Algebraic Metaphysical Semantics Hasen Khudairi Abstract This paper argues that metaphysically fundamental truths ought to be defined within an algebraic language. In the first part of the paper, I provide examples of the algebraic structures used to define models of physical ontology (namely, quantum mechanics and field theory); the mathematical universe (set-theory); modal logic; and the metaphysics of consciousness. I outline, then, some explanatory desiderata concerning the relation between fundamental and derivative truths. I argue that a relation of apriori material implication, i.e. 'scrutability', cannot satisfy the relevant desiderata; and I propose in turn that – given the model-theoretic uniformity between fundamental modal truths and the derivative truths concerning mental representational states – a novel derivability relation can be specified. The relation is unique in having a purely model-theoretic characterization, and I examine the epistemic advantages accruing to the relation's model-theoretic profile. 1 Introduction A contemporary project in metaphysics endeavors to target the fundamental truths of the actual world; to account for the relations which obtain between such truths; and to account for our knowledge thereof (cf. Sider, 2011; Chalmers and Jackson, 2001; Chalmers, 2012; Paul, 2012; and Russell, ms).1 Sider argues that a proposition is fundamental iff it possesses a truth-condition in a 'metaphysical semantics'. A metaphysical semantics is stated in perfectly natural, because metaphysically joint-carving, terms for the sub-propositional entities which comprise the target proposition. The absolute joint-carving terms 1The approach might be interpreted as a metaphysical extension of the project of formal analysis in the late nineteenth and early twentieth centuries, whose proponents familiarly included Frege (1884; 1885); Moore (1899); Russell (1905); Wittgenstein (1921); Carnap (1928); and Stebbing (1932-1933). For further discussion, see the essays in Beaney (2007). 1 are purported to be fundamental because they are structural, and they are taken to include logical vocabulary (including quantifiers), metaphysical predicates such as the mereological parthood relation, and physical predicates. The fundamental-structural truths are purported to be ascertainable via abductive criteria on theory choice. Chalmers argues that the fundamental truths concern phenomenal consciousness, physics, and indexical notions. From the foregoing truths, all other types of truths are argued to be 'scrutable'; i.e., apriori derivable via material implication relations [i.e., (φ → ψ) iff (¬φ ∨ ψ)]. Epistemic truths are argued to track metaphysical truths, where the relation is codified by the epistemic interpretation of multi-dimensional intensional semantics. A formula is epistemically possible (negatively conceivable) iff nothing rules it out apriori (⋄ ⇐⇒ ¬¬). A formula is apriori iff it it is inconceivable for it to be false ( ⇐⇒ ¬⋄¬). A formula can receive its semantic values relative to two parameters, a context and an index. The context ranges over epistemic possibilities and the index ranges over metaphysical possibilities. The value of the formula relative to the context determines the value of the formula relative to the index. Thus – as long as the terms comprising the formula are 'super-rigid', and thus map to the same extension throughout epistemic and metaphysical modal space – conceivability and apriori scrutability can be a guide to metaphysical possibility. Paul argues that the fundamental constituents of reality are properties and relations, and that the fundamental 'world-building' relation is the mereological composition relation (op. cit.: 240-242). Russell argues that the metaphysical semantics appealed to in Sider's account of fundamental-structural truth ought to be regimented using the language of category theory.2 2Cf. Author (ms), for an account of how – via a category-theoretic semantics inspired by Russell's (op. cit.) and Pettigrew's (ms) presentations – mental images, as represented by typed arrows, can be concatenated into propositional form. The category-theoretic semantics can thereby account both for the structure of phenomenal representation and for the relationship between mental imagery and propositional imagination. 2 In this paper, I endeavor to give content to the notion of a metaphysical semantics, and I argue that the metaphysical semantics for fundamental truths ought to be regimented in an algebraic language. Algebraic spaces are availed of in order to countenance the structure of space-time in physics; models of set-theoretic languages; the metaphysics of phenomenal properties; and the model-theory of modality. By contrast to Russell's proposal that structuralfundamental truths ought to be regimented within category theory, the identity of the elements comprising algebraic structures is a crucial aspect of the theories for which they provide a model; the elements are flexible enough to elide the distinction between formulas and terms; and they are thereby flexible enough to elide operators on the formulas with functions on the terms. By contrast to Chalmers' proposal that physical truths, phenomenal truths, and indexical truths are distinct sets of truths comprising a class of base truths, the present proposal targets a more fundamental level of reality, by targeting the fundamental language common to each set of truths in the base. By contrast, finally, to Paul's proposal that the fundamental world-building relation is mereological parthood, the proposal that the language of the fundamental class of base truths is algebraic is corroborated by naturalistic considerations: In mathematics and physics, models of physical theories and set-theory are actually algebraic in form (cf. Halvorson, 2006; Ruetsche, 2011; Jech, 2002; and Bell, 2005). In Section 2, I provide examples of the algebraic structures used to define models of physical ontology (namely, quantum mechanics and field theory); the mathematical universe (set-theory); modal logic; and the property theory for the metaphysics of consciousness. In Section 3, I examine the status of the derivation of non-fundamental truths from the algebraic base truths. I provide reasons adducing against the contention that truths unique to the higher-level 3 sciences such as neuroscience, economics, and sociology are scrutable from the class of fundamental truths. I argue, however, that some truths are yet derivable from the base, where the derivation relation is novel in virtue of being purely model-theoretic. Because possible worlds model theory is availed of in order to characterize both the structure of unconscious perceptual representational states and the speech acts and informational background in natural language semantics, truths about perceptual representation and the values of linguistic operators are model-theoretically derivable from fundamental modal truths. Section 4 provides concluding remarks. 2 Algebraic Metaphysical Semantics for Fundamental Truths In this section, I define the elementary properties of Boolean algebras, and examine four extensions of Boolean algebras to the models of set theory, metaphysical property theory, quantum mechanical kinematics and field theory, and to modal logic.3 2.1 Boolean Algebra A lattice is a non-empty, partially ordered set. Each two element subset has a supremum or join (x ∨ y) and an infimum or meet (x ∧ y). The top element of a bounded lattice is denoted 1, and its bottom element is denoted 0. A lattice is complete if every subset has an infimum and a supremum (Bell, op. cit.: 2). A Heyting algebra is a bounded lattice such that for all element pairs, x,y, x → y iff there is a z such that z ≤ x → y iff z ∧ x ≤ y (3). Pseudocomplementation is 3The material here outlined follows the presentation in Jech (op. cit.) and Bell (op. cit.). 4 an operation mapping each x in the algebra to an element, x' = x → 0 (Bell, op. cit.). A Boolean algebra is a Heyting algebra such that pseudocomplements are complements (x ∨ x' = 1). The algebra has operators, +, •, and –, which satisfy the properties of commutativity (u + v = v + u), associativity [u + (v + w) = (u + v) + w], distributivity [u • (v + w) = u • v + u • w], absorption [u • (u + v) = u], and complementation (u + –u = 1) (Jech, 78). A Lindenbaum algebra is a Boolean algebra satisfying the following operations: With [x] denoting an equivalence class and x an element variable, [φ] + [ψ] = [φ ∨ ψ], [φ] • [ψ] = [φ ∧ ψ], –[φ] = [¬φ], 0 = [φ ∧ ¬φ], and 1 = [φ ∨ ¬φ] (Jech, 79). 2.2 Property Theory and Set Theory Two crucial generalizations of Boolean algebras have been to metaphysics and to mathematics. As a model of metaphysical property theory, the elements of Boolean algebra can be interpreted as phenomenal terms or concepts, the extensions of which are phenomenal properties. With regard to the interaction between Boolean models and foundational mathematical languages, the Stone Representation theorem states, in particular, that every Boolean algebra is isomorphic to an algebra of sets.4 Boolean-valued models of set-theory are crucial, furthermore, to the notion of set-forcing extensions, which have familiarly been availed of in order to prove the independence of the generalized continuum hypothesis – 2אα = אα+1 – from the axioms of Zermelo-Fraenkel set theory (cf. Cohen, 1963, 1964; Kanamori, 2008; and Jech, op. cit.: ch. 14). 4For further discussion, see Stone (1936). 5 2.3 Quantum Mechanics and Quantum Field Theory Boolean algebras also play an ineliminable role in contemporary physics. For example, in the kinematics of quantum mechanics, a separable Hilbert space is a space of countable complex-valued vectors representing physical magnitudes (A,B, . . . ). The inner product of a Hilbert space is a linear functional on vectors which specifies their expectation values. Density operators, f , on Hilbert space map real-valued vectors to complex numbers. The operators are self-adjoint, such that for all one-object categories with a homomorphism, a, from a target category to its underlying set, there is a unique homomorphism, b, such that a is isomorphic to b (cf. Awodey (2006 10, 180-181)); linear [f(A + B) = f(A) + f(B)]; non-negative [f(A) ≥ 0]; normed [f(T) = 1]; and countably additive [if A and B are orthogonal, then f(A ∪ B) = f(A) + f(B)] (cf. Ruetsche, op. cit.: 21-23). In quantum field theory, a unital algebra is closed under the commutative and associative operator, +; scalar multiplication by complex numbers, s.t. for all vectors A,B, and complex numbers cn: c1(A + B) = c1A + c1B; (c1 + c2)A = c1A + c2A; c1(c2A) = (c1c2)A; (c1A)B A(c1B) = c1(AB); and satisfies multiplicative identity, i.e., for an element I, AI = IA = A (op. cit.: 74). A norm on an algebraic model is a function assigning a non-negative real number to each element of the algebra (op. cit.: 75-76). A sequence vn of elements is a 'norm-wise Cauchy sequence' iff for all elements > 0, there is a natural number Ne s.t. ||vi vj|| < e, for all i,j > Ne (76). The model is complete with respect to a norm iff the limit of every norm-wise Cauchy sequence of elements is itself an element (op. cit.). A C*-algebra is defined as an algebra over the complex numbers, complete with respect to a norm, and s.t. ||A x A|| = ||A||2 and ||AB|| ≤ ||A||||B|| (76-77). A von Neumann algebra for quantum fields interprets the 6 elements in a C* algebra as Hilbert space operators whose convergence norm is a topology (op. cit.: sec. 4.5; see also Halvorson, op. cit: 1.1-1.11). 2.4 Modal Algebra Finally, in the topological semantics for modal logic, a frame is comprised of a set of points in topological space, a domain of propositions, and an accessibility relation: F = 〈X, D, R〉; X = (Xx)x∈X ; and R = (Rxy)x,y∈X iff Rx ⊆ Dx x Dx, s.t. if Rxy, then ∃o⊆X, with x∈o s.t. ∀y∈o(Rxy), where points accessible from a privileged node in the space are said to be open.5 A model defined over the frame is a tuple, M = 〈F,V〉, with V a valuation function, such that: ∀x,y∈X[x∈Vx(P) ∧ Rxy → y∈Vx(P)]. Necessity is interpreted as an interiority operator on the space: M,x φ iff ∃o⊆X, with x∈o, such that ∀y∈o M,y φ. In modal algebra, the topological Boolean algebra, A, is formed by taking the powerset of the topological space, X, defined above; i.e., A = P(X). The domain of A is comprised of formula-terms – eliding propositions with names – assigned to elements of P(X). The top element of the algebra is denoted '1' and the bottom element is denoted '0'. We interpret modal operators, f(x), – i.e., intensional functions in the algebra – as both concerning topological interiority, as well as reflecting metaphysical possibilities. A modal-valued algebraic 5The material here follows the presentation in Lando (2015). See McKinsey and Tarski (1944) and Henkin et al. (1971), for further details. 7 structure has the form, F = 〈A, DP (X), ρ〉, where ρ is a mapping from points in the topological space to elements or regions of the algebraic structure; i.e., ρ : DP (X) x DP (X) → A. A model over the modal topological Boolean algebraic structure has the form M = 〈F, V〉, where V(a) ≤ ρ(a) and V(a,b) ∧ ρ(a, b) ≤ V(b). For all xx/a,φ,y∈A: f(1 = 1); f(x ≤ x); f(x ∧ y) = f(x) ∧ f(y); V(a, a) > 0; V(a, a) = 1; V(a, b) = V(b, a); V(a, b) ∧ V(b, c) ≤ V(a, c); V(a = a) = ρ(a, a); V(a, b) ≤ f[V(a, b)]; V(¬φ) = ρ(¬φ) – f(φ); V(⋄φ) = ρφ – f[– V(φ)]; V(φ) = f[V(φ)]. 3 Model-Theoretic Derivability In this section, I examine, finally, the nature of the relevant notion that all truths can be derivable from a base class of fundamental truths. I proffer a particular and a general issue for the approach to derivative and fundamental truths which explains the derivation relation via the notion of scrutability, i.e., the apriori material conditional (cf. Chalmers, op. cit.). The particular issue concerns whether mathematical truths ought to be fundamental, or whether they must be scrutable from physical truths. The general issue concerns the details with 8 regard to whether the truths comprising theories in the higher-level sciences are scrutable from the low-level, fundamental truths comprising the base. I argue that – while the disunity of the sciences remains a live issue – algebraic metaphysical semantics is novel in being able to explain the derivation of the truths about mental representational states from the fundamental truths defined in the algebraic language. The relevant notion of derivability need neither be apriori; take the form of the material conditional; nor be defined as computability (i.e., the equivalence class comprised of partial recursive and lambda-definable functions, as well as the transition functions of finite, discrete-state automata such as Turing machines). Rather, the derivation relation is purely model-theoretic: Because the model-theory of modal logic is defined in algebraic languages, there is prima facie justification to believe that modal truths are fundamental. Because possible worlds model theory is availed of in order to countenance the structure of the truths about mental representational states – such as unconscious perceptual representations, and speech acts in natural language semantics – truths about mental representation are thereby derivable from the fundamental modal truths. 3.1 The Fundamentality of Mathematical Truth There are at least two considerations adducing in favor of the fundamentality of mathematical truths, such that mathematical truths need not be derivable from truths about fundamental physics. The first consideration concerns the disparity between the nature of truth in the foundations of mathematics – e.g., in non-constructive languages such as classical set-theory and constructive languages such as homotopy type theory – and the mathematical vocabulary which is prevalent in physical theories. One major dispute in the setting of set-theory 9 concerns the status of mathematical truth, given the undecidability of propositions such as the continuum hypothesis. A multiverse conception of mathematical truth is in one sense relativist, by arguing that it is innocuous for the truth-value of the continuum hypothesis to vary between distinct extensions of ground models of ZF (cf. Hamkins, 2012). By contrast, a proponent of a cumulative hierarchical conception of the universe of sets will endeavor to augment the theory with large cardinal axioms, such as the existence of a proper class of Woodin cardinals, which will hold invariantly in all set-forcing extensions of the multiverse, and thus provide the foundations for a monadic conception of mathematical truth (cf. Woodin, 2010).6 The status of mathematical truth might thus be unique, because the content of physical truths – such as a Langrangian equation, which codifies the difference between the total kinetic energy of a system and the total potential energy of the system – is orthogonal to the content of the relevant large cardinal axioms and the truth-value of undecidable mathematical propositions. The second consideration adducing against the derivation of mathematical truths from physical truths is more familiar, and targets the status of mathematical ontology. While arguments for platonism can appeal to the indispensability of reference to mathematical objects in physical theories,7 the existence of necessarily non-concrete objects such as numbers, functions, and sets still 6Large cardinal axioms can then be defined as follows. ∃xΦ is a large cardinal axiom, because: (i) Φx is a Σ2-formula; (ii) if κ is a cardinal, such that V |= Φ(κ), then κ is strongly inaccessible; and (iii) for all generic partial orders P∈Vκ, VP |= Φ(κ); INS is a non-stationary ideal; A G is the canonical representation of reals in L(R), i.e. the interpretation of A in M[G]; and L(R)Pmax |= 〈H(ω2), ∈, INS , A G〉 |= 'φ'. P is a homogeneous partial order in L(R), such that the generic extension of L(R)P inherits the generic invariance, i.e., the absoluteness, of L(R). Thus, L(R)Pmax is (i) effectively complete, i.e. invariant under set-forcing extensions; and (ii) maximal, i.e. satisfies all Π2-sentences and is thus consistent by set-forcing over ground models (Woodin, ms: 28). 7See, e.g., Putnam (1971) and Colyvan (2001). 10 cannot be reduced to the truths about the entities postulated in fundamental physics. The argument is then conditional: If mathematical platonism is true, then truths about abstract entities cannot be derivable from truths about the concrete entities postulated in physics. There is thus prima facie support for the contention that mathematical truths are dissociable from physical truths; and the constitutive role of algebraic models in the foundations of mathematics adduces, further, in favor of the thesis that mathematical truths are metaphysically fundamental. 3.2 Explanatory Limits of Scrutability According to the scrutability conception of alethic derivability, the derivation relation is an apriori material conditional. Chalmers (op. cit.: 305) argues that theoretical truths in the higher-level sciences are scrutable from the theoretical truths concerning the fundamental lower-level sciences, in particular, physics. The scrutability relation is supposed to provide a 'transparent bottom-up explanation', where the explanations leave 'no residual mystery about what the higher-level facts are or about how the lower-level facts give rise to them' (op. cit.). Scrutability is further argued to be at least a necessary condition on the reductive explanation between higher-level and lower-level truths (op. cit., 307). Chalmers argues that one means by which scrutability can provide the epistemically transparent explanations alluded to in the foregoing is by targeting a mechanistic analysis of explanation (op. cit.). Thus, e.g., truths about neuroscience will be apriori derivable from truths about physics, by way of a threestep process: First, the 'higher-level explananda are expressed using functional concepts'; second, 'one tells a story about how low-level mechanisms play the relevant [functional] roles'; and third, one matches the roles in the lower-level 11 mechanistic story with the higher-level functional roles (op. cit.). Following Chirimuuta (2014), one can distinguish between three types of functional analysis pertinent to the nature of explanation. Let an 'A-minimal model' satisfy what has been referred to as a 'model-to-mechanism-mapping' requirement, according to which 'a model of a target phenomenon explains that phenomenon to the extent that (a) the variables in the model correspond to identifiable components, activities, and organizational features of the target mechanism that produces, maintains, and underlies the phenomenon, and (b) the (perhaps mathematical) tendencies posited among these (perhaps mathematical) variables in the model correspond to causal relations among the components of the target mechanism' (Kaplan, 2011: 347; Kaplan and Craver, 2011: 611). Let a 'B-minimal model' provide a less coarse-grained level of explanation, where 'the details of the system (those details that would feature in a complete causal-mechanical explanation of the system's behavior) are largely irrelevant for describing the behavior of interest . . . [and] Many different systems with completely different 'micro' details will exhibit identical behavior' (Batterman, 2002: 13). Finally, let a Chirimuuta-, or 'I-minimal model' target a level of explanation which is 'made precise and quantitative by reference to efficient coding principles' (op cit: 143). While A-minimal models target mechanistic explanations, and B-minimal models target non-causal and non-mechanistic, generalizable explanations, the efficient coding explanations in I-minimal models are purported to be more elucidatory, by privileging abstract, computational properties. The coding principles are claimed to be more explanatory in virtue of the conditions on their selection; in particular, their abductive utility (143144). It is unclear whether any of the foregoing functional analyses can satisfy 12 the desiderata proffered by Chalmers, concerning the matching of functional roles across the higherand lower-level sciences. One issue is thus that, even on an I-minimal construal of those functional roles according to which the latter are selected on the basis of their abductive utility, it is unclear why an abductively preferred function in physics – such as the complex-valued wave function in configuration spacetime – ought to be either correlated to, or provide an epistemically transparent explanation of, an abductively preferred function in neuroscience – e.g., the 'Normalization Model' as a canonical computation, according to which 'responses of neurons are divided by a common factor that typically includes the summed activity of a pool of neurons' (Carandini and Heeger, 2011: 51; Reynolds and Heeger, 2009).8 Even within a particular higher-level science such as cognitive neuroscience, the neural localization of psychofunctional properties is currently taken to be widely distributed. Thus, e.g., the retrieval of information from working memory stores is correlated to various brain areas, such as increased blood-oxygenation levels both in V4 and in the dorsal-lateral prefrontal cortex. It is also an open question whether the types of attentional mechanisms can be localized to the frontal eye fields; V4; dorso-lateral prefrontal cortex; lateral intraparietal cortex; or to a unique firing-rate of neural populations. There might thus not be a unique correlation between psychofunctional properties and neurofunctional properties. The absence of a unique correlation might undermine the likelihood that the scrutability of higher-level truths from lower-level truths can be accounted for by the proposed matching of functions. 8The Normalization formula is Ēi(n) = Ei(n) σ2+ ∑ i Ei(n) .σ is a constant that is relevant to the strength of sensory inputs as encoded by Ei(n). 13 3.3 Model-theoretic Derivability Despite the explanatory disunity of the higherand lower-level sciences, the algebraic conception of metaphysical semantics is able to provide a novel account of the derivability of some non-fundamental truths from the base class of fundamental truths. The non-fundamental truths at issue are truths about unconscious mental representational states and the semantic values of linguistic operators. Possible worlds model theory is availed of in Bayesian vision science and the program of natural language semantics in linguistics, in order to regiment the structural content of unconscious perceptual representational states and the speech acts of interlocutors given their shared informational background. In the remainder of this section, I will argue that – because the non-fundamental, representational truths are characterized by modal models, and because the algebraic model-theory of modal logic provides prima facie support for the contention that modal truths are fundamental – the model-theoretic uniformity of the target derivative and fundamental truths can explain in virtue of what the former are derivable from the latter. Bayesian vision science endeavors to answer the problem of under-determination, and thus to account for how retinal lightwave spectra can be transformed into perceptual states with accuracy-conditions. The current model for the constitutive conditions on perceptual representation takes perceptual accuracyconditions to be possible worlds: Given the possibility that light might be emanating from above or might be emanating from below, the visual system computes the likelihood that one of the possibilities is actual (cf. Mamassian et al, 2002; Burge, 2010; Rescorla, 2013). The calculation of which possibility is actual – referred to as the perceptual constancy – places a condition on the accuracy of the attribution of properties, such as boundedness and volume, to 14 distal physical particulars. In the program of natural language semantics in empirical linguistics, modality plays a crucial role, as well, in the characterization of the conversational background presupposed by a community of speakers (cf. Kratzer, 1977, 2012; Stalnaker, 1978). The update effects of various speech acts on the common ground have also been modeled as types of modal operators, whose semantic values are definable relative to an array of intensional parameters. Whereas the speech act of assertion is argued to provide a truth-conditional effect on the shared background of possibilities, the update effects of utterances involving epistemic and deontic modal vocabulary is taken, by contrast, not to be straightforwardly truth-conditional (cf. Yalcin, 2012; Moss, 2015). In the latter case, one might distinguish between the semantic values of subjective and objective deontic speech acts, such that – when it is claimed to be objectively obligatory that φ – the deontic modal can be defined relative to a context ranging over concrete situations (including an agent, location, and time), and – when it is claimed to be subjectively obligatory that φ – the deontic modal can be defined relative to a context as above and an index ranging over the agent's states of information. The philosophical significance of the role of modality in characterizing both perceptual and lingustic mental representations is that the higher-level truths about mental intentional states can be immediately derivable from the fundamental modal truths. The primary virtue of the derivation relation is that it is purely model-theoretic, rather than taking the more coarse-grained form of a material entailment relation stratifying formulas between the higherand lower-level sciences. A further virtue of model-theoretic derivability is that it is neutral with 15 regard to the epistemic profile of the derivation. The derivation is amenable (i) to being justified apriori – where apriority can be defined either extensionally as justification in the absence of experience, or intensionally as the inconceivability of the falsity of the relation's satisfaction; (ii) to being confirmed on the basis of evidence; (iii) and – consistently with the types of truth at issue – to being known by cognitive exercises which take the form of counterfactual presuppositions, where the latter are themselves translatable into modal operators (cf. Stalnaker, 1968, 2011; and Williamson, 2007: 156-158). 4 Concluding Remarks In this paper, I have endeavored to argue that – because algebraic structures are availed of in theories of the structure of space-time in physics, set-theoretic languages, and in the model-theory of modal logic – the metaphysical semantics for fundamental truths ought to be defined in an algebraic language. The limits of competing proposals concerning which truths ought to be fundamental were then examined. I outlined, then, some explanatory desiderata concerning the nature of the relation between fundamental and derivative truths, and I argued that a relation of apriori material implication, i.e. scrutability, could not satisfy the relevant desiderata. Finally, I proposed in turn that – given the model-theoretic uniformity between fundamental modal truths and the derivative truths concerning mental representational states – at least one derivability relation can be specified. 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