§1. Computation and representation
According to Chalmers, a computational model individuates computational states through organizationally invariant properties. Computation in this sense “provides a general framework for the explanation of cognitive processes” (2012, §3.3). A mental state may have many properties that are not organizationally invariant, such as representational, phenomenal, or neural properties. Chalmers concedes that we can
I disagree. Numerous cognitive science explanations individuate mental states through their representational properties. Typically, those properties are not organizationally invariant. By elevating organizational invariance over intentionality, Chalmers flouts contemporary scientific practice. I will present an alternative version of CTM that places representation at center stage. On my approach, computational theories can individuate mental states in representational terms. A computational model can specify a transition function over mental states
§2. Bayesian psychological modeling
Bayesian models of mental activity have proved explanatorily successful, especially within
The perceptual system estimates states of the environment, including shapes, sizes, and motions of distal objects. As Helmholtz emphasized, perceptual estimation faces an underdetermination problem. Proximal sensory stimulation underdetermines distal states. Helmholtz postulated that the perceptual system solves this underdetermination problem through “unconscious inference.” Bayesian models elaborate Helmholtz’s approach (Knill and Richards, 1996). They treat the perceptual system as executing an unconscious
Retinal stimulations underdetermine shapes of perceived objects. For instance, a perceived object might be convex and lit from overhead, or it might be concave and lit from below. The same retinal input would result either way. How do we reliably perceive distal shape based upon inherently ambiguous retinal input? Let
Upon receiving retinal input
Example: Surface color perception
Light reflected from a surface generates retinal stimulations consistent with various surface colors. For instance, a surface might be red and bathed in daylight, or it might be white and bathed in red light. How do we reliably perceive surface colors? To a first approximation, current Bayesian models operate as follows (Brainard, 2009). A surface has reflectance
Cognitive scientists have successfully extended the Bayesian paradigm beyond vision to diverse phenomena, including
What are these “hypotheses”? For present purposes, the key point is that current Bayesian psychological models individuate hypotheses in representational terms. For instance, Bayesian perceptual psychology describes how the perceptual system reallocates probabilities over hypotheses
1The models described in this paragraph assume diffusely illuminated flat matte surfaces. To handle other viewing conditions, we must replace R(ν) with a more complicated property, such as a bidirectional reflectance distribution. My talk about “surface reflectance” should be construed as allowing such generalizations. 2Such models need not identify colors with surface reflectances. For instance, one might combine such models with the familiar view that colors are dispositions to cause sensations in normal human perceivers. 3For extended defense of my analysis, see (Rescorla, forthcoming b). My analysis is heavily influenced by Burge (2010a, pp. 82-101, pp. 342-366).
§3. Organizational invariance?
Many representational properties do not supervene upon the thinker’s internal neurophysiology. The classic illustration is Putnam’s (1975) Twin Earth thought experiment, which Burge (2007) applies to mental content. Quite plausibly, one can extend the Twin Earth methodology to many mental states, including numerous representational mental states cited within Bayesian psychology. For example, Block (2003) mounts a convincing case that the perceptual states of neural duplicates can represent different surface reflectances, if the duplicates are suitably embedded in different environments. Chalmers (2006) himself argues that a brain suitably linked to a Matrix-style computer simulation would not represent distal shapes, reflectances, and so on. I conclude that cognitive science employs an
Neurophysiological duplicates share the same causal topology. If a property does not supervene upon internal neurophysiology, then it does not supervene upon causal topology. So numerous representational properties cited within cognitive science are not organizationally invariant. For that reason, Chalmers’s organizationally invariant paradigm diverges fundamentally from the explanatory paradigm employed within actual cognitive science. Cognitive science may describe certain phenomena in organizationally invariant terms. But it studies numerous phenomena through a representational paradigm
Representational individuation is not essential to Bayesian modeling
Chalmers might suggest that we reinterpret current psychology in organizationally invariant terms. A Bayesian theory
I respond that cognitive science does not simply map sensory inputs to behavioral outputs. It explains mental states
Organizationally invariant theories ignore representational relations to specific distal shapes. More generally, organizationally invariant theories ignore numerous representational properties that figure as
Scientific practice is not sacrosanct. One might argue that scientists are confused or otherwise misguided. In particular, one might attack
I will not review the well-known Quinean arguments that intentionality is illegitimate or unscientific. I agree with Burge (2010a, pp. 296-298) that those arguments are unconvincing. In any event, I favor an opposing methodology. I do not dictate from the armchair how scientific psychology should proceed. Instead, I take current scientific practice as my guide to clarity, rigor, and explanatory success. I examine how science individuates mental states, and I take that individuative scheme as my starting point. This methodology provides strong reason to embrace intentional explanation and scant reason to embrace organizationally invariant explanation, at least for certain core mental phenomena.
In note 6 [added in 2011], Chalmers modifies his position. He concedes that we cannot wholly explain intentional aspects of mental activity within his organizationally invariant framework. Nevertheless, he insists that his framework “can undergird intentional explanation when appropriately supplemented, perhaps by phenomenal and environmental elements.” This modified position seems to allow a valuable explanatory role for organizationally invariant description
In my view, the key question here is whether we have any reason to seek organizationally invariant explanations. Taking current science as our guide, there is excellent reason to believe that a complete psychology will cite representational relations to the environment. A complete theory will also feature non-representational neural descriptions, so as to illuminate the neural mechanisms that implement Bayesian updating (Knill and Pouget, 2004). But neural description is not organizationally invariant. Should we supplement representational and neural description with a third organizationally invariant level of description? We
Science does not usually study organizationally invariant properties. One can specify any physical system’s causal topology, but the result usually lacks scientific interest. To borrow Chalmers’s example, the science of digestion cites phenomena (such as energy extraction) that outstrip any relevant causal topology. In studying digestion, we are not merely studying a causal topology that mediates between food inputs and waste outputs. We are studying
One popular argument emphasizes
This argument faces a serious problem: it overgeneralizes. Analogous arguments would show that we should supplement any other scientific theory, such as our theory of digestion, with organizationally invariant descriptions. Clearly, most sciences do not include such descriptions. Why? Because increased generality is not always an explanatory desideratum. One must isolate
Chalmers provides an argument along these lines (2012, §3.2). He cites functionalism in the style of Lewis (1972): a psychological state is individuated by how it mediates between inputs, outputs, and other psychological states. From his functionalist premise, Chalmers concludes that scientific psychology requires organizationally invariant descriptions.
Lewis offers functionalism as a conceptual analysis of folk psychology. Yet folk psychology routinely cites representational properties that do not supervene upon causal topology. So it is unclear how widely, if at all, Lewis-style functionalist reduction applies to folk psychology. More importantly, folk psychology is not directly relevant to our concerns. Science can consult folk psychology for inspiration — as illustrated by the Bayesian paradigm. But science does not answer to folk psychology. Our question is how scientific explanations should individuate mental states. Our best strategy for answering that question is to examine science, not folk psychology. As I have argued, there are numerous mental phenomena that current science studies by citing representational properties
Chalmers suggests a further argument for embracing organizationally invariant descriptions: they are needed for modeling the mind
4Burge (2010a, pp. 95-101), (2010b) and Peacocke (1994) propose similar treatments of computation.
§4. Computation, syntax, and semantics
A computational model specifies possible states of a system, and it delineates a transition function dictating how the system transits between states. The transition function may be either deterministic or stochastic. Either way, as Chalmers emphasizes, it supports counterfactuals. In the deterministic case, it supports counterfactuals of the form:
In the stochastic case, it supports counterfactuals concerning the probability of transiting from state
According to Chalmers, “computations are specified syntactically, not semantically” (2012, §2.2). To illustrate, consider a Turing machine table that describes how a scanner manipulates strokes on a machine tape. The machine table does not mention semantics. It describes formal manipulation of syntactic items. Chalmers holds that
I agree that some computational models individuate computational states in syntactic, non-semantic fashion. But I contend that
Example: A numerical register machine
A register machine contains a set of memory locations, called
A physical system often represents the same denotation in different ways. For example, “4” and “2+2” both denote the number 4. These two expressions occupy different roles within arithmetical computation. So denotation by itself does not always determine computational role. In general, adequate computational models must address the way that a computational state represents its denotation. To borrow Frege’s terminology, adequate models should individuate computational states by citing
Fodor (1981, pp. 234-241) offers a classic discussion of this question. He considers two ways of taxonomizing mental states:
For example, a transparent scheme type-identifies the belief that Hesperus has craters and the belief that Phosphorus has craters. This approach has trouble explaining why the two beliefs have different functional roles. In contrast, a formal scheme can associate the belief that Hesperus has craters and the belief that Phosphorus has craters with “formally distinct internal representations” (p. 240), thereby explaining why the beliefs have different functional roles. Fodor concludes: “a taxonomy of mental states which honors the formality condition seems to be required by theories of the mental causation of behavior” (p. 241). He posits a
Fodor’s dichotomy between transparent and formal taxonomization ignores a third option. We can postulate MOPs individuated
5For further discussion of numerical register machines, see (Rescorla, forthcoming a). 6In his early work, Fodor (1981, p. 227, p. 240) holds that formal syntactic type determines a unique narrow content but not a unique wide content. His later work, beginning in mid-1990s, abandons narrow content while retaining the emphasis on formal syntactic types that underdetermine wide content.
§5. Semantic permeation versus semantic indeterminacy
To develop my approach, I introduce some terminology. An entity is
Mental representations are types. We cite them to taxonomize token mental states. The types are abstract entities corresponding to our classificatory procedures. A semantically indeterminate type corresponds to a taxonomic scheme that underdetermines representational content. Different tokens of a semantically indeterminate type can express different contents. A semantically permeated type corresponds to a taxonomic scheme that takes representational content into account. Each token of a semantically permeated type expresses a uniform content.
Semantically permeated mental representations are either structured or primitive. Structured representations arise from applying compounding operations to primitive representations. For example, we can postulate a mental representation S that necessarily denotes the successor function and a mental representation + that necessarily denotes the addition function. We can then postulate an infinite array of structured mental numerals that arise from appropriately combining 0, S, and +:
The denotation of a complex numeral follows compositionally from the denotations of its parts:
Each complex numeral necessarily satisfies an appropriate clause from the compositional semantics. Thus, each complex numeral is semantically permeated.
Semantically permeated taxonomization need not be transparent. (SS0 + SS0) and SSSS0 both denote the number 4, but they are distinct types. Likewise, we can postulate distinct but co-referring mental types Hesperus and Phosphorus. Distinct but co-referring types reflect different
A satisfactory development of the semantically permeated approach must elucidate how semantically permeated types are individuated. When do two token mental states share the same semantically permeated type? I want to leave room for conflicting answers to this question. But I follow Burge (2009) and Evans (1982, pp. 100-105) in assigning a central role to
By citing representational capacities, we illuminate what it is for semantically permeated mental representations to have “structure.” Their structure consists in the appropriate joint exercise of distinct capacities. To illustrate, imagine an idealized mathematical reasoning agent with capacities to represent 0, successor, and addition. We posit mental symbols 0 S and + so as to mark the exercise of those three capacities. The agent also has a capacity to apply functions to arguments. These four capacities yield complex capacities to represent natural numbers. Complex mental numerals mark the exercise of the resulting complex capacities. For example, mental numeral SSSS0 marks the exercise of a complex capacity that deploys three capacities:
A mental state is a token of SSSS0 only if it exercises this complex capacity, to do which it must satisfy the appropriate clause from the compositional semantics:
Mental numeral (SS0 + SS0) marks the exercise of a complex capacity that combines the foregoing four capacities in a different way, along with a capacity to represent addition.
As my examples illustrate, a single agent may have different capacities for representing the number 4. Similarly, an agent may have different capacities for representing water (representing it
A complete theory must elucidate the representational capacities through which we individuate semantically permeated types. But I am not trying to offer a complete theory. For present purposes, I simply assume that normal humans have various representational capacities. Current science amply vindicates that assumption. Cognitive science routinely type-identifies mental states through representational capacities deployed by those states. Each semantically permeated mental representation marks the exercise of a particular representational capacity.
7Throughout my discussion, I use outline formatting to signal metalinguistic ascent. For example, “water” denotes the mental representation water, which in turn denotes the potable substance water.
§6. Mental computation over a semantically permeated language of thought
A computational model delineates counterfactual-supporting mechanical rules governing how a computational system transits between states. I claim that,
To illustrate, consider the mathematical reasoning agent introduced in §5. She can entertain infinitely many mental numerals, generated by combining mental symbols 0, S, and +. Let us stipulate that her computations conform to the following symbol transformation rules: for any numerals
where the arrow signifies that one can substitute the right-hand side for the left-hand side. One can compute the sum of any two numbers by applying rules A and B. For example:
A and B are mechanical rules that dictate how to manipulate inherently meaningful mental symbols. They describe transitions among mental states type-identified through the representational capacities that those states exercise.
I stated rules A and B by inscribing geometric shapes on the page. Geometric shapes are subject to arbitrary reinterpretation, so they are semantically indeterminate. But our question concerns the mental symbols that geometric shapes represent. I am
I propose that we take rules A and B as paradigmatic. We should describe various mental processes through mechanical rules that cite semantically permeated mental representations, without any mention of semantically indeterminate syntactic types.
Many readers will regard my proposal warily. How can semantics inform mental computation, except as mediated by formal syntax? My proposal may seem especially suspect when combined with an externalist conception of mental content. As Fodor puts it, “the effects of semantic identities and differences on mental processes must always be mediated by ‘local’ properties of mental representations, hence by their nonsemantic properties assuming that semantics is externalist” (1994, p. 107). Mustn’t computation manipulate mental representations based solely upon their “local” properties, ignoring any relations to the external environment?
I agree that mental representations have local, non-semantic properties: namely,
Current science provides strong evidence that a complete theory of the mind will include at least two levels of description:
I now want to elaborate the semantically permeated approach by examining case studies drawn from CS and AI. In §7, I analyze a powerful computational model of
8Fodor (1981, pp. 226-227) holds that a physical system is computational only if it has representational properties (“no computation without representation”). Chalmers demurs, and rightly so. I do not claim that computation requires representation. I claim that some computational models specify computations representationally, without any mention of formal syntactic types. In this respect, I disagree with both Chalmers and Fodor. Despite their differences, both philosophers agree that every computational model features a level of purely syntactic, non-semantic description. 9The philosophical literature offers additional well-known arguments for postulating formal mental syntax. A complete defense of my approach would scrutinize all such arguments, a task that I defer for another occasion.
§7. The lambda calculus as a programming language
PCF contains primitive symbols, including +, 0, 1, 2, 3, … and devices for generating complex expressions. One notable device is
One can convert these informal clauses into a rigorous
PCF comes equipped with an
where the expression on the right is the result of substituting
PCF’s denotational semantics is no mere rhetorical appendage to its operational semantics. As Scott emphasizes (1993, p. 413), denotational semantics is what elevates PCF from a formal calculus to a model of mathematical reasoning. Denotational semantics provides a standard for evaluating whether the operational semantics is satisfactory. Transformation rules must honor the intuitive meanings of PCF expressions, as codified by the denotational semantics. Two theorems reflect this desideratum (Gunter, 1992, pp. 133-137):
Failure of soundness would entail that our operational semantics yields incorrect results. Failure of computational adequacy would entail that our transformation rules do not generate all the computations we want them to generate.
I have represented PCF expressions through strings of shapes. But PCF expressions are
Yet even this tree diagram contains extraneous detail. It uses arbitrary shapes, which we could vary while representing the same underlying PCF expression. The diagram reads from left to right, but we could instead use a diagram that reads from right to left. Analogous problems arise if we replace tree diagrams with
So what is the underlying expression? In my opinion, CS does not answer this question. CS offers a
On this analysis, PCF expressions are semantically indeterminate. They are formal syntactic items subject to reinterpretation. The operational semantics describes formal syntactic manipulation, without regard to any meanings syntactic items may have.
Alternatively, we can construe PCF in semantically permeated fashion. We can construe it as modeling how an idealized cognitive agent transits among representational mental states:
To illustrate, consider the tree diagram inscribed above:
PCF expressions mark the exercise of representational capacities, so they determine specific representational contents. For example, 2 necessarily denotes 2, while λ
Say that a function is
PCF’s operational semantics is similarly indifferent between semantically indeterminate and permeated taxonomization. We can construe the
A description along these lines is too schematic to favor semantic indeterminacy or permeation. To choose between the two construals, one must say how one interprets tree diagrams. Of course, tree diagrams themselves are semantically indeterminate. But our question is how one should individuate the linguistic items that tree diagrams represent.
Computer scientists usually employ a semantically indeterminate individuative scheme for PCF expressions. They study diverse mathematical structures through which one can interpret PCF (Longley, 2005), (Mitchell, 1996, pp. 355-385, pp. 445-505). The denotational semantics sketched above furnishes one such mathematical structure: the
Nevertheless, semantic indeterminacy is not obligatory. PCF’s operational semantics does not mandate that we individuate PCF expressions entirely through the operational semantics. The operational semantics is consistent with a taxonomic scheme that takes a specific denotational semantics into account. We can employ this taxonomic scheme when modeling the mathematical cognition of an idealized agent. The agent’s computations deploy
My position here goes far beyond the weak claim that we can describe PCF computation in representational terms. Chalmers and Fodor would presumably agree with that weak claim. My position is that nothing about PCF’s operational semantics mandates an explanatory significant role for formal mental syntax. Mechanical transition rules governing mental computation can appeal instead to representational capacities deployed by mental states.
10Technically, PCF extends the pure simply-typed lambda calculus with primitive numerals, primitive Boolean terms, and fixed point operators at each type. 11Several complexities arise in providing a rigorous denotational semantics. First, one must employ familiar Tarskian techniques so as to handle free variables. Second, the language has a type structure, which one must treat more gingerly than I do in my informal exposition. Third, and most seriously, the language expresses recursive definitions through primitive fixed point operators at each type. The semantics for fixed point operators requires serious machinery too complicated to discuss here (Mitchell, 1996, pp. 305-333). Taking such complexities into account would muddy the exposition without affecting my overall argument. 12Almost deterministic. The technical definition of“substitution” engenders a subtle element of indeterminacy (Mitchell, pp. 53-54). 13If we countenance PCF-definability relative to sufficiently deviant semantic interpretations, then intuitively uncomputable functions become PCF-definable (Rescorla, 2007). However, the possibility of deviant semantic interpretations does not militate against semantic permeation. It merely demonstrates that one should not individuate semantically permeated types through a deviant denotational semantics, any more than one should interpret semantically indeterminate types through a deviant denotational semantics.
§8. Computational foundations for Bayesian inference
The previous section focused upon computations that represent the
As many researchers have emphasized, Bayesian inference is often hopelessly inefficient. If the probability domain is continuous, then computing the constant
requires integrating over the domain, a task which may be computationally intractable. Even when the domain is finite but extremely large, computing
Example: Importance sampling
Using the Monte Carlo method, we can approximate conditionalization through an
We now have a list of
- Resample: Draw
One can show that the probability of
The importance sampling algorithm presupposes a capacity to sample from a domain in accord with a prior probability over that domain. The algorithm also presupposes a capacity to assign weights in accord with a prior likelihood, along with a capacity to resample according to the assigned weights. There is no obvious respect in which these capacities require formal syntactic manipulation. There is no obvious respect in which the algorithm mandates that one manipulate formal syntactic types. The algorithm
A similar moral prevails when we implement importance sampling through a high-level programming language. I illustrate by discussing a particularly instructive case study.
Example: A probabilistic extension of the lambda calculus
The language λο, introduced by Park, Pfenning, and Thrun (2008), augments the lambda calculus with resources for representing and manipulating probability functions. To represent a probability function, λο specifies a procedure for drawing samples from the probability domain. λο specifies sampling procedures through
For example, if outcomes of evaluating
The operational semantics for λο resembles that for PCF. The main difference is that λο features new transformation rules governing new vocabulary. In particular, λο features an importance sampling rule that approximates Bayesian updating. To illustrate, consider a prior probability
λο is a purely mathematical language. Thus, it cannot represent a probability function over an empirical domain. It can only represent a probability function over a mathematical domain, such as the real numbers. In itself, then, λο cannot provide a foundation for the Bayesian models postulated by cognitive science. Those models postulate probabilistic updating over hypotheses that represent environmental properties: shapes, reflectances, and so on. To represent the desired probability functions, we must supplement λο with resources for representing desired environmental properties. For example, we can supplement λο with symbols
Once we supplement λο with suitable empirical vocabulary, we can define probability functions over the desired empirical domain. The most straightforward strategy is to assume a fixed mapping from a suitable mathematical domain (e.g.
The operational semantics is compatible with either semantically indeterminate or semantically permeated taxonomization. λο merely supplements PCF-style transformation rules with a few additional rules governing new vocabulary, including a rule that formalizes importance sampling. As with PCF, we can construe the transformation rules as describing how to manipulate formal syntactic items, or we can construe them as describing how to manipulate inherently meaningful mental representations. Under the latter construal, the rules govern transitions among mental states typeidentified by the representational capacities those states deploy. Admittedly, λο’s creators seem to have in mind a semantically indeterminate rather than semantically permeated taxonomic scheme. Nevertheless, both taxonomic schemes are equally consistent with the operational semantics.
To illustrate, imagine a hypothetical Bayesian agent who executes λο computation. She has capacities to represent various environmental properties: shapes, reflectances, etc. We type-identify her mental states by citing these and other representational capacities. We reify the resulting mental state types by positing semantically permeated mental representations.
Thus, prob necessarily satisfies the following compositional clause:
The agent computes over semantically permeated mental representations. She thereby approximately implements Bayesian inference.
More specifically, consider a Bayesian model of surface reflectance estimation. The model postulates prior probabilities over reflectances and illuminants. It postulates a prior likelihood relating reflectances, illuminants, and retinal inputs. It describes the perceptual system as converting these priors and retinal input into a posterior over reflectances. We do not yet know how exactly the human perceptual system executes (or approximately executes) this Bayesian inference. But we can delineate a computational model describing
A model along these lines need not assign any role to formal mental syntax. The transformation rules do not associate mental states with formal syntactic types. Instead, they individuate mental states through the representational capacities that those states exercise.
I have sketched how a hypothetical computational system might approximately implement Bayesian inference. To what extent do the resulting computations resemble
I do not present the model as an empirical hypothesis. I offer it as an existence proof: there exist semantically permeated computational models that tractably approximate Bayesian inference. In principle, then, semantically permeated computation can provide a foundation for Bayesian psychological modeling. Future scientific progress will reveal whether semantically permeated computation provides an
14By using the deliberately vague terms “probability function” and “probability domain,” I conflate
§9. Representational capacities versus causal topologies
Philosophers usually take computational modeling to embody an
You may ask: how can a system “know” whether the symbols it manipulates have appropriate representational properties? Am I presupposing an inner homunculus who inspects a mental symbol’s meaning before deciding how to manipulate the symbol?
I reply that a system can conform to representationally-specified rules even if the system does not interpret the symbols it manipulates. My approach does
Given current scientific knowledge, a semantically permeated version of CTM is speculative. But it is no more speculative than Chalmers’s semantically indeterminate account. Furthermore, it offers a crucial advantage: it preserves the representational explanatory paradigm widely employed within current science. Chalmers replaces that paradigm with an organizationally invariant alternative that finds no echo within Bayesian psychology. I have shown that we can model the mind computationally while avoiding Chalmers’s radical revisionism. We can integrate intentionality directly into computational models of mental activity, without articulating an organizationally invariant level of description.
Chalmers might retort that semantically permeated models are not genuinely
Chalmers and I agree that minds have causal topologies. We agree that minds have representational capacities. Our disagreement concerns whether causal topologies or representational capacities will prove central to a developed science of the mind. I contend that many mental phenomena are best studied by citing representational capacities rather than causal topologies. In contrast, Chalmers elevates causal topologies at the expense of representational capacities. Future scientific developments will settle which paradigm is more fruitful.