Notation and Conventions

This page collects the notation used throughout the book. Mathematical symbols follow standard conventions where they exist; departures are noted.

Logic and Derivability

Symbol	Meaning
$\vdash$	Syntactic derivability: $\Gamma \vdash p$ means $p$ is provable from $\Gamma$
$\nvdash$	Not derivable: $\Gamma \nvdash p$ means $p$ is not provable from $\Gamma$
$\vDash$	Semantic satisfaction: $M \vDash \phi$ means model $M$ satisfies $\phi$
$\bot$	Contradiction / falsity
$\neg p$	Negation of $p$
$p \land q$	Conjunction
$p \lor q$	Disjunction
$p \to q$	Implication

Set and Type Notation

Symbol	Meaning
$\in$	Membership
$\subseteq$	Subset (inclusive)
$\emptyset$	Empty set
$\Pi(p)$	Type of witnesses for proposition $p$
$\pi : \Pi(p)$	Witness $\pi$ has type "evidence for $p$"

Equivalence and Identity

Symbol	Meaning
$=$	Equality in the stated mathematical setting; type-identity judgments are identified where used
$\cong$	Isomorphism: structure-preserving bijection with explicit inverse
$\simeq$	Equivalence in the declared setting: categorical equivalence uses inverse functors up to natural isomorphism; HoTT passages use type equivalence
$\sim_S$	Scoped equivalence: $x \sim_S y$ means $x$ and $y$ are equivalent in scope $S$

Convention: A10/A16 operational equivalence records declare scope and the properties they support. Mathematical equivalences are interpreted under their stated objects, ambient theory and hypotheses.

Category Theory

Symbol	Meaning
$f : A \to B$	Morphism from $A$ to $B$
$g \circ f$	Composition: apply $f$ first, then $g$
$\text{id}_A$	Identity morphism on $A$
$F : \mathcal{C} \to \mathcal{D}$	Functor from category $\mathcal{C}$ to $\mathcal{D}$
$F \dashv G$	Adjunction: $F$ is left adjoint to $G$
$\text{Hom}(A, B)$	Set of morphisms from $A$ to $B$

Schemas and Signatures

Symbol	Meaning
$\Sigma$	Signature: types + predicates + constraints
$(\Sigma, I)$	Signature with constraint set $I$
$M \vDash (\Sigma, I)$	Model $M$ satisfies signature under constraints
$\Sigma \to \Sigma'$	Signature morphism; vocabulary inclusion is a special case
$T(U)$	Local theory at context $U$: $(\Sigma, I, L_U)$ including logic choice

A3 uses the schema shorthand $\Sigma=(T,P,I)$, including integrity constraints. The displayed theory notation $(\Sigma,I)$ exposes that constraint component separately; it does not add a second set of constraints. In comparisons of theories, the vocabulary and governing constraints must both be identified.

Contexts and Covers

Symbol	Meaning
$U, V, W$	Contexts (views, perspectives, local frames)
$\{U_i \to U\}$	Cover of $U$ by family of contexts
$s\vert_{U_i}$	Restriction of section $s$ to context $U_i$
$U_i \cap U_j$	Overlap of contexts (where gluing conditions apply)

Commitment and Provenance

Symbol	Meaning
$C$	Commitment set (propositions the system has asserted)
$C \nvdash \bot$	Commitment set is consistent
$(p, \pi)$	Witnessed assertion: claim $p$ with witness $\pi$
$\Gamma \vdash \pi : \Pi(p)$	Under context $\Gamma$, $\pi$ is evidence for $p$

Anchors (A1–A32)

Formal definitions are marked with anchor identifiers. Later chapters depend on these definitions and on the order in which they appear.

Numbering convention:

• Primary anchors: A1, A2, A3, ...
• Sub-anchors: A2b, A2c, A12b, etc. (refinements or components of a primary anchor)
• Status: formal (definition, written statement or construction), preformal (requirement awaiting a model), or engineering (proposed operational discipline)

The dependency graph is acyclic by construction. Appendix F contains the full anchor registry with formal statements and dependencies.

Touchstones (T1–T10)

The ten touchstones are recurring questions and worked examples, including constructed cases. They connect the definitions to operations a reader can examine. Their conclusions depend on the stated representation and hypotheses; they are not empirical proof that every system needs this architecture.

ID	Name	Domain	One-Line Description
T1	Contradiction	String	Query for impossible objects (primes divisible by 4)
T2	Reference	String	Same referent, different surface (morning star / evening star)
T3	Compositionality	String	Systematic generalization (SCAN-style "add jump")
T4	Exactness	String	Discrete quantities (counting, arithmetic)
T5	Negation/Absence	String	Category boundaries, exceptions (flying mammals)
T6	Predicate Invention	Relational	New dimension on demand ("puffy dress")
T7	Contextual Equivalence	Both	Same sometimes, not always (NYC vs New York City)
T8	Uncertainty/Value	Relational	Non-factual targets ("best restaurant")
T9	Schema Evolution	Relational	Vocabulary drift over time (employees → workers)
T10	Higher-Arity Events	Relational	N-ary meaning ("Alice introduced Bob to Carol")

Each touchstone maps to required artifacts and resolving anchors. The full registry appears in Appendix E.

General Conventions

Written statements, constructions and open problems retain their stated status. Appendix K.8 identifies the machine-checked declarations separately.

Logic choice: Different contexts may operate under different logics (classical vs. intuitionistic). When logic matters, it is stated explicitly. Local theory $T(U) = (\Sigma, I, L_U)$ includes the logic $L_U$.

Running examples: The fashion catalog domain (dresses, predicates like "puffy," equivalences like "blue" vs. "navy") serves as the primary worked example throughout.