Gary, List,
A quick note synthesizing why predication in propositions is inherently indexical — and, in Peirce’s vocabulary, at least minimally indexical–symbolic. * Model-theoretic core (Tarski; Hodges): Predication P(x) is only truth-evaluable relative to an interpretation/model M and a variable assignment s. An open formula is satisfied by a sequence only relative to M and s; a sentence is true in M if satisfied by all sequences in M. Without the index (M, s), predication lacks a determinate extension. See Tarski, The Concept of Truth in Formalized Languages (1935/1956); On the Concept of Logical Consequence (1936). For a modern treatment, see Hodges, Model Theory (1993). * Context and domain restriction (Stanley & Szabó; Bach; von Fintel): Ordinary predicates tacitly depend on a contextually supplied domain/standard/comparison class. The extension of tall, local quantifiers, and many gradable predicates is fixed only after a contextually restricted domain is provided. See Stanley & Szabó, On Quantifier Domain Restriction (1999); Bach, The Myth of Semantic Representations (2001); von Fintel, Restrictions on Quantifier Domains (1994 ms). * Indexicals and parameters (Kaplan; Lewis; Montague; Kratzer): Even non-overtly indexical predicates are evaluated relative to indices: worlds, times, speakers, standards. Kaplan (Demonstratives, 1989) makes explicit the role of contexts and characters; Lewis (Index, Context, and Content, 1980) formalizes multi-dimensional indices; Montague Universal Grammar (1970) and Kratzer’s modal/conditionals work show world–time (and more) parameters as constitutive of predicate interpretation. * Information states and dynamics (Heim; Kamp): Predication updates an anaphorically and situationally constrained information state; felicity and truth conditions are state-indexed. See Heim, The Semantics of Definite and Indefinite Noun Phrases (1982); Kamp & Reyle, From Discourse to Logic (1993). * Proof-theoretic and type-theoretic context (Martin-Löf; Dummett): Judgments (and so predicate application-as-typing) are meaningful only in a context Γ. The well-formedness and applicability of A(x) are Γ-indexed. See Martin-Löf, Intuitionistic Type Theory (1984); Dummett, The Logical Basis of Metaphysics (1991). * Categorical/topos-theoretic truth (Lawvere; Mac Lane & Moerdijk): Predicates are arrows into Ω in a given topos, often evaluated at “stages.” Truth and subobject classifiers are internal to, and hence indexed by, the ambient categorical structure. See Lawvere, ETCS (1964) and hyperdoctrines (1969); Mac Lane & Moerdijk, Sheaves in Geometry and Logic (1992). * Structuralism in mathematics (Benacerraf; Shapiro; Resnik): Properties are defined relative to structures; predication is structure-indexed rather than absolute set-membership. See Benacerraf, What Numbers Could Not Be (1965); Shapiro, Philosophy of Mathematics: Structure and Ontology (1997); Resnik, Mathematics as a Science of Patterns (1997). Putting this together: across semantics, logic, and foundations, predication’s extension is fixed only relative to indices — models and assignments, worlds/times, domains/standards, discourse states, proof contexts, categorical stages, or mathematical structures. That is precisely what “indexicality” amounts to. In Peirce’s terms, because the symbol’s interpretant and object are fixed via such context-anchoring parameters, predication is at least minimally indexical–symbolic: symbolic in its rule-governed form, but irreducibly indexed to a context that determines its application. Cited from Julius because apparently the burden of proof is on the one who claims what is logically necessary (that predication is minimally indexical-symbolic). Anyway, to suggest that predication is not "indexical" (it's not really relevant how much you want to quote Peirce, here, because the logical function of predicates demands it — as can be seen above — is a tiny irritation but one I want to resolve with this post). I consider it resolved - regardless of replies (that is, I know that such has to be secondness in thirdness? — indexical-symbolic, or some-such hybrid, which is only to say that it functions indexically). Plenty agree and those who do, are, as far as I can tell, entirely correct. Best Jack ________________________________ From: [email protected] <[email protected]> on behalf of [email protected] <[email protected]> Sent: Friday, September 5, 2025 4:00 PM To: [email protected] <[email protected]> Subject: RE: [PEIRCE-L] Propositions, Truth, and Experience (was Will and Belief) Perhaps I should apologize for the link I am about to post here, since it may be interpreted by some as a comment on the dialogue between the two main participants in this thread. It was written several years ago, so it is definitely not a response to what has been said in this thread. It does, however, throw some light on the relationship between reality and language, which I take to be an underlying subject of it. https://gnusystems.ca/TS/snc.htm#srd Love, gary Coming from the ancestral lands of the Anishinaabeg } What's good for the gorse is a goad for the garden. [Finnegans Wake, 450] { substack.com/@gnox<https://substack.com/@gnox> }{ Turning Signs<https://gnusystems.ca/TS/>
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