Wednesday, May 27, 2009

Here's a New Project of Mine

A new project of mine has been to attempt to transpose elements of metaphysics into symbolic logic. I don't know how well this works just yet, but I feel that it should be, in principle, possible. Anyway, here's the first part of what I want to call, preliminarily, "Principia Metaphysicae."

1.Set is here taken to indicate not material quantity, but formal differentiation; elements in a set can be materially differentiated.
2.“ε” is the symbol for being or “ens” (from einai).
3.“ψ” is the symbol for “the set of all entities,” which includes sets, elements, and functives.
4.The quantifier “∃x” indicates real existence of x and is convertible with “X → (→ε).”
4.1.“(→ε)” is used to eliminate confusion when speaking about the act of existence itself
5.“→” is meant to indicate only “formal implication.”
6.“∃→” is meant to indicate “existential participation” (The implicandum is the participandum; implicata is participata).
6.1.This is an implicated relationship because, as will be shown, ε is not a set to which elements can belong and participation cannot likewise be a belonging relationship; thus, one might only consider ens as a quasi-superset, which is why I denote this as “implication” rather than belonging.

1.A being exists; ens.
1.⊢ ∃x / (x → ε)
1.1.All things which exist (ψ), including all predicates or sets, are defined as “beings” (ε).
1.1.:= ∀ψ → (→ε)
1.1.1.Non-being (~(→ε)) cannot be a predicate (operator/function), element of a set, or set (non-being is not an empty set)
1.1.1.~ (→ε) → ~∀ψ
1.2.It is false that a demonstration of the existence of at least one being (1) can be made within metaphysics.
1.2.~[d(∃x) ∈ m]
1.2.1.Assume there is such a statement of the demonstration of the existence of at least one being within metaphysics.
1.2.1.Assume: d(∃x) ∈ m
1.2.2.Assume metaphysics is the science of all the possible existential modes of beings.
1.2.2.Assume: m = f(∀ψ)
1.2.3.Assume all possible existential modes of beings are demonstrable if and only if there exists some demonstrable being (of which we are specifying modes of being).
1.2.3.Assume: f(∃ψ) ↔ d(∃x)

1.2.4.Assume: there exists one being whose existence is demonstrable.∃x) demonstration of the existence of one being belongs to the science of metaphysics.∃x) ∈ m metaphysics is true if and only if this demonstration of the existence of said being is true. ↔ d(∃x), a circle ensues such that metaphysics is true if and only if this demonstration is true, and this demonstration is part of metaphysics.[d(∃x) ∈ m] ^ [m ↔ d(∃x)], 1 is unprovable within the science of metaphysics.*** ~[d(∃x) ∈ M]*As such, no strict deductive proof can be proposed for 1 or 1.1. This is an instance of a starting point for any deductive chain of reasoning and is necessarily true. Any such statement in metaphysics is termed an "first principle" and is denoted “⊢.“ While one is unable to strictly demonstrate such a principle, one can show recursively that all reasoning assumes or depends on this principle, as metaphysics underlies all thought of any being whatsoever.

1.3.Nothing is both true and false at the same time and in the same respect; this is an extension of 1 and not strictly distinct [Principle of non-contradiction].
1.3.1.A recursive proof of the first principle: assume the contrary, that: for all X, X and not X are both true.
1.3.1.Assume: ∀x[∃x ^ ~∃x] A is true; instantiation of X as “A.”∃a is not true; instantiation.∃a then either A or Z is true∃a v z thus, we can say that Z is true, from the prior disjunctive (because A is not true), without the principle 1, all statements can be true. (∃a^~∃a) → z***Therefore, that something both be true and false [in the same respect] is false.*** ∀x[~(∃x^~∃x)]
1.4.Every being is undivided from itself; unum.
1.4.∀x(x ↔ ~~x)
1.4.2.By distribution.
1.4.2.*** (∃a) ^ ~~(∃a)
1.5.A definition of equality: if there is no differentia, there is no difference between beings [axiom of extensionality].
1.5.:= ∀x∀y[∀z(z∈x ↔ z∈y)] ↔ x=y

1.6.Definition of difference; every unique being possesses proper aspects which distinguish it from any other being; essentia/quidditas.
1.6.:= ∀x ∃y(∀z(z∈x → ~(z∈y)) ↔ x~=y
1.7.*Formal Causality: Every being might be characterized by those proper aspects which define it as the sort of being that it is (this is equivalent, at this point, with final causality).

2.There is more than one being, at least in some respect.
2.⊢ ∃x∃y(x ~= y)
2.1.A recursive proof: The statement of the existence of the being is not identical with the being asserted.
2.1.f(∃x) ~= ∃x
2.2.But any function of an X exists.
2.2.f(x) → (→ε)
2.3.Thus, the statement about the existence of a being itself exists.
2.4.There exists at least one statement about the existence of one entity and one entity being spoken of.
2.4.[∃x] ^ [∃f(∃x)]
2.5.Therefore, there exists both a statement about the existence of some being and some being, both of which are not strictly identical.
2.5.*** [∃x] ^ [∃f(∃x)] ^ [f(∃x) ~= ∃x]
3.Every being is one being among other beings; aliquid; [axiom of pairing].
3.∀x∀y∃z∀w(w∈z ≡ w=x v w=y)
3.1.The two presumed beings, X and Y, are not identical (according to 2).
3.1.x ~= y
3.2.Difference is defined as possessing some proper characteristic Z which distinguishes X from Y.
3.2.∀x ∃y(∀z(z∈x → ~(z∈y))]

3.3.Therefore, there is some characteristic Z for X to distinguish it from Y.
3.3.x → ∃y(∀z(z∈x → ~(z∈y))
3.4.The same is true of Y; some Z exists to distinguish Y from X
3.4.y → ∃x(∀z(z∈y → ~(z∈x))
3.5.There is thus some Z for which it is the case for all W, such that W belongs to Z if and only if W=X or W=Y.
3.5.∃z∀w(w∈z ≡ w=x v w=y)
3.6.Therefore, for all X and all Y, there exists some Z for all W, such that W belongs to Z if and only if W=X or W=Y.
3.6.*** ∀x∀y∃z∀w(w∈z ≡ w=x v w=y)
4.Ens is thus predicated of existentially diverse instances, although remaining united among all instances.
4.[[∀x → ∃y(∀z(z∈x → ~(z∈y))] ^ [(x ↔ ~~x)]] → ε
5.Ens cannot be the genus of all existing things (set X containing all Phi).
5.~[ε = (∀ψ∈x)]
5.1.Assume ens is a genus (a set X containing all Phi).
5.1.Assume: ε = x(∀ψ ∈ x)
5.1.1.For all X, to be a genus implies that there exists some feature which distinguishes it from other genus and species.
5.1.1.∀x → ∃y(∀z(z∈x → ~(z∈y)))
5.1.2.All beings are defined as ens and if it is not ens it is does not exist.
5.1.2.(∀ψ → ε) ^ (~(→ε) → ~∀ψ )
5.1.3.Any difference in ens would belong to itself.
5.1.3.∀f((→ε)) ∈ (→ε)
5.1.4.Any proper aspect of ens would itself belong to ens.
5.1.4.∀x ∈ (→ε)
5.1.5.But then ens could not have a proper differentia to distinguish it from another set.
5.1.5.∀(→ε) → ~∃y(∀z(z∈x → ~(z∈y)))
5.1.6.Thus, ens would both have proper differentia and not have proper differentia.
5.1.6.(→ε) → [∃y(∀z(z∈x → ~(z∈y)))] ^ [~∃y(∀z(z∈x → ~(z∈y)))]
5.1.7.Therefore, being cannot be a genus.
5.1.7.*** ~[(→ε) = x(∀ψ ∈ x)]
6.There is a real distinction that it is not true for all beings that their “existing” is identical with their proper attributes
6.~∀x [(∃y(∀z(z∈x → ~(z∈y)))) ^ ((→ε) = z)]
6.1.Assume that “existing” (X’s inclusion in ens) is identical with the proper attributes of any being.
6.1.Assume: ∀x [(∃y(∀z(z∈x → ~(z∈y)))) ^ ((→ε) = z)]

6.1.1.There exist at least two beings which are not identical with each other.
6.1.1.∃x∃y(x ~= y)
6.1.2.For all X, there would be some Y such that for all act of existence of Y, “existing” would belong to Y and “existing” would not belong to X.
6.1.2.∀x [(∃y(∀z(z∈x → ~(z∈y)))) ^ ((→ ε) = z)]
6.1.3.For all X and all Y, all Z of X is “existing” and all Z of Y is “existing.”
6.1.3.∀x∀y[∀z((→ε)∈x ↔ (→ε)∈y)]

6.1.4.But, when for all X and all Y, all Z which belong to X also belong to Y, then X is identical with Y.
6.1.4.∀x∀y[∀z(z∈x ↔ z∈y)] → x=y
6.1.5.X is then identical with Y
6.1.6.But, per 6.1.1, X and Y are non-identical beings.
6.1.6.x ~= y

6.1.7.Therefore, proper attributes are not identical with “existence” in all entities.
6.1.7.*** ~∀x[(∃y(∀z(z∈x → ~(z∈y)))) ^ (z=(→ε))]
7.Every being exists either from intrinsic (existence belongs to the set X as a proper part) or extrinsic principles (X is “existentially participative” in Y; it does not “belong to” or make up a “proper part” of Y).
7.∀x [(→ε) ∈ x) v (x ∃→ y)]
7.1.Not all beings can have the property of “existing” as its proper attribute.
7.1.~∀x[(∃y(∀z(z∈x → ~(z∈y)))) ^ (z=(→ε))]
7.2.If there is a being whose proper attribute is identical with existing (Θ), this being is unique (there can only be one such being).
7.2.Θ[(∃y(∀z(z∈ Θ → ~(z∈y)))) ^ (z=(→ε))] → ∃!Θ

7.3.But there are at least two entites which are not identical.
7.3.∃x∃y(x ~= y)
7.4.At least one being does not have “existing” as a proper attribute.
7.4.∃x[~(∃y(∀z(z∈x → ~(z∈y)))) ^ (z=(→ε))]
7.5.That this being (X) does not have existence as its proper attribute is equivalent to saying that the same being lacks “existing” as a proper attribute.
7.5.~∃x [(∃y(∀z(z∈x → ~(z∈y)))) ^ (z=(→ε))] ↔ ~∃x ((→ε) ∈ x)
7.6.But this being (X) exists.
7.6.x → (→ε)
7.7.Thus, there exists at least one being (X) which existentially participates in another being (Y).
7.7.∃x(x ∃→ y)

7.7.1.*Efficient Causality: Beings may be characterized insofar as they are either existential participata or participatum.

7.8.But this other being Y needs to exist as well.
7.8.∀y[((→ε) ∈ y) v (y ∃→ z)]
7.9.If all beings participate existentially to infinity (for all Phi, x existentially participates in x₁…x(n)), the whole series would not exist (all Phi would not possess existence).
7.9.∀ψ(x ∃→ x₁…x(n)) → (∀ψ → ~(→ε))
7.9.1.Assume: All beings existentially participate in each other to infinity.
7.9.1.Assume: ∀ψ (x ∃→ x₁…x(n))
7.9.2.Existence either proceeds from intrinsic or extrinsic principles.
7.9.2.∀x[((→ε) ∈ x) v (x ∃→ y)]
7.9.3.All existentially participating beings are identical with set of all existing beings.
7.9.3.∀(x₁…x(n)) ↔ ψ
7.9.4.All beings would then existentially participate in themselves.
7.9.4.ψ E→ ψ
7.9.5.This implies that all beings cannot have existence as an intrinsic property.
7.9.5.~((→ε) ∈ ψ)
7.9.6.So, the set of all beings would not possess existence.
7.9.6.ψ → ~(→ε)
7.9.7.Therefore, if all beings were to existentially participate in each other, the whole set would not exist.
7.9.7.*** ψ(x ∃→ x₁…x(n)) → (ψ → ~(→ε))
7.10.Thus, there exists one being Θ, whose proper attribute is “to exist” in order to account for the existential participation of all other beings.
7.10.1.In all beings X which are not Θ, there is a real distinction between their existing and proper attributes.
7.10.1.∀x[(x~= Θ) → ((∃y(∀z(z∈x → ~(z∈y)))) ^ (z~=(→ε)))]
8.There is at least one entity X which “changes,” such that this change involves a subset (“accident”) of X, W, changing into Z via existential participation of xC in Y.
8.⊢ ∃x(xC({w} z ∃→ y))
8.1.X and Y are unique entities which possess elements W and Z, respectively.
8.1.∃x∃y((w ∈ x) ^ (z ∈ y) ^ (w ~=z))
8.2.Definition of substantial change: If a change occurs where all the proper subsets of X are eliminated, it becomes identified with whatever proper aspects it now possesses.
8.2.[[xC({w, z} ∃→ y] ^ [∀x∀y[∀z(z∈x ↔ z∈y)] ↔ x=y]] -> xCy
8.3.*The function participates in Y, as the change is a function of the whole X insofar as it acquires or loses an accidental aspect.

9.The change from one state to another in any entity X caused by Y implies a proper aspect in the changing entity to assume the new property (a potentiality) and a proper aspect in the changer to bring about this change (an actuality).
9.xC({w, z} ∃→ y) → ((p(z) ∈ x) ^ (a(z) ∈ y))
9.1.An “actualized” property Z in a being Y is defined as one where there exists some Z which belongs to Y and Z existentially participates in Y.
9.1.:= (∃z(z ∈ y)) ↔ (a(z) ∈ y)
9.2.An “potential” for Z in X is defined as there existing some Z, such that X might acquire Z via existential participation in Y.
9.2.:= [[∃z(xC{z}) ∃→ y] v (a(z) ∈ x)] ↔ (p(z) ∈ x)
9.2.1.A property which is actualized in any entity implies that the entity has a potential for that property.
9.2.1.(a(z) ∈ y) → (p(z) ∈ y)
9.3.It is not true that if some Z belong to Y as a proper aspect that Z is actualized in Y.
9.3.~[(z ∈ y) → (a(z) ∈ y)]
9.4.If there is an X such that X existentially participates in Y, a potential to exist belongs to X.
9.4.x(x ∃→ y) → (p(x→ε) ∈ x)
9.4.1.If X existentially participates in Y, then X has actual existence.
9.4.1.(x ∃→ y) → (a(x→ε) ∈ x)
9.4.2.*** X has a potential to exist.
9.4.2.p(x→ε) ∈ x
9.5.In all entities X, except Θ, there is a real distinction between their existence and their proper aspects.
9.5.∀x[(x~= Θ) → (∃y(∀z(z∈x → ~(z∈y)))) ^ (z~=(→ε))]
9.6.Existence is the actuality of any being whatsoever.
9.6.(→ε) ↔ a(ψ)
9.7.Essence is the basic principle of potentiality in any being whatsoever.
9.7.∀x[(p(x→ε) ∈ x) ↔ (∃y(∀z(z∈x → ~(z∈y)))) ^ (z~=(→ε))]
9.7.1.If one being were such that its proper attribute were “to exist,” it would be absolutely unlimited and possess all possible existential perfections (understood as all actualities of Phi).
9.7.1.((→ε) ∈ x) → (a(ψ) ∈ x)
9.7.2.If some X existentially participates in Y, then some potential to exist belongs to X.
9.7.2.x(x ∃→ y) → (p(x→ε) ∈ x)
9.7.3.For all X, either existence is intrinsic or extrinsic.
9.7.3.∀x [((→ε) ∈ x) v (x ∃→ y)]
9.7.4.For all X, not being identical to Theta implies that X existentially participates in some Y.
9.7.4.∀x[(x~= Θ) → (x ∃→ y)]
9.7.5.There exists some X not equal to Theta.
9.7.5.∃x ~= Θ
9.7.6.For all X, not being identical to Theta is true if and only if a potentiality to exist belongs to X.
9.7.6.∀x[(x~= Θ) ↔ (p(x→ε) ∈ x)]
9.7.7.But this potentiality to exist belongs to X if and only if X has any proper attribute Phi which is not equivalent with “existence.”
9.7.7.*** ∀x[(p(x→ε) ∈ x) ↔ (∃y(∀z(z∈x → ~(z∈y)))) ^ (z~=(→ε))]
10.Every being is “good” or perfected insofar as it possesses existence [For all X, if and only if X exists, for X, there exists some potential Y which belongs to X and actualization of Y belongs to X, and there exists some entity Z such that all Y belong to X and not to Z and Y is not equivalent with existence].
10.∀x[(x → (→ε)) ↔ x((∃p(y) ∈ x) ^ (∃a(y) ∈ x) ^ (∃z(∀y(y∈x → ~(y∈z)))) ^ (y~=(→ε))]
10.1.Assume X exists.
10.1.Assume: x → (→ε)
10.2.If actual existence belongs to X, then potential existence belongs to X.
10.2.(a(x→ε) ∈ x) → (p(x→ε) ∈ x)
10.3.Actual existence belongs to X.
10.3.a(x→ε) ∈ x
10.4.Potential to exist belongs to X.
10.4.p(x→ε) ∈ x
10.5.Potential existence is convertible with having a proper attribute which is not identical with existence.
10.5.∀x[(p(x→ε) ∈ x) ↔ (∃y(∀z(z∈x → ~(z∈y)))) ^ (z~=(→ε))]
10.6.There exists some proper attribute Z of X which is not existence.
10.6.∃y(∀z(z∈x → ~(z∈y)))) ^ (z~=(→ε))
10.7.Therefore, X existing implies that X has potential to exist, and that X actually exists, and that there is some potential existence of X such that Y does not also possess the same potential as X.
10.7.(x → (→ε)) → x((p(x→ε) ∈ x) ^ (a(x→ε) ∈ x) ^ ∃y(∀p(x→ε)(p(x→ε) ∈x → ~(p(x→ε) ∈y)))) ^ (p(x→ε) ~=(→ε))
10.8.But this can be generalized that for all X, if and only if X exists, for X, there exists some potential Y which belongs to X and actualization of Y belongs to X, and there exists some entity Z such that all Y belong to X and not to Z and Y is not equivalent with existence (which is at least the potential of X to exist) [This does not apply to Theta – Theta, as will be shown more fully, possesses fullness of existential perfection and hence is supremely good although not “actualizing” any essence other than its proper attribute of existing, which is a difference from other beings but which is not a potentiality or limitation on its being.]
10.8.*** ∀x[(x → (→ε)) ↔ x((∃p(y) ∈ x) ^ (∃a(y) ∈ x) ^ (∃z(∀y(y∈x → ~(y∈z)))) ^ (y~=(→ε))]
10.9.*Final Causality: Every being might be characterized inasmuch as any proper aspect of its being is actualized or unactualized.

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