PET Symbol Dictionary

This page defines all symbols used in the PET axiom system (ax1–ax14). For the axioms themselves, see PET Axioms ax1–ax14. For derived results, see PET Theorems.

Entities and Variables

Symbol	Name	Meaning	Technical context
G	God	The distinguished divine entity	Distinguished constant
W	The World	Totality of all finite/created entities	Distinguished constant
\(G_n\)	Necessary divine aspect	The abstract, unchanging divine nature that exists in every possible world	Component of dipolar decomposition (ax11)
\(G_c\)	Contingent divine aspect	God’s concrete experience, which varies depending on which world exists	Component of dipolar decomposition (ax11)
\(G_c(w_i)\)	Subworld divine experience	God’s contingent experience specific to subworld \(w_i\)	Functional structure added in strengthened ax11 (lines 3–4)
\(R\)	God’s self-knowledge	The set of true propositions about God	Defined so that ax12 is tautological by design; substantive work shifts to ax14
\(p, q\)	Propositions	Statements that can be true or false	Propositional variables
\(x, y\)	Entities	Parts of God or the world	Individual variables
\(w_i\)	Subworld	A part of the world W (i.e., \(w_i \leq W\))	Used in ax11 to index divine experience

Relations and Predicates

Symbol	Name	Meaning	Technical context
\(\leq\)	“is part of”	Mereological parthood: reflexive, transitive, antisymmetric	Mereology (part-whole logic)
\(<\)	“is proper part of”	\(x \leq y\) and \(y \nleq x\) (part of, but not identical to)	Derived from \(\leq\)
\(P(x, y)\)	“x is present to y”	A relation of immediate awareness or access	Primitive relation (axiomatically introduced, not further reduced)
\(S(x, y)\)	“x sustains y”	y’s continued existence depends on x	Primitive relation
\(\text{Pos}(\varphi)\)	“φ is a positive property”	A perfection in Gödel’s sense	From Gödel’s ontological framework; listed but unused in ax1–ax14
\(\text{claim}(p)\)	“p is claimed divine”	A human claim that proposition p is divinely revealed	Introduced in ax14 (Revelation Claims Test)

Logical Operators

Symbol	Name	Meaning	Technical context
\(\Box\)	Necessarily	True in every possible world	Modal logic S5
\(\Diamond\)	Possibly	True in at least one possible world	Modal logic S5
\(\forall\)	For all	Every entity satisfies the condition	First-order logic (universal quantifier)
\(\exists\)	There exists	At least one entity satisfies the condition	First-order logic (existential quantifier)
\(\exists!\)	There exists exactly one	Exactly one entity satisfies the condition	First-order logic (uniqueness quantifier)
\(\wedge\)	And	Both conditions hold simultaneously	Propositional logic (conjunction)
\(\vee\)	Or	At least one condition holds	Propositional logic (disjunction)
\(\neg\)	Not	The condition does not hold	Propositional logic (negation)
\(\rightarrow\)	Implies / If…then	If the first condition holds, then the second must hold	Propositional logic (material conditional)
\(\oplus\)	Mereological sum	The combination of parts into a whole	Mereology
\(\in\)	Is a member of	The element belongs to the set	Set theory
\(\neq\)	Is not equal to	The two entities are distinct	Standard mathematics

Note

Modal logic S5 is the system where “possibly necessary” implies “necessary.” This means the accessibility relation between possible worlds is an equivalence relation: every world can “see” every other world. S5 is the standard choice for reasoning about metaphysical necessity.