2.1. The Meaning of Proposition (3) in Problem of Change in Knowledge

In the places where Avicenna talks about God’s knowledge of particulars, he does not present a straightforward and comprehensive account of what the occurrence of change among particulars means. While in certain theological contexts he does talk about just one state of affairs, which falls under the concept of change, by looking at other places, we can extract a straightforward account of the occurrence of change that incorporates all things that we intuitively think should be instances of change. In the theological context, the state of affairs that he considers to be an instance of change is a thing coming in or going out of existence. He writes:

Let us capture this idea as follows:

Avicenna in non-theological contexts introduces two other states of affairs that fall under the notion of change. He writes:

According to this text, Avicenna says that if an object receives at least one new property or loses at least one of its properties, then that object has changed. In order to get a precise account of this definition, we should note that Avicenna distinguishes between two kinds of properties: primary properties and secondary properties. The former, as Rāzī^{Footnote 16} interprets Avicenna, are non-relational properties and the latter are the relational properties. For example, ‘being red’ and ‘being 1 km away from London’ are non-relational and relational properties, respectively. As is clear, this distinction between primary properties and secondary properties corresponds to the distinction between intrinsic properties and extrinsic properties. By having this distinction in hand, we can say:

It seems that the conjunction of these three states of affairs, i.e., (i) coming or going out of existence, (ii) receiving or losing an intrinsic property by an object, and (iii) receiving or losing an extrinsic property by an object accommodate all cases that we are inclined to consider as cases of change. Therefore, the definition of ‘occurrence of a change,’ according to Avicenna’s point of view, among particulars or, in short, the ‘definition of change’ (DC) would be as follows:

Where P_i and P_e represent an intrinsic and an extrinsic property, respectively. So, according to this definition, if we combine hydrogen and oxygen in a chemistry lab, and water is produced, and if when undergoing electrolysis, water is broken down into hydrogen and oxygen, then a thing, namely water, comes in to or goes out of existence, respectively, and so change (i) occurs among particulars. Or if the colour of a yellow apple on a tree turns red, then a thing, i.e., an apple, has lost and gained an intrinsic property, i.e., its colour, and hence change (ii) has occurred among particulars. In the case of change (iii), if I move my philosophy book located on the desk to my bag located on the ground, then a thing, i.e., my book, has lost and gained an extrinsic property, namely its specific location. In this case, we should note that change in an extrinsic property of the book entails occurrence of a change in the world (i.e., motion of the book), not a change in the book itself.

So far, I have only extracted Avicenna’s definition of occurrence of a change among particulars. Now, let me clarify what that natural and common way of modeling knowledge of the occurrence of a change among particulars consists in. Let us briefly call this way the ‘natural and common way’ (NCW). In the following, I will present four texts from Avicenna and on the basis of them extract NCW. Let us begin with the first text:

In this text, Avicenna claims that if God’s knowledge is temporal knowledge (‘ilman zamānīyan), then his knowledge changes. Avicenna clarifies that by ‘temporal’ in the term ‘temporal knowledge,’ he means a time that is pointed out (zamānan mushāran ilayhu). This kind of knowledge, as his examples show, is knowledge of a proposition that contains the adverb ‘this time’ or ‘now.’ Avicenna considers knowing propositions such as ‘O does not exist in this time,’ ‘O does not exist now,’ and ‘O exists tomorrow’ as examples of this kind of knowledge. Then, he presents a brief argument to show why applying this kind of knowledge to God is problematic. His argument presents a dilemma: given that object O does not exist today and does exist tomorrow, on the one hand, if God’s set of knowledge today includes knowing the proposition ‘O does not exist now,’ but also includes it tomorrow when O does exist, then tomorrow God’s knowledge turns out to be false because we have assumed that tomorrow O exists. On the other hand, if tomorrow, God’s set of knowledge does not include ‘O does not exist now,’ then God’s set of knowledge today and tomorrow are not the same and therefore God’s set of knowledge has changed.

In another place, Avicenna writes:

In this text, Avicenna declares that God’s knowledge of particulars is not temporal knowledge. In explaining what he means by ‘temporal knowledge,’ Avicenna says that temporal knowledge is the kind of knowledge that includes adverbs such as ‘now,’ ‘past’ and ‘future.’

Now, let us have a look at the following text:

In this text, Avicenna claims that there are two different ways of knowing the occurrence of some change among particulars, such as an eclipse. Since the first way of knowing has no relevance to my current purpose, I put it aside here. The second way is what Avicenna calls ‘knowing in a temporal way.’ To explain what he means by temporal knowledge, Avicenna declares that this manner of knowing is knowing propositions such as ‘the eclipse is occurring now,’ ‘the eclipse has occurred before now,’ or ‘the eclipse will occur after now.’ From Avicenna’s point of view, this kind of knowledge appears and disappears by the appearing and disappearing of the eclipse. This contrasts with how a knower knows the occurrence of an eclipse in the first way which, unlike the knowing in the second way, remains fixed during, before, and after the occurrence of the eclipse.

Let us look at the next text:

In this text, Avicenna asserts that it is impossible that God knows changeable particulars by certain temporal knowledge (‘ilman zamānīyan mushakhkhaṣan). Then, he claims that it is not possible that God knows temporally, i.e., at one time that ‘O exists’ and at another time that ‘O does not exist.’ As we saw in previous texts, by ‘temporal knowledge,’ Avicenna means knowing propositions that contain adverbs such as ‘now.’ Therefore, we can restate his claim as it is impossible that God at one time knows that ‘O exists now’ and at another time that ‘O does not exist now.’ For then, as the end of the passage states, for each two previous propositions, there would be two different and distinct forms that are inconsistent with each other.

Now, I want to draw two important conclusions from the four texts presented above that play a very important role in justifying proposition (3) for any change, C, among particulars and any knower, S, if S knows the occurrence of C, then S’s set of knowledge changes. First, if we look at all these four texts, we will realize that, from Avicenna’s point of view, God’s knowledge of the occurrence of a change among particulars should not be modeled by using propositions in form of ‘O exists now,’ ‘O does not exist now,’ and ‘the eclipse is occurring now.’ If we consider these propositions with a subtle scrutiny, we will realize that all have one thing in common. What is common among them is that they are a certain kind of indexical propositions, i.e., propositions that contain the adverb ‘now.’^{Footnote 21} Avicenna’s mentioning it in a theological context indicates that he thinks that a common and natural way to model knowledge of the occurrence of a change among particulars is by means of indexical propositions. Interestingly, however, Avicenna realizes that applying this model of knowledge of the occurrence of a change among particulars to God is problematic and he therefore looks for an alternative, i.e., one without indexical propositions. Avicenna’s rejection of applying this model (and not the other one) to God indicates that it is a common and natural way to model one’s knowledge of occurrence of a change among particulars. Therefore, according to DC and this point, we can present NCW (natural and common way of modeling knowledge of the occurrence of a change among particulars) as follows:

Second, in TEXT (6) and TEXT (9), Avicenna has presented a brief argument which, in part, shows why we should not model God’s knowledge of the occurrence of a change among particulars by indexical propositions, that is, why we should not apply NCW to God. This argument, I think, will lead us to establish the truth of (3). I will talk about this argument later.

So far, I have clarified what the antecedent of proposition (3), i.e., knowledge of the occurrence of a change among particulars (NCW), means. Now, let us clarify what its consequent, i.e., change in a knower’s set of knowledge means. In Avicenna’s philosophical jargon, the term ‘knowledge’ refers to two different, but closely related things. The first account of the term ‘knowledge,’ which is a normal account of this term, refers to a proposition known by S. That is, according to this account of the term ‘knowledge,’ to say S has knowledge is to say that S knows that P. We should note that this account of the term ‘knowledge’ is silent regarding what metaphysically happens for S to come to know that P. Let us call this account of the term ‘knowledge’ the ‘epistemological account.’^{Footnote 22} The following text is an example that shows Avicenna’s use of this account of the term ‘knowledge’:

The second account of the term ‘knowledge’ is Avicenna’s standard definition of ‘knowledge,’ according to which knowledge is the occurrence, i.e., existence, of a form, in some sense, in the knower.^{Footnote 24} He writes:

Indeed Avicenna defines ‘knowledge’ by means of some relation between the knower and a form that represents the content of knowledge. We should note that the term ‘knowledge’ here refers to Avicenna’s explanation of how S knows that P. Let us call this account of the term ‘knowledge’ the ‘metaphysical account.’ Therefore, when we talk about S’s knowledge, we mean the existence of a form in S. Therefore, we should distinguish between the metaphysical and the epistemological account of the term ‘knowledge.’ That is, metaphysically speaking, S’s knowledge refers to the existence of a form, F, in S, and epistemically speaking, S’s knowledge refers to S’s knowing that P. But we should not forget that these two accounts of the term ‘knowledge’ are closely related. From Avicenna’s point of view, there is a one-to-one relation between a form, F, which exists in S and the proposition, P, which S knows on the basis of F. Let me support my claim. Let’s look at the two following texts:

These texts support my claim in two different ways. First, according to these texts in which Avicenna talks about the structure of an argument, he claims that knowing each premise of an argument is distinct from knowledge of the other premises and knowing them is distinct from knowing the conclusion. Now, we can ask what Avicenna means by the term ‘knowledge’ here. There are two possibilities: either the term ‘knowledge’ has used in its epistemological sense or in metaphysical sense. But, it cannot be used in its epistemological sense. For if it is used in its epistemological sense, then Avicenna’s claim turns out to be that knowing the premises and conclusion of an argument is knowing distinct propositions, which is trivially obvious and there is no need to mention it. Therefore, Avicenna, by the term ‘knowledge’ here, intends its metaphysical usage. If so, Avicenna, in these texts, claims that for each premise and conclusion there is a distinct form which exists in the knower.

Second, in the TEXT (14), Avicenna claims that knowing the premises and the conclusion are distinct knowledge and distinct relations to the soul. Now, if we suppose for knowing all premises and their conclusion there exists only one form in the soul, then only one relation would occur for the soul, not different relations. This is so because any relation is determined by its two relata. Since one relatum, i.e., the soul is constant here, for occurring different relations to the soul, the other relatum for each relation should be different and distinct. Therefore, for each premise and conclusion, there is a distinct form that exists in the knower. Therefore, we can say that, from Avicenna’s point of view, for knowing any proposition, P, there exists a distinct form, F, for P in the knower, S. We can break up this claim into two other claims: (i) for any proposition, P, there exist a form, F, in the knower, S, and (ii) it is not the case that for knowing two propositions there exist just one form in the knower. That is, the existence of different forms in the knower entails that the knower knows different propositions, and knowing different propositions by the knower entails that there exist different forms in the knower. Therefore, if S knows, for example, just three propositions, P₁, P₂, and P₃ then there exist three forms F₁, F₂, and F₃ in S and vice versa. Let us formulate this as follows:

We can also show the above formulation as the following diagram:

As is clear, the term ‘knowledge,’ in both its metaphysical and epistemological senses, does not necessarily refer to just one form or just one proposition, respectively. Therefore, let us talk about S’s knowledge, in both its metaphysical and epistemological senses as a set. Let us call the propositions that S knows ‘S’s set of knowledge’ (or ‘S’s knowledge-set’) and designate it by ‘K’ and the forms that exist in S, ‘S’s set of forms’ and designate it by ‘K*.’ Five points are worth noting about K and K*. First, members of K are the propositions that S knows and the member of K* are the forms that exist in S and cause S to come to know some proposition. Second, it is clear that sets K and K* might have just one member, which means that there exist only one form in S. Hence, S would know just one proposition. It is even possible that K and K* are empty sets, which means that no form exists in S and hence S does not know any proposition. Third, there is a one-to-one relation between K and K*, which means that for any member in K there is one and only one member in K* and vice versa. Fourth, since sets change by changing their members, K and K* change by changing their members. Accordingly, since the members of K are the propositions that S knows and the members of K* are the forms that exist in S, we can formulate change in K and K* as follows:

Accordingly, we can say:

Fifth, since, according to the third point, there is a one-to-one relation between K and K*, every change in K entails a change in K* and vice versa. Therefore, we can say that:

Now by clarifying what knowledge of a change among particulars and what change in a knower’s set of knowledge means, the meaning of proposition (3) for any change, C, among particulars if S knows the occurrence of C, then S’s set of knowledge changes becomes clear. The next step is dealing with how Avicenna justifies proposition (3) in PCK.

2.2. Avicenna’s Justification for Proposition (3) in PCK

As I previously said, Avicenna, in TEXT (6) and TEXT (9), has presented a brief argument which, in part, shows that why we should not model God’s knowledge of the occurrence of a change among particulars by indexical propositions; that is, why we should not apply NCW to God. This argument will lead us to justify (3) in PCK. In light of NCW, we can restate Avicenna’s argument as follows.

Suppose that an Object, O, at t₁ exists and at t₂ goes out of existence. Therefore, the propositions ‘O exists now’ and ‘O does not exist now’ are true at t₁ and at t₂, respectively and hence according to DC, we can say that a change has occurred among particulars. Suppose, also, that a knower, S, has knowledge of the occurrence of this change. Therefore, according to NCW, at t₁ S knows that ‘O exists now’ and at t₂ knows that ‘O does not exist now.’ It is clear that propositions ‘O exists now’ and ‘O does not exist now’ cannot be true at the same time without resulting in a contradiction. Therefore, no one can hold them together.^{Footnote 30} Since one cannot hold the propositions ‘O exists now’ and ‘O does not exist now’ at the same time, then S’s set of knowledge at t₂ includes ‘O does not exist now’ and does not include ‘O exists now.’ According to the definition of change in S’s set of knowledge, we can say on the one hand that S’s set of knowledge has changed. For S’s set of knowledge at t₁ includes ‘O exists now’ and at t₂ it does not. On the other hand, S’s set of knowledge at t₁ does not include ‘O does not exist now’ and at t₂ does. Therefore, if O exists at t₁ and goes out of existence at t₂ and at t₁ S knows that ‘O exists now’ and at t₂ S does not know that ‘O exists now,’ then S’s set of knowledge has changed. We can formulate this argument as follows:

1. 1) At t₁ S knows that ‘O exists now’ and at t₂ S knows that ‘O does not exist now.’ [Assumption]

2. 2) If S at t₁ knows that P and at t₂ knows that ∼P, then S at t₂ does not know that P

3. 3) Therefore, at t₂ S does not know that ‘O exists now.’ [Modus ponens 1&2]

4. 4) At t₁ S knows that ‘O exists now.’ [Simplification 1]

5. 5) At t₁ S knows that ‘O exists now’ and at t₂ S does not know that ‘O exists now.’ [Conjunction 4&5]

6. 6) S’s set of knowledge, K, changes if and only if there is a proposition, P, such that (i) at t₁ S knows that P and at t₂ (t₁<t₂) S does not know that P or (ii) at t₁ S does not know that P and at t₂ S knows that P. [Definition]

7. 7) Therefore, S’s set of knowledge has changed. [Modus ponens 5&6]

8. C) If at t₁ S knows that ‘O exists now’ and at t₂ S knows that ‘O does not exist now,’ then S’s set of knowledge has changed. [Conditional reasoning 1-7]

But there is an important problem about A₁ with regard to proposition (3) in PCK. This argument, as stated, proves change in S’s set of knowledge only for a certain kind of change among particulars, i.e., part (i) in DC, not all parts of it, i.e., part (ii) at t ₁ O has P _i and at t ₂ O does not have P _i and vice versa and part (iii) at t ₁ O has P _e and at t ₂ O does not have P _e and vice versa. In other words, this argument, as stated, does not show that if a change occurs by occurrence of part (ii) or (iii) in DC and S knows it, then S’s set of knowledge changes. Indeed, Avicenna’s argument proves that if a specific kind of change, i.e., the part (i) in DC, occurs and S knows it, then S’s set of knowledge changes. If so, A₁ does not establish the truth of (3) whose domain covers all kinds of change, i.e., parts (i), (ii), and (iii) in the definition of occurrence of a change among particulars (DC).

However, A₁ can be easily adapted to deal with parts (ii) and (iii) of DC as well. So, we can advance the similar argument for (ii) and (iii) and through them establish the truth of (3). This is why I previously said that A₁ leads us to establish the truth of (3). In the following, I advance just such an argument for parts (ii) and (iii) in DC.

Let us suppose that an object, O, at t₁ has an intrinsic property, P_i, and at t₂ loses that property (or vice versa). For example, suppose that the colour of a yellow apple on the tree turns to red after a week or my height which is 180 cm increases during one year and reaches 190 cm. Also, suppose that an object, O, at t₁ has an extrinsic property, P_e, and at t₂ loses that property (or vice versa). Suppose, for example, that I move my hand from left to right such that its spatial relation with all things in the world changes. The argument that proves that S’s set of knowledge changes if S knows the occurrence of this kind of changes is as follows:

1. 1) At t₁ S knows that ‘O has P_i/ P_e now’ and at t₂ S knows that ‘O does not have P_i/ P_e now.’ [Assumption]

2. 2) If S at t₁ knows that P and at t₂ knows that ∼P, then S at t₂ does not know that P.

3. 3) Therefore, at t₂ S does not know that ‘O has P_i/ P_e now.’ [Modus ponens 1&2]

4. 4) At t₁ S knows that ‘O has P_i/ P_e now.’ [Simplification 1]

5. 5) At t₁ S knows that ‘O has P_i/ P_e now’ and at t₂ S does not know that ‘O has P_i/ P_e now.’ [Conjunction 3&4]

6. 6) S’s set of knowledge, K, changes if and only if there is a proposition, P, such that (i) at t₁ S knows that P and at t₂ (t₁<t₂) S does not know that P or (ii) at t₁ S does not know that P and at t₂ S knows that P. [Definition]

7. 7) Therefore, S’s set of knowledge has changed. [Modus ponens 5&6]

8. C) If at t₁ S knows that ‘O has P_i/ P_e now’ and at t₂ S knows that ‘O does not have P_i/ P_e now,’ then S’s set of knowledge has changed. [Conditional reasoning 1-7]

This argument shows that if a change of parts (ii) or (iii) occurs among particulars and S knows it, then S’s set of knowledge changes. We should note that since arguments A₁ and A₂ are structurally similar, Avicenna just mentions one of them, i.e., A₁.

So far, I have argued that for each part of DC, i.e., parts (i), (ii), and (iii): S’s knowledge in each case entails that S’s knowledge-set changes. Since there is not a fourth option for capturing the definition of change, that is, since (i), (ii), and (iii) are mutually exclusive and jointly exhaustive, we can conclude that:

From the above, we see how Avicenna supports proposition (3) of PCK. Indeed, Avicenna believes that applying the NCW model to the knowledge of the occurrence of a change among particulars entails that one’s set of knowledge changes. Let us now mention two related points that can be insightful to understanding arguments A₁ and A₂. First, these arguments are consistent with both gradual changes—which Avicenna calls ‘motion’—and instantaneous changes. According to Avicenna, a change or instability among particulars comes in two varieties. One type of change, motion, is a change or instability that occurs in just four accidental categories, i.e., quality (kayfīyah), quantity (kammīyat), place (ayn), and position (waḍ‘)^{Footnote 31} such that the first two are non-relational and the last two are relational properties.^{Footnote 32} For example, when the colour of a yellow apple on the tree begins to change and after, say, three weeks becomes red, or when I move my hand for 10 seconds from point M₁ to point M₂, it is true to say that there is a motion in the quality, i.e., colour of the apple and in the place of my hand, respectively. Another type of change or instability among particulars occurs only in substance and in the accidental categories other than the previous four accidental categories in which motion occurs,^{Footnote 33} such that some of them are relational and others are non-relational properties.

The second point is that if a knower, S, knows the occurrence of a motion among particulars during (t₁,t₂), not only S’s set of knowledge changes but also S’s set of knowledge, during (t₁,t₂), frequently changes. Suppose, for example, an object, O, changes its place from point M₁ to point M₂ in a certain period of time (t₁,t₂). Then, we can define this motion as follows:

On the basis of this definition and in light of NCW, we can define S’s knowledge of the occurrence of this motion as follows:

According to this definition, S’s knowledge of the occurrence of a motion among particulars causes S’s set of knowledge to change frequently.

So far, I have shown how Avicenna justifies proposition (3) in PCK on the basis of NCW. Now, let us present a detailed version of PCK by arguing why Avicenna thinks that proposition (4) God’s set of knowledge does not change is true.

In order to present a detailed version of PCK, I need a principle in Avicenna’s philosophy, which I call the ‘principle of the necessity of a necessary being’s attributes’ (NNBA), according to which a necessary being, possesses his attributes necessarily. That is, if a being is a necessary being and is, for example, simple, then that being, necessarily, is simple. This means that it is metaphysically impossible for a necessary being to lack this attribute. Avicenna writes:

We can formulate this principle as follows:

The Principle of NNBA

Since one of the possible attributes of a being is having knowledge and we talked about knowledge in terms of sets, we can, on the basis of NNBA, say that:

Having this proposition in hand, we can formulate the detailed version of PCK as follows:

PCK (detailed version)

1. 1) God is omniscient.

2. 2) Therefore, for any change, C, among particulars, God knows the occurrence of C.

3. 3) For any knower, S, if S knows the occurrence of C, then S’s set of knowledge changes.

4. 4) If S’s set of knowledge changes, then there are two times t₁ and t₂ such that S’s set of knowledge at t₁ is K₁ and S’s set of knowledge at t₂ is K₂ and K₁ ≠K₂.

5. 5) If K₁ ≠K₂, then either (a) K₂ is a subset of K₁ or (b) K₁ is a subset of K₂ or (c) neither of them is a subset of the other.

6. 6) If K₁ is a subset of K₂, then at t₁ S is not the owner of K₂.

7. 7) If K₂ is a subset of K₁, then at t₂ S is not the owner of K₁.

8. 8) If neither of them is a subset of the other, then at t₁ S is not the owner of K₂ and at t₂ is not the owner of K₁.

9. 9) If (i) X is a necessary being and (ii) X is the owner of K₁, then X is necessarily the owner of K₁.

10. 10) If (i) X is a necessary being and (ii) X is the owner of K₂, then X is necessarily the owner of K₂.

11. 11) God is the owner of K₂.

12. 12) God is the owner of K₁.

13. 13) God is a necessary being.

The main idea of this detailed version of PCK is this: according to Avicenna, God is omniscient and therefore knows the occurrence of any change among particulars. But, as I previously discussed, by knowing the occurrence of a change among particulars, God’s set of knowledge changes and therefore God’s set of knowledge at t₁, i.e., K₁, is different from God’s set of knowledge at t₂, i.e., K₂. Now recall the definition of change in S’s set of knowledge, according to which:

According to the two parts of this definition, i.e., (i) and (ii), three states are possible. First, (i) is the case, second, (ii) is the case, and third, both (i) and (ii) are the case. If (i) is the case, then knowledge of at least one proposition has been subtracted from S’s set of knowledge at t₁, i.e., K₁, which means that S’s set of knowledge at t₂, i.e., K₂, is a subset of K₁. This is just like when we subtract the number 3 from the set {1, 2, 3} and get {1, 2}. In this case, the latter is a subset of the former, which means that the latter, and not the former, exists, in some sense, at both times t₁ and t₂. If (ii) is the case, then knowledge of at least one proposition has been added to S’s set of knowledge at t₁, i.e., K₁, which means that S’s set of knowledge at t₁, i.e., K₁, is a subset of S’s set of knowledge at t₂, i.e., K₂. This is similar to when we add the number 4 to the set {1, 2, 3} and get {1, 2, 3, 4}. In this case, the former is the subset of the latter, which means that the former, and not the latter, exists, in some sense, at both times t₁ and t₂. Finally, if both (i) and (ii) are the case, then knowledge of at least one proposition has been subtracted from, and knowledge of at least one proposition has been added to, S’s set of knowledge at t₁, i.e., K₁, which means that S’s set of knowledge at t₂, i.e., K₂, is not a subset of K_1, nor is K₁ a subset of K₂. Just like when we subtract the number 3 from the set {1, 2, 3} and add the number 4 to it to get {1, 2, 4}. In this case, {1, 2, 3} is not a subset of {1, 2, 4}, nor is {1, 2, 4} a subset of {1, 2, 3}, which means that neither the latter nor the former exist, in some sense, in both times t₁ and t₂.

Now, if (i) is the case (i.e., state (a) in proposition 5 above), which means that K₂ is a subset of K₁, then at t₂ K₁ does not exist,^{Footnote 37} which means that at t₂ God is not the owner of K₁—which means that it is not the case that God is necessarily the owner of K₁. But, according to NNBA, if God, a necessary being, is the owner of K₁—which is the case at t₁—then God is necessarily the owner of K₁. And this contradicts our earlier result. Now, if (ii) is the case (i.e., state (b) in proposition 5 above), which means that K₁ is a subset of K₂, then at t₁ K₂ does not exist,^{Footnote 38} which means that at t₁ God is not the owner of K₂—which means that it is not the case that God is necessarily the owner of K₂. But, according to NNBA, if God, a necessary being, is the owner of K₂—which is the case at t₂—then God is necessarily the owner of K₂. And this again contradicts our earlier result. Finally, if both (i) and (ii) are the case (i.e., the state (c) in proposition 5 above), which means that neither K₁ nor K₂ are subsets of each other, then K₂ does not exist at t₁, nor does K₁ exist at t₂—which means that at t₁ God is not the owner of K₂ and at t₂ God is not the owner of K₁. And this means that it is not the case that God necessarily is the owner of K₁ and K₂. But again, according to NNBA, if God, a necessary being, is the owner of K₁ and K₂—which is the case at t₁ and t₂, respectively—then God is necessarily the owner of K₁ and K₂. And this contradicts our earlier result. Therefore, the set of propositions (1)—(13) is inconsistent and hence one cannot hold all of them.

The detailed version of PCK, I think, represents one of the problems with which Avicenna was engaged and his theory of God’s knowledge of particulars is a solution to this problem.

In this section, I first presented a simple version of PCK. In order to justify proposition (3) for any change, C, among particulars and any knower, S, if S knows the occurrence of C, then S’s set of knowledge changes of this simple version, I first clarified what is meant by (3) and then extracted two arguments from Avicenna’s philosophy, i.e., A₁ and A₂, in order to justify (3). What is important in both A₁ and A₂ is using indexical propositions for formulating knowledge of the occurrence of a change among particulars. Then, by showing why proposition (4) God’s set of knowledge does not change in this simple version is true, I presented a detailed version of this problem. The gist of PCK can be stated as follows: God’s knowledge of the occurrence of a change among particulars entails a change in God’s knowledge-set, which in turn entails that God is not a necessary being. But, God, according to Avicenna, is a necessary being. And so a contradiction results. In the next section, I will discuss another problem with which Avicenna was engaged, i.e., PCE.