.. meta::
   :description: Every symbol in Pet axioms ax1--ax14 defined: G (God), W (World), mereological parthood, S5 modal operators, presence, sustaining, and dipolar structure.
   :keywords: Pet symbols, notation, G God, W World, mereological parthood, S5 modal operators, presence predicate, sustaining, Gn, Gc, dipolar structure
   :author: Yah, Yas, everyone, LLoL as Laurence Loewe of Laodicea, ClaudeOp46Max, Anthropic, and Spirit of Boolean Truth
   :og:card:title: Pet Symbol Dictionary<br>Notation for ax1--ax14
   :og:card:description: Reference for every formal symbol in the Pet axiom system: entities, mereological relations, modal operators, and derived predicates.

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   OO :description: Complete symbol dictionary for PET axioms ax1--ax14, covering entities, mereological relations, modal operators, and derived predicates.
   OO :keywords: matheology, PET, symbol dictionary, notation, mereology, modal logic, formal symbols, God, World, parthood, S5, mathematical theology
   OO :og:card:title: PET Symbol Dictionary<br>Formal Notation ax1--ax14
   OO :og:card:description: Every symbol in the PET axiom system defined: God, World, mereological parthood, modal necessity, and all derived predicates.
   PP :description: Every symbol in Pet axioms ax1--ax14 defined: G (God), W (World), mereological parthood, S5 modal operators, presence, sustaining, and dipolar structure.
   PP :keywords: Pet symbols, notation, G God, W World, mereological parthood, S5 modal operators, presence predicate, sustaining, Gn, Gc, dipolar structure
   PP :og:card:title: Pet Symbol Dictionary<br>Notation for ax1--ax14
   PP :og:card:description: Reference for every formal symbol in the Pet axiom system: entities, mereological relations, modal operators, and derived predicates.

.. SOCIAL-CARD-REVIEW --- generated by Claude Opus 4.6, 2026-03-26
   dv_ClaOp46_PP_2026m03d26 --- max-effort rewrite, read full page.
   :description: 150 chars | :og:card:title: 42 chars (excl <br>)
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PET Symbol Dictionary
*****************************

This page defines all symbols used in the PET axiom system (ax1–ax14).
For the axioms themselves, see :doc:`/matheology/pet/axioms`.
For derived results, see :doc:`/matheology/pet/theorems`.


.. PET-SYMBOLS-CONTENT-START

Entities and Variables
========================

.. list-table::
   :widths: 10 20 45 25
   :header-rows: 1

   * - Symbol
     - Name
     - Meaning
     - Technical context
   * - **G**
     - God
     - The distinguished divine entity
     - Distinguished constant
   * - **W**
     - The World
     - Totality of all finite/created entities
     - Distinguished constant
   * - :math:`G_n`
     - Necessary divine aspect
     - The abstract, unchanging divine nature that exists in every possible world
     - Component of dipolar decomposition (ax11)
   * - :math:`G_c`
     - Contingent divine aspect
     - God's concrete experience, which varies depending on which world exists
     - Component of dipolar decomposition (ax11)
   * - :math:`G_c(w_i)`
     - Subworld divine experience
     - God's contingent experience specific to subworld :math:`w_i`
     - Functional structure added in strengthened ax11 (lines 3–4)
   * - :math:`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
   * - :math:`p, q`
     - Propositions
     - Statements that can be true or false
     - Propositional variables
   * - :math:`x, y`
     - Entities
     - Parts of God or the world
     - Individual variables
   * - :math:`w_i`
     - Subworld
     - A part of the world W (i.e., :math:`w_i \leq W`)
     - Used in ax11 to index divine experience


Relations and Predicates
==========================

.. list-table::
   :widths: 10 20 45 25
   :header-rows: 1

   * - Symbol
     - Name
     - Meaning
     - Technical context
   * - :math:`\leq`
     - "is part of"
     - Mereological parthood: reflexive, transitive, antisymmetric
     - Mereology (part-whole logic)
   * - :math:`<`
     - "is proper part of"
     - :math:`x \leq y` and :math:`y \nleq x` (part of, but not identical to)
     - Derived from :math:`\leq`
   * - :math:`P(x, y)`
     - "x is present to y"
     - A relation of immediate awareness or access
     - Primitive relation (axiomatically introduced, not further reduced)
   * - :math:`S(x, y)`
     - "x sustains y"
     - y's continued existence depends on x
     - Primitive relation
   * - :math:`\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
   * - :math:`\text{claim}(p)`
     - "p is claimed divine"
     - A human claim that proposition p is divinely revealed
     - Introduced in ax14 (Revelation Claims Test)


Logical Operators
====================

.. list-table::
   :widths: 10 20 45 25
   :header-rows: 1

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

.. PET-SYMBOLS-CONTENT-END