Multi-Dimensional Truthmaker Semantics Hasen Khudairi Abstract This paper endeavors to establish foundations for the interaction between hyperintensional semantics and multi-dimensional indexing. I examine the significance of the semantics, by developing three, novel interpretations of the framework. The first interpretation provides a characterization of the distinction between fundamental and derivative truths. The interaction between the hyperintensional truthmaker semantics and modal ontology is further examined. The second interpretation demonstrates how the elements of decision theory are definable within the semantics, and provides a novel account of the interaction between probability measures and hyperintensional grounds. The third interpretation concerns the contents of the types of intentional action, and the semantics is shown to resolve a puzzle concerning the role of intention in action. 1 Introduction Philosophical applications of multi-dimensional intensional semantics have demonstrated that an account of representation which is sensitive to an array of parameters can play a crucial role in explaining the values of linguistic expressions (Kamp, 1967; Kaplan, 1979); the role of speech acts in affecting shared contexts of information (Stalnaker, 1978; Lewis, 1980,a/1998; MacFarlane, 2005); the relationship between conceivability and metaphysical possibility (Chalmers, 1996); the limits of subjectivism in ethics (Peacocke, 2003); and the viability of modal realism (Russell, 2010). In order to circumvent issues for the modal analysis of counterfactuals (2012,a, 2012,b), and to account for the general notion of aboutness and a subject matter (2015), a hyperintensional, 'truthmaker' semantics has recently been developed 1 by Fine (2017a, 2017b). In this essay, I examine the status of multi-dimensional indexing in truthmaker semantics, and specify the multi-dimensional profile of the grounds for the truth of a formula (Section 2.2). I proceed, then, to outline three novel interpretations of the multi-dimensional, hyperintensional framework, beyond the interpretations of multiply indexed intensional semantics that are noted above. The first interpretation provides a formal setting in which to define the distinction between fundamental and derivative truths (Section 3.1). The second interpretation concerns the interaction between the multi-dimensional profile of the verifiers for a proposition, subjective probability, and decision theory (Section 3.2). Finally, a third interpretation of the multi-dimensional hyperintensional framework concerns the types of intentional action. I demonstrate, in particular, how multiply indexed truthmaker semantics is able to resolve a puzzle concerning the role of intention in action (Section 3.3). Section 4 provides concluding remarks. 2 Multi-dimensional Truthmaker Semantics 2.1 Intensional Semantics In his (1979), Evans endeavors to account for the phenomenon of the contingent apriori by distinguishing between two types of modality. In free logic, closed formulas may receive a positive, classical semantic value when the terms therein have empty extensions (op. cit.: 166). Informative identity statements – such as that 'Plotinus = the author of The Enneads' – are thus taken to be 'epistemically equivalent' to vacuously true identity statements – e.g., 'Plotinus = Plotinus' (op. cit.: 177). The apriority of the vacuously true identity statement 2 is thus argued to be a property of the informative identity statement, as well.1 However, the informative identity statement is contingent. For example, it is metaphysically possible that the author of The Enneads is Plato, rather than Plotinus. Evans argues that the foregoing 'superficial' type of contingency at issue is innocuous, by distinguishing it from what he refers to as a 'deep' type of contingency according to which a sentence is possibly true only if it is made true by a state of affairs (185). The distinction between the types of modality consists in that superficial contingency records the possible values of a formula when it embeds within the scope of a modal operator – e.g., possibly, x is red and possibly x is blue – whereas deep contingency records whether the formula is made true by a metaphysical state of affairs. In light of the approach to apriority which proceeds via the free-logical, epistemic equivalence of vacuous and informative identity statements, a formula may thus be apriori and yet superficially contingent.2 Evans (op. cit.: 183-184; 2004: 11-12) goes further and – independently developing work in multi-dimensional intensional semantics by Kamp (1967), Vlach (1973), and Segerberg (1973) – treats the actuality operator as a rigidifier, such that the value of actually φ determines the counterfactual value of possibly φ. 1A premise in the argument is that definite descriptions are non-referring, although – in free logic – still enable the sentences in which they figure to bear a positive, classical value. See Evans (op. cit.: 167-169). 2Evans' approach is defined within a single space of metaphysically possible worlds. However, one may define the value of a formula relative to two spaces: A space of epistemic possibilities and a space of metaphysical possibilities. By contrast to securing apriority by (i) eliding the values of informative and vacuous identity statements in a free logic within a single space of metaphysical possibilities, and then (ii) arguing that apriori identity statements are superficially contingent because possibly false, an alternative approach argues that an identity statement is contingent apriori if and only if it is (i) apriori, because the statement is necessarily true in epistemic modal space, while the statement is (ii) contingent, because possibly the statement is false in metaphysical modal space (cf. Chalmers, 2006). However, a point of convergence between the above approaches is that neither avails of the notion of deep contingency – i.e., truthmaking – in order to reconcile the apriority of a formula with the possibility of its falsehood. 3 Multi-dimensional intensional semantics provides a framework for regimenting the thought that the value of a formula relative to one parameter determines the value of the formula relative to another parameter. The semantics assigns truth-conditions to formulas, and semantic values to the formula's component terms. The conditions of the formulas and the values of their component terms are assigned relative to the array of intensional parameters. So, e.g., a term may be defined relative to a context; and the value of the term relative to the context will determine the value of the term relative to an index. For example, according to Kaplan (1979), an utterance's character is a mapping from the utterance's context of evaluation to the utterance's content. According to Stalnaker (op. cit.; 2004), having distinct functions associated with the value of an utterance provides one means of reconciling the necessity of a formula presupposed by speakers with the contingency of the values of assertions made about that formula. According to Chalmers (op. cit.), the value of a formula relative to a context, which ranges over epistemically possible worlds, determines the value of a formula relative to an index, which ranges over metaphysically possible worlds. According to Lewis (op. cit.), the context may be treated as a concrete situation ranging over individuals, times, locations, and worlds; and the index may be treated as ranging over information states relative to the context. According to MacFarlane (op. cit.), formulas may receive their value relative to a context ranging over two distinct agents; the context determines the value of an index ranging over their states of information; and the value of the formula may yet be defined relative to a third parameter ranging over the states of an independent, third assessor. Finally, in decision theory, the value of a formula relative to a context, which ranges over a time, location, and agent, constrains the value of the formula relative to a first index on which a space of the agent's 4 possible acts is built, and the latter will subsequently constrain the value of the formula relative to a second index on which a space of possible outcomes may be built. Generally, a proposition, P, is such that P is true in a Model, M, relative to a context, c, and an index, i: • M P iff ∀c∃c'JPKc,c ′ = 1 and ∃iJPKc,i = 1. 2.2 Truthmaker Semantics A hyperintensional, 'truthmaker' semantics has recently been developed by Fine (ms,a, ms,b).3 Truthmaker semantics has been applied, in order to explain the verification-conditions which ground the truth of parts of propositions, rather than of the propositions in their entirety.4 Truthmaker semantics is defined over a state space, F = 〈S, ⊏〉, where S is a set of states comprising a world, and ⊏ is a parthood relation on S comprising a partial order, such that it is reflexive (a ⊏ a), anti-symmetric [(a ⊏ b) ∧ ¬(b ⊏ a)], and transitive (a ⊏ b, b ⊏ c; a ⊏ c) (ms,a: 19). A proposition P ⊆ S is verifiable if P is non-empty, and is otherwise unverifiable (20). s,t ⊏ S are compatible if their fusion s ⊔ t is a possible state of S (op. cit.). s overlaps t if ∃u(u ⊏ S), s.t. u ⊑ s and u ⊑ t, denoted s ⊓ t (op. cit.). A model, M, over F is a tuple, M = 〈F,D,V〉, where D is a domain of closed formulas (i.e. propositions), and V is an assignment function mapping propositions P∈D to pairs of subsets of S, {1,0}, i.e. the verifier and falsifier of 3The logic for the semantics is classical. Fine (2014) develops a truthmaker semantics for intuitionistic logic. 4Anticipations of the theory of partial content are advanced by van Fraassen (1969) and Angell (1989; cf. Fine, 2015). Intensional interpretations of subject matters, which anticipate Fine's hyperintensional approach, are developed in Lewis (1988/1998) and Yablo (2014). 5 P, such that JPK+ = 1 and JPK− = 0 (35). The verification-rules in truthmaker semantics are then the following: s ⊢ P if s∈JPK+ (s verifies P, if s is a truthmaker for P i.e. if s is in P's extension); s ⊣ P if s∈JPK− (s falsifies P, if s is a falsifier for P i.e. if s is in P's anti-extension); s ⊢ ¬P if s ⊣ P (s verifies not P, if s falsifies P); s ⊣ ¬P if s ⊢ P (s falsifies not P, if s verifies P); s ⊢ P ∧ Q if ∃t,u, t ⊢ P, u ⊢ Q, and s = t ⊓ u (s verifies P and Q, if s is the fusion of states, t and u, t verifies P, and u verifies Q); s ⊣ P ∧ Q if s ⊣ P or s ⊣ Q (s falsifies P and Q, if s falsifies P or s falsifies Q); s ⊢ P ∨ Q if s ⊢ P or s ⊢ Q (s verifies P or Q, if s verifies P or s verifies Q); s ⊣ P ∨ Q if ∃t,u, t ⊣ P, u ⊣ Q, and s = t ⊓ u (s falsifies P or Q, if s is the state overlapping the states t and u, t falsifies P, and u falsifies Q); s exactly verifies P if and only if s ⊢ P if s∈JPK; s inexactly verifies P if and only if s ⊲ P if ∃s'⊏S, s' ⊢ P; and s loosely verifies P if and only if, ∀t, s.t. s ⊔ t, s ⊔ t ⊢ P (35-36). Differentiated contents may be defined as follows.5 A state s ⊑ S is differentiated only if s is the fusion of distinct parts, s.t. s = s1 ⊔ s2. s is thereby 5Fine (op. cit.: 8, 12) avails of product spaces in his discussion of content and subject matter, though we continue here to work with a single space for ease of exposition. 6 comprised of three parts: An initial state, s1; an additional state, s2; and a total state, s. The three states correspond accordingly to three contents: The initial content s1 ⊢ P1; the additional content, s2 ⊢ P2; and the total content, s ⊢ P1,2 (ms,b: 15). The subtraction of Q from P, P – Q, is defined such that p – q = p – (q ⊓ p) (17). Finally, subject matters may be defined as follows. A positive subject matter, p+, expresses a verifiable proposition, JPK+ (2021). A negative subject matter, p−, expresses a falsifiable proposition, JPK− (21). A comprehensive subject matter expresses the fusion of the subject matters both verified and falsified by the fusion of a number of states: p1,+,− = p1,+ ⊓ p1,− = 〈s ⊢ P ∧ s ⊣ P〉; p2,+,− = p2,+ ⊓ p2,− = 〈s ⊢ P2 ∧ s ⊣ P2〉; such that, p1,2,+,− = p1,2,+ ⊓ p1,2,− = 〈s ⊢ P1,2 ∧ s ⊣ P1,2〉 (op. cit.). A differentiated subject matter expresses the fusion of the subject matters that are either verified or falsified by the fusion of a number of states: p1,+/− = p1,+ ⊔ p1,− = 〈s ⊢ P ∨ s ⊣ P〉; p2,+,− = p2,+ ⊔ p2,− = 〈s ⊢ P2 ∨ s ⊣ P2〉; such that, p1,2,+/− = p1,2,+ ⊔ p1,2,− = 〈s ⊢ P1,2 ∨ s ⊣ P1,2〉 (op. cit.). Informally, propositions P and Q express the same subject matters, p and q, when the following conditions hold: P is exactly about Q if p = q; P is partly about Q if p and q overlap, such that ∃u⊏S(u ⊢ R); ∀s1,s2⊑S, s1 ⊢ P, s2 ⊢ Q; and u = s1 ⊓ s2, such that R = P ∩ Q; P is entirely about Q if p ⊆ q; and 7 P is about Q in its entirety if p ⊇ q (5). 2.3 Multi-dimensional Truthmaker Semantics In order to account for multi-dimensional indexing, we augment the model, M, with a second state space, S*, on which we define both a new parthood relation, ⊏*, and partial function, V*, which serves to map propositions in D to pairs of subsets of S*, {1,0}, i.e. the verifier and falsifier of P, such that JPK+ = 1 and JPK− = 0. Thus, M = 〈S, S*, D, ⊏, ⊏*, V, V*〉. The multi-dimensional hyperintensional profile of propositions may then be recorded by defining the value of P relative to two parameters, c,i: c ranges over subsets of S, and i ranges over subsets of S*. (*) M,s∈S,s*∈S* ⊢ P iff: (i) ∃csJPK c,c = 1 if s∈JPK+; and (ii) ∃is∗JPK c,i = 1 if s*∈JPK+ (Distinct states, s,s*, from distinct state spaces, S,S*, provide a multidimensional verification for a proposition, P, if the value of P is provided a truthmaker by s. The value of P as verified by s determines the value of P as verified by s*). The foregoing provides a multi-dimensional hyperintensional semantic framework within which to interpret the values of a proposition. In order to account for partial contents, we define the values of subpropositional entities relative again to tuples of states from the distinct state spaces in our model: s is a multi-dimensional exact truthmaker of P if and only if (*); s is a multi-dimensional inexact truthmaker of P if and only if ∃s'⊏S, s→ s', s' ⊢ P and such that 8 ∃cs′JPK c,c = 1 if s'∈JPK+, and ∃is∗JPK c,i = 1 if s*∈JPK+; s is a multi-dimensional loose truthmaker of P if and only if, ∃t, s.t. s ⊔ t, s ⊔ t ⊢ P: ∃cs⊔tJPK c,c = 1 if s'∈JPK+, and ∃is∗JPK c,i = 1 if s*∈JPK+. • JPKc,i is exactly about JQKc,i if f1−1[p c,i ⇐⇒ qc,i] (Suppose that the values of P and of Q are multi-dimensionally determined, as above. Then P is exactly about Q if there is a bijection between the multi-dimensionally individuated subject matters that they express); • JPKc,i is partly about JQKc,i if p and q overlap, s.t. ∃u⊏S, s.t. u ⊢ R, and ∀s1,s2⊑S, s1 ⊢ P, s2 ⊢ Q, and u = s1 ⊓ s2 such that R c,c = P ∩ Q. A neighborhood function, A, maps u to a state s* in i where s* ⊢ Rc,i. • JPKc,i is entirely about JQKc,i if pc,i ⇐ qc,i (Suppose that the values of P and of Q are multi-dimensionally determined. Then P is entirely about Q if there is a surjection from the subject matter of Q onto the subject matter of P); • JPKc,i is about JQKc,i in its entirety if pc,i ⇒ qc,i (Suppose that the values of P and of Q are multi-dimensionally determined. Then P is about Q in its entirety if there is an injection from the subject matter of P onto the subject matter of Q). 9 3 New Interpretations The multi-dimensional account of truthmaker semantics provides a general framework in which a number of interpretations of the state spaces at issue can be defined. The framework may accommodate, e.g., the so-called 'semantic' and 'metasemantic' interpretations of the framework. The semantic interpretation targets, as noted, the bearing of contextual parameters on the values of terms, and provides an account of validity relative to the spaces defined in the term's intensional parameters (cf. Kaplan, op. cit.). The metasemantic interpretation accommodates, by contrast, the update effects of contingently true assertions, with the necessary propositions of which speakers' shared information states might be comprised (cf. Stalnaker, op. cit.). The framework may further be provided an 'epistemic' interpretation, in order to countenance hyperintensional distinctions in the relations between conceivability, i.e. the space of an agent's epistemic states, and metaphysical possibility, i.e. the state space of facts (cf. Chalmers, op. cit.).6 In this section, I advance three novel interpretations of multi-dimensional semantics, as witnessed by the new relations induced by the interaction between multi-dimensional indexing and hyperintensional value assignments. The three interpretations concern (i) the distinction between fundamental and derivative truths; (ii) probabilistic grounding in the setting of decision theory; and (iii) the structural contents of the types of intentional action. 3.1 Fundamental and Derivative Truths The first novel interpretation concerns the distinction between fundamental and derivative truths. In the foregoing model, the value of the subject matter ex6Cf. Khudairi (2017) for further discussion. 10 pressed by a proposition may be verified by states in a first space, which determine, then, whether the proposition is verified by states in a second space. Allowing the first space to be interpreted so as to range over fundamental facts and the second space to be interpreted so as to range over derivative facts permits a precise characterization of the determination relations between the fundamental and derivative grounds for a truth. Suppose, e.g., that the fundamental facts concern the computational characterization of a subject's mental states, and let the fundamental facts comprise the first state space. Let the derivative facts concern states which verify whether the subject is consciously aware of their mental representations, and let the derivative facts comprise the second state space. Finally, let φ be a formula in an experimental task which expresses that there is a particular valence for the contrast-level of a stimulus. The formula's having a truthmaker in the first space – where the states of which range, as noted, over the subject's psychofunctional facts – will determine whether the formula has a truthmaker in the second space – where the states of which range over the mental representations of which the subject is consciously aware. If the deployment of some attentional functions provides a necessary condition on the instantiation of phenomenal awareness, then the role of the state of the attentional function in the first space in verifying φ will determine whether φ is subsequently verified relative to the second space. Intuitively: Attending to a stimulus with a particular value will constrain whether a truthmaker can be provided for being consciously aware of the stimulus. If the computational facts at issue are fundamental, and the phenomenal facts at issue are derivative, then a precise characterization may be provided of the multi-dimensional relations between the verifiers which target fundamental and derivative truths. 11 Note that hyperintensional truthmaker semantics is consistent with a necessitist modal ontology, according to which necessarily everything is necessarily something. An apparent tension might be thought to arise by augmenting a first-order formal language with both a truthmaker principle – to the effect that a proposition is true only if there is something which entails it [A → ∃x[∃y(x = y → A)]] – and the principle of the necessary necessity of being – which codifies the thought that necessarily everything is necessarily something [∀xy(x = y)]. The apparent tension is that conjoining the truthmaker principle to the necessity of being entails the necessity of truth (cf. Williamson, 2013: 400). The contingency of truth is belied by there necessarily being a (possible) object which serves as either a truthmaker or a falsifier for every proposition. However, the foregoing language is consistent with there being contingent truths; e.g., formulas comprised of non-logical vocabulary, such as that there are tigers. Williamson replies by arguing against the truthmaker principle (op. cit.). He observes, however, that a hyperintensional version of the truthmaker principle is consistent with the necessary necessity of being – A → ∃P(A because P) – which is an instance of the principle of the priority of being to truth: namely, that for all formulas πA, [Tπ(A) because A] (400-401). Consistently, then, with the necessary necessity of being, hyperintensional truthmaker semantics is one means of interpreting the explanatory relation – A 'because' B – in the principle of the priority of being to truth. 3.2 Decision Theory A second novel interpretation of multi-dimensional truthmaker semantics concerns the types of intentional action, and the interaction of the latter with decision theory. As noted in the foregoing, multi-dimensional intensional semantics 12 may be availed of in order to explain how the value of a formula relative to a context ranging over an agent and time will determine the value of the formula relative to an index ranging over a space of admissible, actions made on the basis of the formula, where the value of the formula relative to the context and first index will determine the value of the formula relative to a second index, ranging over a space of outcomes. One notable feature of the decision-theoretic interpretation is that it provides a natural setting in which to provide a gradational account of truthmaking. A proposition and its component expressions are true, just if they are verified by states in a state space, such that the state and its parts fall within the proposition's extension. In decision theory, a subject's expectation that the proposition will occur is recorded by a partial belief function, mapping the proposition to real numbers in the {0,1} interval. The subject's desire that the proposition occurs is recorded by a utility function, the quantitative values of which – e.g., 1 or 0 – express the qualitative value of the proposition's occurrence. The evidential expected utility of a proposition's occurrence is calculated as the probability of its obtaining conditional on an agent's action, as multiplied by the utility to the agent of the proposition's occurrence. The causal expected utility of the proposition's occurrence is calculated as the probability of its obtaining, conditional on both the agent's acts and background knowledge of the causal efficacy of their actions, multiplied by the utility of the proposition's occurrence. There are three points at which a probabilistic construal of the foregoing may be defined. One point concerns the objective probability that the proposition will be verified, i.e. the chance thereof. The second point concerns subjective probability with which a subject partially believes that the proposition will obtain. A third point concerns the probability that an outcome will occur, 13 where the space of admissible outcomes will be constrained by a subject's acts. An agent's actions will, in the third case, constrain the admissible verifiers in the space of outcomes, and thus the probability that the verifier for the proposition will obtain as an outcome.7 In order formally to countenance the foregoing, we define a probability measure on a state space, such that the probability measure satisfies the Kolmogorov axioms: normality [Pr(T) = 1]; non-negativity [Pr(φ) ≥ 0]; additivity [For disjoint φ and ψ[Pr(φ∪ψ) = Pr(φ) + Pr(ψ)]]; and conditionalization [Pr(φ|ψ) = Pr(φ∩ψ) / Pr(ψ)]. In order to account for the interaction between objective probability and the verification-conditions in truthmaker semantics, we avail, then, of a regularity condition in our earlier model, M, in which the assignment function, V, maps propositions P∈D to pairs of subsets of S, {1,0}, i.e. the verifier and falsifier of P, such that JPK+ = {0,1} and JPK− = 1 – P. In our gradational truthmaker semantics, a state, s, verifies a proposition, P, if the probability that s is in P's extension is greater than or equal to .5: s ⊢ P if Pr(s∈JPK+) ≥ .5. A state, s, falsifies a proposition P if the probability that s is in P's extension is less than .5 iff the probability that s is in P's anti-extension is greater than or equal to .5 s ⊣ P if Pr(s∈JPK−) ≥ .5 7A proponent of metaphysical indeterminacy might further suggest that the verifiers are themselves gradational; thus, rather than target the probability of a verifier's realization, the proponent of metaphysical indeterminacy will suggest that a proposition P is made true only to a certain degree, such that both of the proposition's extension and anti-extension will have nonnegative, real values. One objection to the foregoing account of metaphysical indeterminacy for truthmakers is, however, that the metalogic for many-valued logic is classical (cf. Williamson, 2014). A distinct approach to metaphysical indeterminacy is proffered by Barnes and Williams (2011), who argue that metaphysical indeterminacy consists in persistently unpointed models, i.e. a case in which it is unclear which among a set of worlds is actual, even upon filtering the set with precisifications. A proponent of metaphysical indeterminacy for probabilistic truthmaker semantics might then argue both that the realization of a verifier has a gradational value and that it is indeterminate which of the states which can verify a given formula is actual. 14 iff Pr(s∈JPK+) < .5. The subjective probability with regard to the proposition's occurrence is expressed by a probability measure satisfying the Kolmogorov axioms as defined on a second state space, i.e., a space whose points are interpreted as concerning the subject's states of information. The formal clauses for partial belief in truthmaker semantics are the same as in the foregoing, save that the probability measures express the mental states of an agent, by being defined on the space of their states of information. Finally, the interaction between objective and subjective probability measures in hyperintensional semantics may be captured in two ways. One concerns the interaction between the chance of a proposition's occurrence, the subject's partial belief that the proposition will occur, and the spaces for the subjects actions and outcomes. The formal clause for the foregoing will then be as follows: M,s ⊢ JPKc(c ′,a,o) > .5, where c ranges over the space of physical states, and a probability measure recording objective chance is defined thereon; c' ranges over the space of an agent's states of information, and the value of P relative to c' determines the value of P relative to the space of the agent's acts, a, where the latter determines the space of admissible outcomes concerning P's occurence, o. Thus, the parameters, c',a,o possess a hyperintensional multi-dimensional profile, whereas the space of physical states, c, constrains the verification of the proposition's occurrence, without determining the values of the subject's partial beliefs and their subsequently conceivable actions and outcomes. Directly accounting for the relation between c and c' – i.e., specifying a norm on the relation between chances and credences – is the second means 15 by which to account for how objective gradational truthmakers interact with a subject's partial beliefs about whether propositions are verified. Following Lewis (1980,b/1987), a candidate chance-credence norm may be what he refers to as the 'principal principle'.8 The principal principle states that an agent's partial belief that a proposition will be verified, conditional on the objective chance of the proposition's occurrence and the admissible evidence, will be equal to the objective chance of the proposition's occurrence itself: Prs(P | ch(P) ∧ E) = ch(P). 3.3 Intentional Action A third novel interpretation of multi-dimensional intensional semantics provides a natural setting in which to delineate the structural content of the types of intentional action. For example, the mental state of intending to pursue a course of action may be categorized as falling into three types, where intending-that is treated as a modal operator defined on an agent's space of states of information. One type targets a unique structural content for the state of acting intentionally, such that an agent intends to bring it about that φ just if the intention satisfies a clause which mirrors that outlined in the last paragraph: • JIntenton-in-Action(φ)Kw = 1 only if ∃w'JφK w′,c(=t,l),a,o = 1. A second type of intentional action may be recorded by a future-directed 8See Pettigrew (2012), for a justification of a generalized version of the principal principle based on Joyce's (1998) argument for probabilism. Probabilism provides an accuracy-based account of partial beliefs, defining norms on the accuracy of partial beliefs with reference only to worlds, metric ordering relations, and probability measures thereon. The proposal contrasts to pragmatic approaches, according to which a subject's probability and utility measures are derivable from a representation theorem, only if the agent's preferences with regard to a proposition's occurrence are consistent (cf. Ramsey, 1926). Probabilism states, in particular, that, if there is an ideal subjective probability measure, the ideality of which consists e.g. in its matching objective chance, then one's probability measure ought to satisfy the Kolmogorov axioms, on pain of there always being a distinct probability measure which will be metrically closer to the ideal state than one's own. 16 operator, such that an agent intends to φ only if they intend to pursue a course of action in the future, only if there is a world and a future time relative to which the agent's intention is satisfied: • JIntention-for-the-future(φ)Kw = 1 only if ∃w'∀t∃t'[t< t' ∧ JφK w′,t′ = 1]. Finally, a third type of intentional action concerns reference to the intention as an explanation for one's course of action. Khudairi (op. cit.) regiments the structural content of this type of intention as a modal operator which receives its value only if a hyperintensional grounding operator which takes scope over a proposition and an action, receives a positive semantic value. The formal properties of the grounding operator follows the presentation in Fine (2012,c), according to which the operator corresponds to relations of parthood, which satisfy the logical properties of reflexivity and permit bijections between the states at issue, namely, actions and intentional descriptions. The varieties of subject matter, as defined in multi-dimensional truthmaker semantics, can be availed of in order to enrich the present approach. Having multiple state spaces from which to define the verifiers of a proposition enables a novel solution to issues concerning the interaction between action and explanation. The third type of intentional action may be regimented, as noted, by the agent's reference to an intention as an explanation for her course of action. The foregoing may also be availed of, in order to provide a novel solution to an issue concerning the interaction between involuntary and intentional action. The issue is as follows. Wittgenstein (1953/2009; 621) raises the inquiry: 'When I raise my arm, my arm goes up. Now the problem arises: what is left over if I subtract the fact that my arm goes up from the fact that I raise my arm?' Because the arm's being raised has at least two component states, namely, the arm's going up and whatever the value of the variable state might be, the answer 17 to Wittgenstein's inquiry is presumably that the agent's intentional action is the value of the variable state, such that a combination of one's intentional action and one's arm going up is sufficient for one's raising one's arm. The aforementioned issue with the foregoing concerns how precisely to capture the notion of partial content, which bears on the relevance of the semantics of the component states and the explanation of the unique state entrained by their combination. Given our multi-dimensional truthmaker semantics, a reply to Wittgenstein's inquiry which satisfies the above desiderata may be provided. Let W express a differentiated subject matter, whose total content is that an agent's arm is raised. W expresses the total content that an agent's arm is raised, because W is comprised of an initial content, U (that one's arm goes up), and an additional content, R (that one intends to raise one's arm). The verifier for W may be interpreted as a multi-dimensional loose truthmaker. Let c range over an agent's motor states, S. Let i range over an agent's states of information, S*. We define a modal operator for intentional action on the state space of the agent's motor actions. The value of the modal operator for intentional action is positive just if a selection function, g, is a mapping from the powerset of motor actions in S to a unique state s' in S. This specifies the initial, partial content, U, that one's arm goes up. An intention may then be defined as a unique state, s*, in the agent's state of information, S*. The state, s*, specifies the additional, partial content R, that one intends to raise one's arm. Formally: s ⊢ U only if ∃s'⊏S, such that g: s→ s', s.t. s' ⊢ U, ∃s*, s* ⊢ R, and 18 W = U ⊔ R. The multi-dimensional loose truthmaker for one's arm being raised may then be defined as follows: ∃cs→s′JWK c,c = 1 if s'∈JWK+, and ∃is∗JWK c,i = 1 if s*∈JWK+. Intuitively, the value of the total content that one's arm is raised is defined relative to a set of motor states – where a first intentional action selects a series of motor states which partly verify that one's arm goes up. The value of one's arm being raised, relative to (the intentionally modulated) motor state of one's arm possibly going up, determines the value of one's arm being raised relative to the agent's distinct intention to raise their arm. The agent's first intention selects among the admissible motor states, and – all else being equal – the motor states will verify the fact that one's arm goes up. Recall that the value of a formula relative to a context determines the value of the formula relative to an index. As follows, the priority of the motor act to the subsequent intention to raise one's arm is thus that it must first be possible for one's arm to go up in order to determine whether the subsequent intention to raise one's arm can be satisfied.9 The fusion of (i) the state corresponding to the initial partial content that one's arm goes up, and (ii) the state corresponding to the additional partial content that one intends to raise one's arm, is sufficient for the verification of (iii) the state corresponding to the total content that one's arm is raised. 9The role of the first intention in acting as a selection function on the space of motor actions corresponds to the comparator functions stipulated in the cognitive science of action theory. For further discussion of the comparator model, see Frith et al. (2000) and Pacherie (2012). 19 4 Concluding Remarks In this essay, I have endeavored to establish foundations for the interaction between multi-dimensional indexing and hyperintensional semantics. I examined, then, the philosophical significance of the framework by developing three, novel interpretations of multi-dimensional truthmaker semantics, in light of the new relations induced by the model. The first interpretation enables a rigorous characterization of the distinction between fundamental and derivative truths. The second interpretation evinces how the elements of decision theory are definable within the multi-dimensional hyperintensional setting, and a novel account was then outlined concerning the interaction between probability measures and hyperintensional grounds. The third interpretation of multi-dimensional hyperintensional semantics concerns the structural content of the types of intentional action. 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