is α (∈ L) such that neither it is true that α nor it is true

that not-α. (OP) There is α (∈ L) such that neither it is true that it has been the case

that α nor it is true that it has been the case that not-α. The same is true for the past analogue to oF.

Supposing that everything what is necessary is just, evil is possible only as a result of some free deeds. The omniscient and omnipotent free agent G can decide about any possible deed. If G does it and if it is absolutely just, no evil is possible. But we see a lot of evil. Thus evil is done by free agents that are not omniscient or absolutely just. We can ask why G, the omnipotent and omniscient free agent – if G exists – does not complete all the empty places, all the places open for a free agent.^{8} The “empty

8 The problem of evil concerns the contradiction, or apparent contradiction, in holding the following pair of propositions

1. God exists and is almighty, omniscient, and perfectly just;

2. Evil exists. Cf. Mar, G., On not Multiplying Depravity Beyond Necessity, www.sunysb.edu//philosophy//faculty//gmar//evil.txt. In our setting of the problem there is a question why G left uncompleted some places for other free agents. Even G does not know in advance

places” not completed by some free agents give the opportunity to other free agents.

In the case of OA, α can be present in sense, i.e. no tense operators occur in it. The proposition in OP and oP can also be past in sense, e.g. tense operators do not occur in α. Such a situation can take place if some past facts are forgotten. The past can be changed only due to some future facts (that are actualized in the presence).

The language of tense logic consists of an infinite set S of propositional letters, logical connectives (negation: ¬, disjunction: ∨, conjuction: ∧, implication: →, biconditional: ↔), unary tense operators: P (past tense operator – it has been the case that), F (future tense operator – it will be the case that), H (it has always been the case that), G (it will always be the case that) and parenthesis as punctuation marks. Usual formation rules are applied.

A time-frame (a time) is a structure T [= (T, R)], where T is a non- -empty set (of moments of time) and R (⊆ T × T) is a binary relation of precedence (earlier-later) on it.

A valuation V is a function: V (t) ∈ 2^{S}, i.e. a subset of the set of propositional letters is assigned to any t. Each valuation can be uniquely extended for any formula α (∈ L) and t (∈ T).

A model M is a pair (T , V).

M, t ⊧ α is intended to mean that α is true in the model M at the moment t.

T ⊧ α iff for any t(∈ T) and any V: M, t ⊧ α.

Let C be a class of time-frames. C ⊧ α iff for any T (∈ C): T ⊧ α. Time (T, R) is linear in the past (branching in the future) iff for any

t, t_{1}, t_{2}(∈ T): if t_{1}Rt and t_{2}Rt, then t_{1} = t_{2} or t_{1}Rt_{2} or t_{2}Rt_{1}.

Time (T, R) is linear in the future (branching in the past) iff for any t, t_{1}, t_{2}(∈ T): if tRt_{1} and tRt_{2}, then t_{1} = t_{2} or t_{1}Rt_{2} or t_{2}Rt_{1}.

Time (T, R) is linear iff it is linear in the past and in the future. Linear time does branch neither in the past nor in the future. A branch b_{t} is a maximal linear subset of T such that: t ∈ b_{t}. The tense operators can be defined as follows:

(F_{c}) M, t ⊧ F α iff there is t_{1}(∈ T), tRt_{1}: M, t_{1} ⊧ α. (P_{c}) M, t ⊧ P α iff there is t_{1}(∈ T), t_{1}Rt: M, t_{1} ⊧ α.

how free deeds will be done by other free agents. To prevent evil G as omniscient and omnipotent could complete all the places in that evil is possible. But would it be in such a world any possibility of free acts? Thus we can only wonder why the omnipotent G left something to do to others.

(G_{c}) M, t ⊧ Gα iff for each t_{1}(∈ T), tRt_{1}: M, t_{1} ⊧ α. (H_{c}) M, t ⊧ Hα iff for each t_{1}(∈ T), t_{1}Rt: M, t_{1} ⊧ α.

For the above defined tense operators: 1’. α → HF α holds. The formula expresses the intuitive meaning of (1), thus it should be rejected to avoid the theological fatalism. Moreover, in such a case for non-ending time [if for any t(∈ T) there is t_{1}(∈ T) such that: tRt_{1}]:

3. F α ∨ F ¬α holds, too. It means that in such a world there is no place for free deeds. It contradicts to FC (and oF, OF).

If we base on the classical logic to avoid both the consequences, (1’) and (3), the future tense operator F has to be defined in another way. (F_{p}) M, t ⊧ F α iff for any branch b_{t} (∈ T) there is t_{1} (∈ b_{t}), tRt_{1}, such

that: M, t_{1} ⊧ α. The operator F is modalized and means: it necessarily will be the case

that. For this Perceian notions if the time is branching in the future,

4. F (α ∨ β) → (F α ∨ F β) does not hold. In particular, for some α it could be that: neither F α nor F ¬α.

The modal operator can be separated from the tense operator. In order to have the Occamists’ solution one branch has to be distinguished (prima facie). Let it be b′_{t}. Now: (F_{o}) M, t ⊧ F α iff for b′_{t} there is t_{1} (∈ b_{t′}), tRt_{1}, such that: M, t_{1} ⊧ α. (N) M, t ⊧ NF α iff for any branch b_{t} (∈ T) there is t_{1} (∈ b_{t}) tRt_{1}, such

that: M, t_{1} ⊧ α. Now for non-ending time:

5. F (α ∨ β) → (F α ∨ F β) holds, but

6. NF (α ∨ β) → (NF α ∨ NF β) and

7. F Nα → NF α do not hold.

In any of the both solutions, the tense logic is based on the classical logic. It is possible to construct a tense logic based on the intuitionistic logic. We have to modify semantics according to intuitionistic requirements.

Instead of one structure (T, R) we have a partially-ordered (≤) set of times (T_{γ}, R_{γ}), γ ∈ Γ. It is supposed that:

if γ ≤ ϕ, then T_{γ} ⊆ T_{ϕ} and R_{γ} ⊆ R_{ϕ}. The valuation is such that:

if γ ≤ ϕ, then V_{γ}(t) ⊆ V_{ϕ}(t). The semantics has the following intuitive motivation. At any moment t

there is given partially-ordered set of “states of knowledge”. Within each state-of-knowledge there is a set of times and a temporal ordering. The information in a lesser states-of-knowledge is retained in a greater states-of-knowledge.

The function V can be uniquely extended for any formula. The definition of the future and past tense operators are essentially the same as F_{c} and P_{c}, respectively. (F_{i}) M_{γ}, t ⊧ F α iff there is t_{1}(∈ T), tR_{γ}t_{1}, such that: M_{γ}, t_{1} ⊧ α. (P_{i}) M_{γ}, t ⊧ P α iff there is t_{1}(∈ T), tR_{γ}t_{1}, such that: M_{γ}, t_{1} ⊧ α.

There is a difference in the case of G and H: (G_{i}) M_{γ}, t ⊧ Gα iff for each ϕ, γ ≤ ϕ, for each t_{1}(∈ T_{ϕ}), tR_{ϕ}t_{1}:

M_{ϕ}, t_{1} ⊧ α. (H_{i}) M_{γ}, t ⊧ Hα iff for each ϕ, γ ≤ ϕ, for each t_{1}(∈ T_{ϕ}), t_{1}R_{ϕ}t:

M_{ϕ}, t_{1} ⊧ α. In the case of intuitionistic tense logic:

1’. α → HF α holds. It means that the arguments for theological fatalism remains. But, because:

3. F α ∨ F ¬α does not hold, it is a logic of a world in which there are possible some free deeds. The logical structure of the world allows some free deeds, but theses deeds could be done as free only by the omniscient being G.

So far we have showed logical compatibility of worlds in that: I. the argument for theological fatalism is valid and no free deeds are

possible, II. the argument for theological fatalism is not valid and there are possible

some free deeds; III. the argument for theological fatalism is valid and there are possible

some free deeds – in this case the free deeds could be done only by the omniscient being. It remains the fourth combinatorial possibility:

IV. the argument for theological fatalism is not valid and no free deeds are possible. In both the logics in which F α ∨ F ¬α does not hold – tense logic of

time branching in the future and intuitionistic tense logic – if we suppose F α ∨ F ¬α, in consequence we receive: α → HF α. It means that in the case of completely determined world, the argument for theological fatalism is valid.

The fact that there are possible some free deeds does not contradict to the possibility that any contingent α sooner or later is actualized (The principle of plenitude). This kind of fatalism was already considered in antiquity. The famous Master Argument of Diodorus Cronus is based on the definition of possibility as something that is or will be. The question of this kind of fatalism does not have direct connection with problems considered in this paper.