Transitive Inference over Affective Representations in Non-Human Animals

Abstract

The mainstream philosophical approach to inference, which insists on sentence-like representations and a linguistic capability, excludes non-human animals as possible agents capable of making inferences. However, an abundance of studies show that many animal species exhibit behaviors that seem to rely on some kind of reasoning. My focus here are the transitive inference tasks, which most species solve quite successfully. These findings put pressure on the mainstream views, and still lack a convincing explanation. I introduce the concept of affective representations, which satisfy the semantic and structural requirements for participating in inferences. I also argue for a broader conception of inference, and show how we can apply this view to explain the results of the transitive inference studies. Finally, I suggest it is more useful to think of flexibility of thought in terms of a continuous range, rather than a dichotomy of flexible vs. inflexible.

1 Introduction

How do we explain behaviors of animals that seem to rely on making transitive inferences? What is the underlying mechanism of their transitive behavior? Answering this question should contribute to wider questions: What is the nature of non-human animals’ reasoning? Is there a way to conceive inferential reasoning without linguistic vehicles? I will argue against the traditional view that animal minds cannot support inferential reasoning, and provide a positive proposal of how animals can make inferences by means of affective representations.

Philosophers of mind have traditionally denied that non-human animals are capable of inferential reasoning. Variations of arguments from Descartes (1637), Stich (1979), Davidson (1982), etc. that significantly influenced the discussion on animal thought, take the lack of language in non-human animals as one of the main indicators of their incapability of thought and reasoning. Contemporary epistemological accounts of inference reinforce this view by defining inference as an essentially linguistic activity, or one that at least requires language-like mental representations (Broome 2013; Boghossian 2014, 2018; Quilty-Dunn and Mandelbaum 2017; Valaris 2014, 2017).

This view is challenged by the overwhelming empirical evidence of the behaviors of non-human animals that seem to require inferential reasoning. In reaching their goals, animals often act as if they were capable of different forms of reasoning, such as logical (Call 2004, 2006; Marsh and MacDonald 2012; Erdőhegyi et al. 2007), causal (Blaisdell et al. 2006; Völter and Call 2017; Schloegl and Fischer 2017), and transitive reasoning (Moon and Zeigler 1979; Vasconcelos 2008; Lazareva 2012; Gazes, and Lazareva 2021).

Here I focus on the ability of animals for transitive inference (TI). In technical terms, TI is a transition in which if, given aRb and bRc, where R is a transitive relation, the subject infers aRc (Evans et al. 1993; Goodwin and Johnson-Laird 2008). For example, given A > B and B > C, and since ‘ > ’ or ‘is bigger than’ is a transitive relation, we can infer A > C. Studies on transitive inference are interesting because human TI is considered a classic case of logical inference.

Experimental studies have shown that many animal species such as non-human primates, crows, pigeons, rats, etc. are capable of behaviors that seem to require transitive inference (Moon and Zeigler 1979; Vasconcelos 2008; Lazareva 2012; Gazes, and Lazareva 2021). However, it is still debated whether the animals are truly performing TI to solve the tasks, or whether they are successful on account of some other, possibly simpler mechanism. Resistance toward attributing TI to animals is not surprising, since the conceptual framework for explaining inference comes from an anthropocentric perspective: it essentially involves abstract notions such as concepts, symbols, propositions, etc., and it is unclear how this framework can be meaningfully applied to non-human animals.

Some attempts to remedy this involve attributing a language of thought (LOT) to non-linguistic animals, e.g., in baboons (Cheney and Seyfarth 2007). This strategy, however, faces its own difficulties, such as explaining what exactly LOT implies in this context, and most of these discussions remain vague. It also leaves a mismatch between the postulated language-like thoughts and the lack of a public language (Camp 2009a, b), which was the original inspiration for the language of thought hypothesis (e.g., Fodor 2008). There is also a difficulty of how the thoughts involved in the LOT can be implemented, as some authors explicitly argue that in human thought they are always implemented as linguistic sentences (Bermúdez 2003, p. 159–160), but this does not seem necessarily true (Beck 2018).¹

Partly in response to these difficulties, and in attempt to explain complex intelligent behaviors of non-linguistic animals, an alternative to LOT in animals was proposed, according to which animals’ thoughts can be represented spatially, through map-like or diagrammatic representational systems (e.g., Camp 2009a, b).² Camp criticizes the neglect of alternative representational possibilities in describing animal thought that results from holding onto the dichotomy between imagistic and sentence-like formats.³ She suggests that a useful way to contribute to this discussion is to investigate the combinatorial and referential principles for different representational systems, especially systems that mix the elements of these two respective systems. This paper is a contribution precisely in this direction.

I propose another form in which non-linguistic thought can occur, which supports the relevant functional relations needed for some kinds of inference – affective representations (AR). I introduce ARs as a type of representations that differ significantly from the representations assumed by the LOT hypothesis in that they are not abstract nor fully systematically recombinable, and thus do not support the combinatorial richness characteristic for LOT representations. However, they are also unlike spatial or pictorial representations, first, in the obvious sense that they do not rely on spatial or visual imagery, but also regarding the way they are structured and in the compositionality and flexibility that their structure allows, which seems to share some (but not all) features with sententially structured thought. I discuss this in detail in Section 3.2.1. The format of ARs, as construed in this paper, does not fall neatly into either side of the dichotomy between sentence-like and iconic representations. Importantly, this way of describing animal thought need not be incompatible with the attribution of thinking through spatial representations – both kinds of representations may be tokened in different cases, or even in parallel, complementing each other.

To approach these issues, I first question the mainstream understanding of the nature of inference, as this seems to be the foundation for a priori denying the possibility of animal inference. I show that the existing definitions of inference, which depend on sentence-like representations are anthropocentric and limited by unjustified assumptions. I argue that inferences need not be limited to sententially structured representations. It is plausible to broaden our understanding of the nature of inference to be able to include additional types of formats aside from the one assumed so far. In particular, I will argue that inferences can be performed over affective representations as well. This type of representation can unproblematically be attributed to both humans and many species of non-human animals, and can help bring more coherence to the explanations of similar human and non-human behaviors. The main goal of the paper is describing how inference over affective contents can be used to explain the behavior of animals in solving in the TI task.

The paper is structured as follows. In Section 2 I present the main assumptions of the armchair view on inference and extract the underlying assumptions that jointly imply an inability of animals to make inferences. In Section 3.2 I present the empirical results that challenge this view, and I show why the existing accounts fail to give a satisfying explanation of the empirical results. In Section 3.2.1 I introduce affective representations and show that they satisfy the requirements for representation and for being suitable to participate in inferences, and finally I show how they can be applied to provide an explanation of the results of the TI task.

2 The Mainstream View: The Intellectual Nature of Inference

Here I present the basic assumptions of the dominant view on inference, and their implications for the possibility of animal inference. Most of the weight of the view that animals cannot make inferences is carried by assumptions concerning the nature of representations involved in inference.

Inference in general is standardly understood as a reason-dependent transition between mental states. Inferential transitions essentially involve an epistemic connection between the states that function as premises and the state that functions as the conclusion, such that the premises support the conclusion. This is what differentiates inferential from non-inferential transitions (e.g., associative transitions), or ‘arbitrary mental jogging’ (Broome 2013; Quilty-Dunn and Mandelbaum 2017), which do not involve such an epistemic connection, e.g., a cocktail makes me think of the summer, which makes me think of the beach. There are variations to the mainstream view whose details are not necessary for the purpose of this paper. One distinction worth laying out is between the normative and the naturalistic approach to determine the nature of inference, as proposed by Quilty-Dunn and Mandelbaum (2017). The normative approach is primarily led by an interest in explaining how inferential transitions transmit epistemic warrant from premises to the conclusion (e.g., Boghossian 2014, 2018; Broome 2013). The aim seems to be finding an internalist justification of a warranted inferential transition – what is it that the agent does that guarantees the transition to be warranted. The naturalistic approach, on the other hand, abandons the normative requirements and aims instead at a psychological description of inferential transitions, i.e., explaining the empirical cases of inferential behaviors, whether they are successful or not. Proponents of this approach rightly point out the difference between misinference – an unsuccessful inference, and non-inferential or arbitrary transitions (Quilty-Dunn and Mandelbaum 2017). They allow for both ‘bare inferential transitions’, which are determined by rules of mental logic built into the cognitive architecture, but do not require explicit representation of the rules (and can thereby also be unconscious), and ‘rich inferential transitions’, which involve explicit ‘taking’ the premises to support the conclusion (Quilty-Dunn and Mandelbaum 2017).

The account I will put forward in the later part of this paper is a naturalistic account of inference, such that also aims to explain the empirically observed behaviors. Its scope, however, is wider than the existing naturalistic accounts, since I aim to include seemingly inferential behaviors of animals. The existing accounts, be they naturalistic or normative, exclude animal inference either because they explicitly take language as a necessary requirement, or because of the underlying assumptions about the kinds of representations and the kinds of processes necessarily involved in reasoning (Munroe 2021). For example:

“In…reasoning, you operate on the marked contents of your conscious attitudes, following a rule. The marked contents are complex. They have a syntactic structure, and the rules you apply in operating on them depend on their structure. In operating on them, you have to hold them in your consciousness, maintaining an awareness of their syntactic structure. Language is well suited to doing that. It has a meaning that can represent the semantic elements of the marked contents, and it has a syntax that can represent their syntactic structure. It is plausible that, without the help of language, you could not keep the marked contents properly organized in your consciousness.” (Broome 2013, p. 267).

“Rational thought and inference are assumed to require implementation with representations that are internally structured and systematically interconnected; but only one sort of structure is assumed to suffice. In one version of the view, found in philosophers like Rey (1995), Bermúdez (2003), and Devitt (2006), genuinely logical thought is assumed to require a specifically linguistic medium” (Camp 2018).

To gain more clarity about which parts of the dominant view of inference I argue against, I will now reconstruct the main assumptions upon which it rests. Some of the assumptions are explicitly stated, while others I extract as tacitly accepted, since they seem necessary for adding up to the final conclusion. Importantly, an overarching assumption of the dominant view seems to be that, while representations participating in the inferences may have affective or associative content (Mandelbaum 2013, 2015), that content is not what is able to ‘drive’ the inference.

• 1) Elements of inference must be sententially structured

In order to be suitable for the kind of ‘mental manipulation’ that allows for flexibility of thought required for making inferences, elements of inference must be sententially structured, i.e., have a subject-predicate form. Logical rules are crucially connected to the structure of representations featuring in the transitions, because logical rules are rules that are sensitive to constituent structures of the representations (Fodor 2008; Quilty-Dunn and Mandelbaum 2017; Rescorla 2019).

• 2) Only linguistic vehicles can have sentential structure

Most accounts of inference assume that the representations over which inferential transitions operate are sentential in the sense that they are language-like (Broome 2013; Boghossian 2014, 2018; Quilty-Dunn and Mandelbaum 2017). Bermúdez (2003) argues that all the sententially structured thoughts that we consciously introspect take the form of sentences in a public language, and denies we are ever conscious of sententially structured thoughts that do not have linguistic vehicles (p. 159–160). This is a stronger assumption than 1) in that the elements of inference are not just sententially structured, but they necessarily take a linguistic form. He also explicitly states that human language is not only sufficient for providing the kind of structure necessary for inference, but that it is also necessary (p. 158).

• 3) Second-order processes are a necessary part of inference

Many accounts of inference construe it as essentially requiring second-order or reflective processes (although some reject this claim, e.g., Kornblith 2012). This is especially endorsed by the normative accounts, which justify it by the need for some kind of rational evaluation or self-monitoring that differentiates inferences from arbitrary transitions. Explicitly taking the premises as supporting the conclusion is what makes the transition a rational one (Boghossian 2014, 2018; Valaris 2014, 2017). Naturalistic accounts include this requirement only for the ‘rich’ inferential transitions, but not for ‘bare’ ones (Quilty-Dunn and Mandelbaum 2017). Bermúdez (2003) argues that inferences necessarily involve two levels of processes, such that the second-order processes involve considering the epistemic relations between the first-order thoughts and their truth-values. There is an additional assumption tied to this one according to which only linguistic formulation can provide the stable structure that allows for this second-order dynamics.

• 4) Language structure reflects thought structure

Insisting on the relevance of language for inference seems to also rely on the assumption that thoughts are formatted like sentences that express them, and that external communication structurally mirrors internal thought. And since animal communication doesn’t seem to be structured in the rich ways in which human languages are, it is assumed that their cognition also lacks sententially structured representations, and thereby cannot support inferential transitions (for a commentary, see Beck 2018).

The argument can most comprehensibly be summarized as follows. Since animal communication is not structured similarly to human languages, and since external communication structurally mirrors the internal mental contents, the contents of animal cognition lack such structure too. And since such structure is necessary for inferential transitions and for the second-order processes required for inferences, animal cognition doesn’t support inferences.

Although empirical evidence of intelligent behaviors of animals challenges the mainstream view, these assumptions can be questioned on their own.⁴ This paper presents a case against the view that inferential capability requires linguistic capability. Although it makes sense that human language is sufficient for forming representations adequately structured for allowing inferences, the main weight of the additional claim that language is also necessary relies on the fact that we are not aware of alternative representations that can have structure that allows them to participate in inferences. In the positive part of the paper, I aim to provide such an alternative in the form of affective representations. I will show that affective representations are a viable alternative to language-like representations in that they have the appropriate structure and can participate in inferences. There are two ways in which my proposal can be fleshed out, either as affective representations having sentential structure, which will make this a case against Assumption 2, or as affective representations involving attribution, but allowing for compositionality that makes them suitable for participating in inferences, which will make it a case against Assumption 1. I remain agnostic on which one is the case. I will explain this in detail in Sect. 3.2.1.

3 The Challenge: Explaining Results of Transitive Inference Studies

The observed success of non-human animals in many reasoning tasks seems to pose a challenge to the traditional view. For example, there is now overwhelming empirical evidence of the ability of animals to solve many varieties of tasks that requires transitive inference (TI). After presenting the relevant evidence from the TI studies, I diagnose the source of the problems of the existing attempts to explain the psychological mechanism underlying the TI success.

3.1 The Test

The standard test for laboratory investigation of TI in animals is the 5-item procedure. The animals are trained on these 5 items, organized in 4 preferential pairs: A + B-, B + C-, C + D-, and D + E-, where ‘ + ’ means that choosing this item is rewarded, and ‘-’ means choosing the item is not rewarded. Once the animals reach a certain level of correct performance on the so-called ‘premise pairs’, they are then tested with a novel pair BD. In the training sessions, B was rewarded exactly as frequently as D (50% of the time on average). The choice in the novel pair is meant to test whether the animals are able to make a transitive inference from B > C and C > D to a pair of items B and D that they have never encountered together before. Every species tested so far (with the single exception of honeybees) has been successful at the task, with the lowest average success score in a large set of experiments with different experimental procedures being 80% (Lazareva 2012; Gazes, and Lazareva 2021).

3.2 Existing Accounts

It is still debated whether the animals are truly performing inferential reasoning in order to solve the tasks, or if they are successful on account of some other, simpler mechanism. There are two general strategies to explain the transitive behaviors in these and similar studies. One strategy – commonly named the cognitive or relational accounts, is to argue that animals are able to solve the TI task because they form a representation of the ordered series of the items, including the relations between them, and flexibly compare the items in the series. The other strategy is taken by the associative accounts – explains them on the basis of conditioning through reinforcements (rewarding the ‘correct’ choices) and associative values attached to the items in the task. The animals’ choices are based on simply responding to the item with higher associative value, without invoking any explicit representation of the entire series.

3.2.1 Cognitive Accounts

This approach aims to explain the animals’ transitive success by attributing them some kinds of mental models or mental representations of the logical order of the stimuli. The representation is commonly understood spatially, as a ‘mental line’ along which the elements are ‘placed’. Proponents of this kind of account argue that the subjects integrate the independently learned training pairs into a linear mental representation of the whole series (e.g., De Soto et al. 1965; Huttenlocher 1968; Riley and Trabasso, 1974; Trabasso et al. 1975; Sternberg, 1980; Breslow, 1981; McGonigle and Chalmers 1986; Acuna et al. 2002). The representation is formed in an end-inward fashion in the following way. The subjects first locate the ends of the series, isolating items A and E, because they are always positive or always negative. Then, the other, ‘inner’ items gradually become ordered between the ends. Once the representation is complete, the subjects can answer any relational questions about the items using the order information that has been built into the linear representation, and their choices are governed by a search along the mental line. The assumed process that mediates between the representation and performance vary in different theories (for a review, see Banks and Flora 1977).

An interesting recent account by Aguilera (2016), similar in spirit to the mental line view, offers an innovative development of the idea of a ‘tree map’ representation proposed by Cheney and Seyfarth (2007) and Camp (2009a, b). The initial proposal by Cheney and Seyfarth and Camp was that baboons represent social dominance relations through mental structures such as tree maps.⁵ Tree maps are composed of nodes connected by branches, such that nodes represent sets or individuals. Branches (or lines) represent relations between the nodes. In this way, tree maps, being hierarchically organized representational systems, are thus adequate for representing real-world domains characterized by transitivity. Aguilera’s account further develops this by adding the claim that branches that connect the nodes function as predicates that can be applied to different individuals in largely the same way in which linguistic predicates work. In this account, tree maps are proposed as representational systems that can incorporate a predicative structure which sustains inferences that depend on the internal structure of representations (i.e., branches and nodes). The branches function as “predicate of arity 2 denoting the property of being higher than, that is, a binary relation which can range over different pairs of elements” (p. 358.). When applied to the TI studies, this can be translated to the spatial representation of the whole series such as a mental line in which the lines between the nodes denote the predicate ‘ > ’ or ‘is better than’ (since choosing the ‘better’ item results in receiving the reward).

While I am in favor of the general idea that maps can incorporate predicates for the reasons Aguilera offers in her paper, this account faces similar problems as other cognitive accounts when it is applied to explain TI behavior in experimental studies. Several serious criticisms were proposed against cognitive accounts that appeal to spatial representations (and I am not aware of non-spatial kinds of cognitive accounts proposed in the literature).

One of the main problems with cognitive accounts is that they are both theoretically and empirically underspecified in at least three ways. First, it is not clear enough what exactly the mental lines are and how exactly are they formed through the processes of reinforcement and non-reinforcement. The majority of the proponents assume that the representation is formed from end-items toward the middle, but it is not explained why the formation proceeds in that way. It is especially problematic given that in most of the studies, the subjects are trained through a sequential procedure: in the first phase the animals learn the pair AB until they reach 80% success rate, then they are trained with BC, and proceed further to the final pair DE. It is not clear how and why the end items would be the first to be placed on the mental line. An additional problem is that the sequential procedure has been shown to be the easiest for the animals in learning the training pairs, which clashes with the assumption that the end items tend to be learned first. An intermixed procedure, where all the training pairs are learned simultaneously in the training phase should in that case be easier. Second, the cognitive accounts are static – they specify the subjects’ performance given the already formed linear representation, but they do not deal with intermediary phases nor with important procedural details. They are ill-equipped to explain differences in the learning phase, for example, how different training procedures might influence learning the training pairs and performance in the test phase, which is where their rival theories shine. Third, they also do not specify the process of information retrieval in the test phase. Do the subjects ‘scan’ the mental line from one end to another, or do they rely on a different process? Specifying the retrieval process in different ways would lead to different predictions regarding the reaction times of choosing in the test phase. An important related empirical problem is that these accounts have trouble explaining some behavioral patterns observed in TI studies, such as systematic differences in accuracy and speed of solving particular pairs in the series. I will elaborate on this in the subsection on behavioral effects below (for more details on these criticisms, see Wynne 1995; Vasconcelos 2008; Lazareva 2012).

3.2.2 Associative Accounts

The associative accounts explain away the seemingly complex transitive-like behavior as a result of simple associative mechanisms, without the need for mental representations or inferences, thus accounting for the animals’ success within the traditional view. The animals’ choices are determined by the associative values of each stimulus, and values are in turn crucially determined by the reinforcement (rewarding) history. Below are some of the most prominent variations of the associative account.

Value Transfer Theory

Initially put forward by von Fersen et al. (1991), the value transfer theory (VTT) was one of the first quantitative accounts of TI (Vasconcelos 2008). According to VTT, associative value transfers between stimuli that are presented together in a pair. For instance, in the AB pair, B gains value due to its association with the always rewarded stimulus A. This results in each stimulus having composite value – a combination of direct value from its own reward history, and indirect (transferred) value acquired from the other stimulus in the pair (Vasconcelos 2008; Clement et al., 1998). A number of experimental studies provide independent evidence for the occurrence of value transfer in general (Vasconcelos 2008; Zentall and Sherburne, 1994; Cohen et al. 2001; Siemann et al. 1996; Zentall et al., 1996b), supporting its role the TI behaviors. However, applying VT as the main mechanism of TI leads to problems that render it an unlikely explanation of all transitive behaviors in animal studies. While the theory’s predictions fit with the results of intermixed training, in other training procedures, e.g., sequential training (and most often the animals are trained sequentially), the VTT assumptions produce false predictions, such as B acquiring even higher value than A, or non-transitive behavior in the test trials with BD, which is directly opposite to the results (Vasconcelos 2008).

Reinforcement-Based Models

Reinforcement-based models (RBM) also rely on the stimuli’s rewarding history, but with an attempt to overcome the limitations of VTT models and make accurate predictions in both sequential and intermixed training procedures. The RBM assume that both reinforcement and non-reinforcement have an effect on the associative value of stimuli: reinforcement increases it and non-reinforcement decreases it. The overall value of a stimulus is determined by its reward/non-reward ratio. Wynne’s (1995; 1998) configural RBM⁶ assumes that each stimulus differs functionally depending on the context, i.e., when paired with different stimuli. The model assigns both (context-independent) elemental and (context-dependent) configural values to the stimuli, and updates each value on a trial-by-trial basis after reinforcement or non-reinforcement, based on different equations.

The configural model can explain success in standard 5- and 6-item transitive tests, but faces problems with the bias-reversal modification of the experiment. This modification involves making the item D (which stands ‘lower’ in the series) associated with more rewards. After the standard training with the four pairs (AB, BC, CD, DE), the pair DE is trained additionally a large number of times, therefore producing richer reinforcement history for D than for B. Here the RBMs predict an anti-transitive choice in the test pair BD, however, both pigeons and crows continued to exhibit transitive behavior by choosing B, and not the most rewarded item D (Lazareva et al. 2004; Lazareva and Wasserman 2006; Lazareva 2012). The problem seems to arise from the assumption that both elemental and configural value of D are updated in overtraining with DE. This causes D to surpass B in associative value and leads to the incorrect D > B prediction. If the model updated only configural value of DE, it would yield correct predictions post-bias reversal. It seems that some additional assumptions are needed in order to adjust the model so that it covers this modification of the experiment, while keeping its other advantages (see Lazareva 2012).

Neural Network Models

A special subclass of RBM includes neural network models. A group of authors developed a biologically inspired neural network model based on the functioning of different areas of the medial temporal lobe, aiming both to explain the behaviors in the studies and to offer insights into the role the hippocampus in the transitive behavior (Van Elzakker et al. 2003; Frank et al. 2003).

The model relies crucially on higher-order, conjunctive representations of the stimuli, and the blocking effect that results from them. While the processes in the learning phase involve sophistication and complexity seemingly beyond what is typical of associative accounts, the choices in the test phase rely on simple associative strategies: selecting the stimulus with the highest associative value. In the training phase, the hippocampus encodes conjunctive representations of the stimuli that appear together (e.g., AB, BC…) and associates them with particular rewarding outcomes. In training with end-items, this creates a blocking effect: in A > B, since A is reliably positive, it hinders negative conditioning to B. The subjects can only focus on A to get the reward, and need not remember to avoid B, resulting in little or no increment in negative value associated with B. And since training B > C leads to creating a positive association for B, B gets an overall stronger positive association than it would without the blocking effect. The same (just inverse) effect occurs for stimuli at the other end of the series, D and E. This leads to distributing the associative values in a way that makes B have a much stronger positive association than D, even though B, C, and D are nominally rewarded equally often.

The advantage of this account is that it explains through a unified mechanism the differential behaviors of rats with both an intact and lesioned hippocampus, as well as in both the 5- and 6-item tests. The model also doesn’t seem to have a problem incorporating different training procedures, giving it a clear advantage over the VTT models. Although it was not tested directly against bias reversal, overtraining of DE should not increase the value of D, due to the blocking effect. This would give the model advantage over other reinforcement-based models. Further support comes from the fact that the blocking effect is already known in learning theory (Kamin 1969; Wagner and Rescorla 1972), giving legitimacy to this account. However, other TI studies with hippocampal lesions in mice and pigeons provide conflicting evidence either for or against this account. This variability might indicate a difference in the neural underpinnings of TI across species, but certainly suggests that further research is needed to gain clarity about the role of the hippocampus in TI behavior (Van der Jeugd et al. 2009; DeVito et al. 2010; Lazareva 2012). Furthermore, this model seems unsuitable to explain performance tasks with longer series. The model predicts non-transitive behavior, or even below chance success for any non-adjacent pair that does not involve next-to-end-stimuli (e.g., CE in a 7-item task). This is directly disconfirmed by findings that different species make a transitive choice in CE and other pairs in the 7-item TI task, e.g., corvids: Bond et al. 2003; Bond et al. 2010; rhesus monkeys: Merritt and Terrace 2011; Rapp et al. 1996; Treichler and Raghanti 2010; Treichler et al. 2007; Treichler and Van Tilburg 1996; pigeons: Von Fersen et al. 1991.

List-Linking

Another important challenge for the account proposed by Frank et al. is the list-linking variation of the TI experiment. In this modification of the experiment, the subjects learn two lists independently: e.g., A > B > C > D > E, and 1 > 2 > 3 > 4 > 5, and then the lists are ‘linked’ by introducing the pair E > 1. This variation of the experiment tests whether the linking will lead to integrating all of the previously learned pairs into a representation of a single, larger list. The versions of this modification vary from combining two 5-items list into a single 10-items list, tested with macaque monkeys (Treichler and Van Tilburg 1996) and pinyon jays (Wei et al. 2014), to three 5-item lists into a 15-items list (Treichler 2007; Treichler et al. 2003). The common results in these studies show that, after the ‘linkage’, e.g., learning the pair E > 1, both monkeys and pinyon jays show significant success in the between-list pairs, which supports the possibility that the subjects formed an integrated representation of a longer list. Especially important are the pairs of stimuli that occupied the same position in their original lists e.g., B > 2, C > 3, etc. and the pairs where the stimulus higher in the linked list occupied a lower position in the original list than the lower stimulus, e.g., D > 3. It also seems to suggest that the associative values of the items lower in the list somehow depend on the values of the items higher above them. The fact that the pairs in the second list were affected by learning only E > 1 seems to indicate storing some kind of relational information about the stimuli that were not presented together. It is unclear how these results could be explained if the animals are only relying on individual absolute associative values to make choices in the test phase. Therefore, all associative models proposed so far still need to find a way to explain the results of list linking studies.

3.2.3 Behavioral Signatures Observed in TI Studies

Symbolic Distance Effect (SDE)

Theories of TI must aim to explain success in solving the tasks, but also the specific behavioral patterns observed during training and testing. These patterns include systematic differences in accuracy and response time for particular items or pairs within a series. A particularly relevant pattern is the symbolic distance effect (SDE): the phenomenon in which faster reaction times and higher accuracy correlate with the larger distance between stimuli on a judged dimension. (Moyer and Bayer 1976). In TI tasks, subjects typically choose items farther apart in the series (e.g., BF) more quickly and accurately than those closer together (e.g., BD). The SDE is found in humans, in various tests that require comparing stimuli along a judged dimension, such as size, height, numerosity, but also symbolically represented numerals or letters of the alphabet (e.g., Moyer and Landauer 1967; Bryant and Trabasso 1971). In TI studies with non-human animals, clear SDE was found in tasks with 6 and 7 items and in list linking studies with 10 and 15 items (Von Fersen et al. 1991; Bond et al. 2003; Treichler and Van Tilburg 1996; Van Elzakker et al. 2003).⁷

Many authors interpret the SDE as evidence of integration of items into an ordered series (Chalmers and McGonigle, 1984; Gazes et al. 2012; Bond et al. 2003; D’Amato 1991; Maclean et al. 2008; Merritt and Terrace 2011) or of forming an inferred linear representation (Gazes and Lazareva 2021) and thus as supporting cognitive accounts. The closer the items are in a series, the closer they are in a mental line, which makes discriminations more challenging. However, the support for cognitive models is not straightforward. The SDE implies the existence of noise in the representations, which is what makes discrimination of stimuli of similar magnitudes prone to errors (Gallistel and Gelman 2005; Beck 2012). How exactly the mental line accounts accommodate noise, and how they determine the distances between the items requires further clarification in order to make the SDE genuinely explained by the cognitive accounts. Some authors doubt there could be a viable way to provide such details without resorting to associative values and undermining the starting assumptions of spatial accounts (Vasconcelos 2008; Van Elzakker et al. 2003). Furthermore, models based on differential associative values can also explain the SDE by predicting easier discriminations between items with larger differences in associative strength than those with similar values. Numerical analyses support this, indicating that associative models, whether cast as VTT or RBM, predict the SDE based on the differential associative strength (Vasconcelos 2008; Lazareva 2012). Overall, it seems that differences in associative values more easily explain the SDE than distances on a mental continuum.

Serial Position Effect (SPE)

Another behavioral pattern observed in TI studies is the serial position effect (SPE). Accuracy and speed follow a U-shaped function, with pairs at the ends of the series, e.g., AB and DE, being better solved than interior ones such as BC and CD (Bryant and Trabasso 1971; Woocher et al. 1978). This effect is not surprising, since the pairs involving end-items are the easiest to solve due to consistent reinforcement. The SPE was observed across species, including young children (Bryant and Trabasso 1971) human adults (Greene et al. 2001), and non-human animals (Dusek and Eichenbaum, 1997; Gillan, 1981; Higa and Staddon, 1993; Lazareva et al. 2004; Lazareva and Wasserman 2006; McGonigle and Chalmers, 1977, 1992; Treichler and Van Tilburg 1996; Von Fersen et al. 1991; Wynne 1997). Overall, the SPE seems almost as prevalent in TI studies as transitive performance itself (Vasconcelos 2008), so a plausible account of TI must also account for the ubiquitously observed SPE. All associative models accommodate the SPE through differential associative values. End-items receive unequivocal associative values: either exclusively positive for the first item or solely negative for the last, unlike the internal items, which receive both reinforcement and non-reinforcement depending on their pairing.

Cognitive accounts may struggle to explain the SPE and in the best case need additional assumptions to accommodate it. Most accounts postulate that the linear series is mentally constructed from the ends inward, with the always rewarded and always non-rewarded stimuli placed first, and the internal ones then ordered between them. Once formed, the mental line is used for solving tasks through a spatial search along the line, making the pairs with end-items the easiest to solve. However, this assumption clashes with sequential training procedures, which introduce one end of the series last. One way to deal with this is to claim that the premise representations are integrated into a mental line only during the testing phase, but this is proposed only by a minority of these accounts (Vasconcelos 2008). Overall, further assumptions and details are needed for the cognitive accounts to plausibly explain the SPE.

3.3 Diagnosis

Both types of accounts struggle to explain the behaviors observed in TI studies. In short, cognitive models lack specificity about how the linear representations are acquired through reinforcement, how the information gets retrieved in the test phase, and how different training procedures influence performance in training and test phases. They also face empirical problems, such as the assumption of end-inward line formation clearly conflicting with the sequential training procedures. They also struggle to explain how the linear arrangement of stimuli predicts behavioral effects such as SDE and SPE, and which retrieval processes give rise to these effects.

I will argue that the problem with cognitive accounts stems from their main claim that the relational representations should be conceived spatially, such that elements or nodes on the mental line are understood as lacking individual values. Therefore, these explanations cannot appeal to individual value strengths of the stimuli. However, the reinforcement procedures quite plausibly create positive or negative associations of varying strengths, contributing to the observed behaviors. In the positive part of the paper, I propose an alternative understanding of a relational representation – a non-spatial, affective representation that stores information both about the individual values of the elements and the relations between them. This approach would resolve problems with explaining SDE and SPE, since these would be accounted for by the individual values of the elements stored in the representation, in the same way in which associative models account for them.

Concerning the associative models, even the most successful among them seem to be in conflict with one or another piece of empirical data. While they explain the SDE and SPE without difficulties, and are able to accommodate some of the results of TI studies (especially those they were initially devised to account for), they make false predictions in studies with different procedures. The VTT fails in sequential training procedures, the RBMs in bias reversal modification, and the NN models in longer series tasks such as 7-item and list-linking tasks. This might indicate a more general problem with these kinds of accounts. First, each model seems to be limited to one or two principles previously known from psychological research. Both value transfer mechanism and the blocking effect have been found in other work, but each of the proposed models rely mostly on one such mechanism. It seems reasonable to assume that multiple mechanisms might be at work, either simultaneously, or in different settings, or in different phases of learning. This has already been suggested in the literature, although without further specifications (Bond et al. 2010; Moses et al. 2006). Second, the RBMs, including Frank et al.’s NN model assume that the mechanism driving learning continues remains uniform throughout later stages. It seems worth exploring the possibility that after a certain ‘critical’ point when the training pairs are memorized well enough, this results in a formation of a stable representation, or a ‘relational model’ that could alter the cognitive dynamics. In such case, for example, overtraining of particular pairs such as DE would not cause a bias reversal, but would instead merely confirm the already formed model of the relations among these stimuli. Supplementing the associative models with the possibility of a stable relational representation based on prior associatively-learned values could also explain the list-linking results. It would explain how learning only the linking pair E > 1 enables comparing items C and 3 even though nothing new about these items was learned.

To sum up, the positive account I will now present differs from the cognitive accounts in not relying on a spatial understanding of relation and in including values for individual elements. It also differs from the associative accounts in that the updating of associative values doesn’t proceed indefinitely in a uniform way, and that it might result in forming stable representations that store the information about the relations between the individual items.

4 Affective Representations and Inference over Affects

4.1 Affective Representations between Association and Cognition

As noted above, a significant aspect of the discussion on transitive inference is cast as a debate on whether the observed behaviors are better explained by an appeal to associative or cognitive mechanisms. This is part of a more general practice (the ‘Standard Practice’) in comparative psychology concerning many observed intelligent behaviors of non-human animals such as causal cognition, mindreading and other capacities, for which it has also been debated whether the results of empirical research are best explained by appeal to associative or cognitive mechanisms. This practice has already been criticized for being oversimplified, relying on conceptual confusions, and leading to unproductive discussions (e.g., Allen 2006; Papineau and Heyes 2006; Buckner 2011, 2018; Dacey 2016, 2017). One of the central assumptions of this practice that has led to many such disputes is that associative and cognitive processes are mutually exclusive alternatives.

Without going into too much detail, one of the reasons for criticizing this assumption is directly related to the latest generation of associative models – such as Frank et al.’s (2003) neural network model presented above – which seem to resist being neatly classified as either cognitive or associative models (Penn and Povinelli 2007; Buckner 2011). While the model is initially offered as a deflationary associative alternative to cognitive, relational accounts of TI, since the animals make choices based on differential associative values of individual stimuli, the underlying mechanisms it postulates rely on encoding of conjunctive representations of stimuli that have co-occurred together in the training phase, which enables resolving the ambiguity of the associative values of inner stimuli (such as B, C, and D in a 5-item task). The complexity of the mechanisms seems open to allowing for some forms of flexibility, which is one the central features of cognitive accounts (Buckner 2011; Buckner 2015), although the model requires some improvements to also accommodate longer series tasks such as list-linking. The features of models such as this give rise to the question whether the model actually implements an integrative, cognitive explanation of transitive behaviors, pitched at a lower level of analysis, that is, whether it provides a neural implementation of a cognitive explanation, rather than being a deflationary alternative to it (Buckner 2011; Buckner 2015).⁸ While the idea that cognitive processes may be simultaneously described by associative models at a lower level of analysis is not new (Fodor and Pylyshyn 1988; Buckner 2015; Buckner 2018), the question remains how exactly the cognitive processes driving the observed transitive behaviors are connected to the associative processes of the lower level. The previous passages give some reasons to think it might be worth to explore the possibility of relational representations that also store information about the individual values of elements in the TI series, acquired through one or another kind of associative process. Here I argue that, if we substitute spatially interpreted cognitive accounts with the one postulating affective representations, there is a much clearer path from associative mechanisms to an integrated representation that allows more flexibility and that can explain success in longer series such as 7-item or list linking tasks. I will not try to devise a lower-level explanation of how such representations are formed and through which exact mechanisms. I believe that a further, improved version of a biologically inspired model like the one proposed by Frank et al. might provide that. I am here interested in exploring an alternative relational representation than the ones proposed in the literature, such that could emerge on the basis of reinforcement associative learning, and investigating how, once such representations have supposedly already been formed, they could be further specified and situated within higher level explanations. Importantly, contrary to the discussions about how exactly we can differentiate associative from cognitive processes, I am trying to define the middle ground, or the connection between them. This is where I introduce affective representations (AR) as an intermediary concept with features that enable it to be a common ingredient to both kinds of accounts. I have indicated in Sect. 3.2.3 in what way ARs can arise from the associative reinforcement history and figure in associative explanations. Here I turn to their possible role in cognitive explanations.

4.2 Representational Properties of Affects (Semantic Properties)

Affective representations are not widely recognized in the literature on mental representations, nor are they discussed in animal cognition research. On the other hand, in the philosophical literature aimed at defining emotions and affects, it is uncontroversial that most affective states are experiences that involve an intentional object (they are object-directed) and an appraisal or evaluation of that object, often cast in representational terms.⁹ The experience of affective states consists in experiencing the intentional object in evaluative terms. Affects in general are characterized by valence – positive vs. negative. They qualify their intentional objects in a particular way – as good vs. bad, or as attractive vs. aversive, or in more complex emotional terms.¹⁰ This involves assessments of environmental or bodily events, and their relevance either to previously formed goals, or to the underlying values stored subcortically as dispositional properties of reward-systems in the basal ganglia. And they can also involve reappraisals of the same intentional objects (De Sousa 1987; Roberts 2001; Döring 2007; Gunther 2004; Deonna and Teroni 2015; Carruthers 2017; Gross 2015; Scarantino and de Sousa 2022). For example, sadness is considered as being crucially connected to the evaluation or a representation of its object as a loss, fear as connected to the evaluation of the object as threatening (Deonna and Teroni 2012).¹¹ The metaphor ‘wearing rose-colored glasses’ hints at how affective states towards someone or something makes us evaluate the intentional object differently than if there was no affective state, or if there was a different affective state involved. One reason in support of thinking of affective states as representational is that they are generally outward focused, in the sense that the focus of the subject of the affective state is targeted on the intentional object.¹² For example, if someone is afraid of a bear, it is the threatening aspect of the bear that makes the frightened person have the negative evaluation of it – it is the bear that seems threatening (Carruthers 2017).¹³

Another argument in favor of an evaluative understanding of affective states is the recognized crucial role they play in human practical reasoning and decision making (Carruthers 2017; Damasio, 1994; Gilbert and Wilson 2005, 2007; Buckner 2010; Seligman et al. 2013). When choosing between different options, for example, we respond to the options affectively, with positive or negative valence directed at the alternatives. The options are evaluated as good or bad, or better or worse, which provides intrinsic motivation to pursue or avoid them (Carruthers 2017). Coming back to the transitive inference studies, note that this is also complimentary with the fact that associative values of the items in the series have an important role in the choices made between them. This shows a potential ‘double role’ of affective representations, such that they can participate both in associative accounts in a more elementary way, as well as in cognitive accounts, where they can have more complex roles. I will provide support for the latter type of roles in the following section.

4.3 Suitability for Participating in Inferences: Structure (Syntactic Properties)

Affective experiences can be analyzed in terms of structure because they have distinct components: intentional objects and evaluative properties. These two components correspond to the object and the properties assigned to it, and jointly constitute the content of a particular affective representation.¹⁴ Affective representations can thus be understood as structured, complex states, which allows them to be constituents of inferential transitions and a viable alternative to sententially structured representations.

As discussed in Sect. 2, an important structural feature of representations considered necessary for inferential transitions is having sentential, or subject-predicate form.¹⁵ The subject is something that the representation is ‘about’. The predicate functions to qualify the subject in a particular way. This is taken as a paradigmatic type of structure for sentential representations, and as such they are suitable for participating in inferences. Episodes of affective experience, on the other hand, most often involve qualifying or appraising their intentional object as ‘being in a particular way’. As already noted, affects in general are characterized by a positive or negative valence, in the sense that they qualify their intentional objects, for example, as attractive or aversive, etc. The valences are, further, characterized by having a degree – affects are felt with a positive or negative valence at a particular degree of strength. They can thus qualify their objects as, say, more or less attractive, or more or less aversive. Valence and degree importantly determine the evaluative character of affects, since it is through them that the subjects experiencing affects qualify the intentional objects in a way distinct from other kinds of representations.

There are seemingly two possible strategies in how to translate these features of affects into the discussions about mental representation and inference. One is to translate the intentional object and its appraisal into a sentential structure as subject and predicate. In this case, it would mean claiming that affects involve predication and have sentential form. Alternatively, the appraisal component can be claimed to function as an attribute, which would mean that affects are more similar to noun-phrase-like structures. These structures can be formalized in the following way. Affects represent the object of representation (o), evaluated as having a value (V) with a particular degree (D), forming a noun-phrase-like structure ‘D(V(o))’ in the attributive interpretation, or forming a sentence-like structure ‘O is D(V)’ in the predicative interpretation. I am staying neutral regarding these two interpretations, as the choice might depend on how exactly we understand the requirements for predication. Regardless, both interpretations allow drawing the conclusions needed for my proposal about making inferences that involve affective representations. Coming back to the initial assumptions of the traditional view, whether the main critical target of this analysis is Assumption 1 (‘elements of inference must be sententially structured’), or Assumption 2 (‘only linguistic vehicles can have sentential structure’) depends on whether the attributive or the predicative interpretation is adopted. If we adopt the attributive interpretation, this analysis shows that inferences can be performed over representations that are not sententially structured (against Assumption 1), but we remain silent on whether sententially structured representations are necessarily linguistic. Alternatively, we can claim that sententially structured representations do not necessarily require linguistic capability (against Assumption 2), but do not question whether inferences can only be performed over sentential representations. The choice of strategy here is determined by how ‘predicate’ is understood, and whether we take the appraisals to be bounded by the context, and also how we understand context-boundness (Burge 2010). I do not consider this distinction as crucial for the rest of the argument. In either case, affects seem structured in a way that allows assigning a feature to an entity without needing a linguistic expression. Another important feature of representational systems that can sustain inferences is compositionality. Representations need to be able to systematically assign the same property to different objects. Due to their structure that can involve different evaluations and have different entities as objects, affects have the kind of structure that permits assigning the same property to different objects (‘A is good’, ‘A is bad’), or assigning different properties to the same object (‘A is good’, ‘B is good’). Affects thus satisfy the condition of compositionality that is important for making inferences regardless of whether the appraisal is understood as predication or attribution. Admittedly, affects may be unable to form elaborate structures or reach a degree of flexibility that linguistic expressions or LOT representations can support, but the particular structure revealed here should suffice for sustaining at least some kinds of inferences as well as allowing some degree of flexibility. I will now present how ARs as analyzed here can be applied to describe the transitive inference task, and then come back to a few remarks on their potential for flexibility.

4.4 Affective Representations and Dual Information

While the associative accounts postulate only individual values associated with items (e.g., V(A) = 5; V(B) = 4; V(C) = 3), and the cognitive accounts postulate a mental line that stores only the relations between the items without the individual values (A > B > C), the AR view postulates representations that store both kinds of information. This ‘dual information’ view makes sense for several reasons.

First, the structure of the task is such that individual items are orderable along a dimension that automatically contains the relations between them. Individual values associated with the items and the relations among items co-constitute each other: by virtue of having degrees, these values already stand in relations of ‘higher than’, ‘lower than’ (or ‘equal to’), and therefore give preferential relations among items ‘for free’. In other words, attributing values of, say, 5, 4, and 3 to items A, B, and C, respectively, implies a preference of A over B, and B over C, because higher valued items are preferred over the lower ones. Conversely, although we can imagine an abstract relation of preference that is not instantiated in any individual values of the elements, if an agent is exhibiting a consistent preference ordering across contexts, there are most probably some individual values that encode this ordering. Preferring A over B seems to imply that there are some individual values of A and B such that A’s value is higher than B’s value. Different individual values and the corresponding preferential relations are thus two sides of the same coin.

Second, there is empirical evidence of multiple encodings of the same information in both humans and some non-human animal species. Studies on SDE in human comparisons of items along a judged dimension (e.g., circles of different sizes) show that mental representation underlying comparisons preserves not only ordinal, relational information, but also absolute individual values (absolute sizes of the circles). The individual values of the symbols in the series, even though not necessary for success in the task, were still stored along with the relational information (Moyer and Bayer 1976; see also Moses et al. 2006). Studies on SDE in monkeys (D’Amato and Colombo 1990) and pigeons (Lazareva and Wasserman 2006) also support the possibility of multiple encodings in non-human species. Based on these studies and their own TI studies with corvids, Bond and colleagues (2010) suggest that “stimulus relationships in sequential learning designs are probably multiply encoded, and most species would be expected to make some use of both associative and relational representations in operant transitive inference” (p. 285), and that “[m]ost animals probably develop both associative and relational representations of pairs of stimuli, perhaps acquiring them simultaneously as two different aspects of a single encoding process” (p. 290). Bond et al. do not specify how exactly this multiple encoding would look like. I am here offering a way in which both individual and relational information is cashed out in terms of value.

Finally, as I argued in Sect. 3.2.3Diagnosis, both associative and cognitive accounts lack one type of information to be able to accommodate all the results. Introducing representations that contain dual information makes sense because such representations are able to account for both the standard TI test performance, but also the variations such as bias reversal and the list-linking results, as well as the behavioral effects (SDE and SPE). In short, the relational information keeps the representation safe from bias reversal: overtraining D > E merely confirms the model A > B > C > D > E instead of raising D’s value. Relational information is also crucial for linking two separate lists into one and explains how learning only E > 1 leads to transitive behavior across lists. Finally, individual values explain straightforwardly easier discriminations between items with a bigger difference in value (SDE) as well as easier solving of the end-pairs (SPE).

4.5 Affective Representations in Transitive Inference

Here I put forward an account of the animals’ behavior in the TI task that involves inference over affective representations. We can plausibly apply ARs to explain animal behaviors in the TI studies as a result of representing objects as attractive or aversive, and by representing one object as more attractive or aversive than another object, thereby representing relations between the objects (‘more attractive than’) that are transitive in nature. In the following analysis I adapt Burge’s (2010) description of the structure of perceptual experience and propositional thought, in order to provide a formal description of the role of affective representations in TI, step by step in the experiment.

• 1) Perception: (A B)

The first step in the TI task is perceiving the stimuli presented. The animals first perceive two cups, and these perceptions have the form: ‘That₁M_A’ and ‘That₂M_B’, where ‘That’ indicates a perceived object, and these objects are marked (M) with different colors or shapes, which range from A, B, C, D and E, and these different marks are used as the labels of individual cups. For example, for the cups differing in colors, if cup A is marked with yellow color, and B with blue, etc., we will say that M_A is yellow, M_B is blue, etc. Perceiving ‘That₁M_A’ and ‘That₂M_B’ means that the animals perceive two objects as different, and differing by their marks, in this case color. Throughout the experiment, the animals consistently exhibit differentiation between the cups according to their marks and the reinforcement associated with them. Perception of objects and detection of their features should be uncontroversial for all existing accounts (both associative and cognitive). This stage involves only perceptual contents, and does not require affective representations, representations of relations between the cups, nor inference. The animals then interact with the presented cups: they approach one of them, and either get a reward or they do not.

• 2) Representation in memory and affective representation: (A > B)

As soon as some choices start granting rewards, associations are formed. The dual information view is applicable already in this step. First, positive associations are formed toward the stimuli whose choice is rewarded. The positive association is reinforced with each reward for the same stimulus. When the subjects become consistent in choosing the ‘correct’ stimulus A, this is, I argue, when both a memory representation of ‘M_A’ and a positive affective representation ‘V + (M_A)’ are formed. In other words, the rewarded item gains individual value. Second, the task requires learning the correct preference, represented by the relation A > B. However, in this step there might not yet be a representation of M_B, nor of the relation between M_A and M_B. The subjects might only direct attention to M_A and choose it without considering the alternatives (consistent with the blocking effect described above). This could also be described as learning a preferential relation ‘M_A > _’, where only the positively reinforced item is remembered, while the other item’s features (M_B) are less salient or not salient at all. Framed in terms of the dual information view, because of the asymmetric salience of the two items (or the blocking effect), information about the preferential relation (A > _) collapses into the information about the individual value of the salient item (A +).

The same description applies to learning the second ‘premise’ (B > C). Since M_B probably didn’t receive much attention in learning A > B, training with B > C leads to learning the preference ‘M_B > _’, such that a representation of M_B is formed in memory as well as a new affective representation ‘V + (M_B).’

• 3) Learning degrees and/or explicit relations through intermixing: (B > C, A > B)

In the most common TI learning procedures, after 80% success at B > C, the experimenters standardly introduce another part of the procedure: intermixing premises AB and BC until both are solved with 80% success rate. This imposes new requirements on the subjects: perceiving cup B paired with A prompts one action (choosing A), yet perceiving B paired with C prompts a different action (choosing B). Making differential choices depending on the context requires representing relational information about B and A, as well as about B and C. These transitions can be described as conforming to the rules:

$$\begin{array}{c}If\;{M}_{B}\;or\;{M}_{A},\;choose\;{M}_{A}\\ If\;{M}_{B}\;or\;{M}_{C},\;choose\;{M}_{B}\end{array}$$

Even though the mental transitions conform to such rules, it doesn’t mean that the animals are representing the rules in this conditional form. In line with the criticism of the Assumption 2 above regarding the second-order dynamics, there is no a priori reason to insist that an agent cannot make an inference without explicitly representing the rules or the epistemic relations between the representations involved. Making differential choices in AB and BC does not necessitate an explicitly represented rule. It can also be guided (or constrained) by an implemented rule.

There is another way to describe a possible implemented rule, without appealing to conditionals. This rule is significantly connected to how the items are affectively represented. The intermixing procedure ‘adjusts’ the order of values of the items A, B and C, to conform to the rule:

$$V({M}_{A})>V({M}_{B})>V({M}_{C})$$

To conform to these rules, the subjects need either to represent both relations M_A > M_B and M_B > M_C, or to assign different individual values, for example, V(M_A) = 5, V(M_B) = 4, and V(M_C) = 3. I argue that, due to the co-constitution between individual values and preferential relations discussed above, both kinds of information are encoded in the affective representations. Even though not both are necessary in this step – one would suffice, it makes sense that both are present anyway, for the reasons explained in 4.4.¹⁶

• 4) Further representations of objects and relations: (C > D, D > E)

Learning the rest of the premises (C > D and D > E), doesn’t seem to require anything additional from what was used in the previous steps. These steps might only be more demanding for their memory, since they are acquiring further representations of objects and additional affective representations, as well as relations between the objects.

• 5) The test task: (B D)

Finally, the test task seems to impose new conditions, since the subjects are presented with two familiar stimuli that they never encountered together before. They consistently choose B, which is a transitive choice in light of the other, directly acquainted relations. How exactly do they solve this step of the task, which is the only one that supposedly requires transitive inference? First of all, it is unlikely that they are performing classical stepwise transitive inference of the form:

$$\begin{array}{c}B>C\\ \text{C}>\text{D}\end{array}$$

Therefore, B > D.

Such stepwise inference would contradict the finding of the SDE: the farther the items are in the series, the more steps are needed to reach the transitive conclusion, resulting in longer response times and more chances for errors. In other words, the classical stepwise TI would yield a reverse SDE, which is directly opposite to the findings. Therefore, the conclusion needs to be reached by a different kind of transition.

Different degrees of individual values suffice to solve this task after a standard training procedure, as most associative accounts have already shown. However, I argue here that all the previous steps together with the intermixing procedures lead to an integration of different pieces of information: from separately learned A > B, B > C, C > D and D > E to an integrated A > B > C > D > E. This integrated affective ‘relational model’ contains both preferential and individual value information: ‘V₅(M_A) > V₄(M_B) > V₃(M_C) > V₂(M_D) > V₁(M_E)’. I will shortly summarize the support for the claim that this representation a) contains dual information, and b) integrates information from all the premises. First, I have already presented the main reasons for dual information view in Sect. 3.2.3.1., where I explain why even though both kinds of information are not necessary for solving this task, they seem to be present. Additionally, they both seem to be available in the modified, more demanding versions of the task such as bias reversal and list-linking. It seems more likely that the information is already stored before the modifications, rather than the subjects are suddenly adding relational information of the whole series when the linking pair E > 1 is introduced, or when they start receiving additional training with D > E. Second, the reason for the affective representation being integrated rather than split into several separate relations was stated above: the stepwise inferences based on non-integrated separate relations predict the reverse SDE, opposite to the findings in all the longer series versions of the task (including list-linking).

But is this an Inference?

Finally, is the mental transition described in step 5) an inference? It definitely doesn’t fit the classical stepwise characterization of TI nor even more generally the traditional syllogistic inference over sententially structured representations. However, if we don’t limit ourselves to the assumptions I criticized in Sect. 2, we can consider it an inference in a broader sense. The transition from having learned the separate premises A > B, B > C, C > D and D > E to the conclusion B > D is characterized by a degree of indirectness: the conclusion itself is not contained in any of the premises nor is it directly observable by itself. It is instead gathered or ‘synthesized’ from other pieces of information. Separate premises are integrated into an affective representation that stores dual information about the whole series. This AR acts as an intermediary element that connects the information contained in separate premises to the information contained in the conclusion, and it mediates the transition from premises to the conclusion. Admittedly, the BD task alone is explainable also by simpler alternative accounts that might not need this intermediary step. But again, when we look at more demanding versions of the TI task (e.g., C > 3 in the list-linking study), characterization of the mental transition as mediated by an integrated representation seems more clearly convincing. Broadening the conception of inference to include indirect informational transitions such as this one is crucial for better understanding of a diversity of cases of non-linguistic reasoning, but also the nature of inference.

4.6 Flexibility of Affective Representations

An important feature of all representational systems is their potential for flexibility. Behavioral flexibility is often looked at as a key piece of evidence meant to distinguish cognitive from merely associative psychological processes. Here I argue that affective representations allow for a degree of flexibility that is sufficient for many kinds of inferences, including the more complex variations of transitive inference, but also that this degree of flexibility is not the full-blown flexibility enabled by language or language-like representations. In line with the criticism of the cognitive vs. associative dichotomy, I also suggest it is more useful to think of flexibility in terms of a continuous range, rather than a dichotomy of flexible vs. inflexible. I will first indicate the ways in which ARs support flexible thought and reasoning, and then proceed to indicate the limits of their potential for full flexibility.

First, in 4.4 and 4.5, I presented the dual information view according to which ARs store information about absent entities and relations. For example, solving the transitive task or the list-linking tasks relies on storing information about items and relations that are not currently shown to the subjects. In other words, ARs have a certain degree of stimulus independence. This transcendence of current circumstances yields a degree of flexibility in the sense that ARs can be deployed in a broad range of situations, without being tied to perceiving all the relevant affectively represented entities. Second, in 4.3 I proposed an analysis of ARs that reveals them as structured and composed out of distinct, recombinable components. This makes them apt for flexibility in the sense of combining different components in an inferential process similarly to inferences over sententially structured representations. Third, subjects can apply ARs to an infinite variety of intentional objects, evaluating different objects in the same way or in different ways, or evaluating one and the same object in different ways. This implies a degree of flexibility in potential objects of representation. Fourth, ARs are revisable in light of new evidence. As we have seen in the list-linking studies, learning the linking pair E > 1 leads to an update of the values of the items in the lower list in a way that seems to adapt to this new evidence. Finally, because ARs allow integration of different pieces of information into a complex representation, they are capable of context sensitivity. Intermixing of premises AB and BC in 4.5 shows that ARs do not limit the subjects to uniform reactions to the same stimuli, but allow differential reactions to the same items in different contexts. Once they integrate the learned information into A > B > C, the subjects show consistent selective reactions to B depending on whether it is presented with A or with C.

Additional support to this view comes from recent neuroscientific and cognitive science research which suggests a novel conception of the affective system as a flexible learning system. The affective system is understood as involving reward, emotion, mood, affectively charged or valenced memory. Even though there are still disagreements over the contribution of particular brain regions to these functions, there seems to be evidence for the view that both in animals and humans the affective system functions as a flexible, experience-based information-processing system, capable of tracking statistical dependencies and of guiding behaviors in a way that is strikingly similar to how rational procedures for decision making are understood in philosophy (Railton 2015; Schultz 2002; Quartz 2007; Schwarz and Clore 1983, 2007; Carver and Scheier 1990; Craig 2009; Pessoa 2008; Zahn et al. 2009). Based on this research, Railton (2015) considers the information structures learned by the affective system a representation, arguing that “it exhibits information-value-sensitive processing that is substantially independent of current experience or context” (p.838).

On the other side, inherent properties of ARs limit their flexibility in at least two ways. As Camp (2009a, b) notes when discussing the flexibility of concepts, there are two senses of being free or under stimulus control. One sense concerns transcending current circumstances, and ARs seem free from stimulus control in this sense, as I explained above. Still, ARs seem limited regarding which of the potential representations can be formed by the subjects. Since they are by definition based on experience and/or experiential history, which of all the potential representations are formed depends on the actual experience of the subject. I call this the ‘experiential’ limitation. While in principle any object can be affectively evaluated, the subjects need to be prompted by external circumstances and internal reactions for the ARs to exist. For example, an actual affective response to a novel object, say, a spiral shape¹⁷ is necessary to prompt an AR of that object. This makes ARs systematically recombinable only in a causal counterfactual way, to use Camp’s terminology: should the occasion arise, they could be applied to infinitely many new objects, but the combination of the particular intentional object and its evaluation needs to happen in experience.

The second limitation concerns a closely related aspect of flexibility: systematic recombinability. I call this the ‘material’ or ‘semantic’ limitation. Due to the fact that affective experiences are concrete and embodied (and not abstract like linguistic representations), and that they are inherently valenced, not all combinations of object and evaluation are (equally) possible, even though infinitely many intentional objects can be affectively represented. For example, while it is possible to make affective evaluations such as FEAR(spider), FEAR(pain), FEAR(dog), JOY(dog), JOY(food), etc., it is not impossible, but it is very difficult to affectively represent FEAR(food), or JOY(illness). Some potential intentional objects have a predetermined positive or negative influence on the subjects, which could make them incompatible with an evaluation of the opposite valence. I wouldn’t go so far to claim that such valence-conflicted combinations are impossible, but I would argue that it is highly difficult to form them, even if they were prompted by the appropriate experience. In comparison to linguistic representations, this would be similar to not (or hardly) being able to combine particular nouns with particular adjectives or verbs. We might not ever need to combine them or they might not fit well together semantically, but whenever we want to, we are perfectly able to make false or nonsensical combinations. Affective representations, on the other hand, seem constrained by material factors such as the valence and the influence of the intentional object and the compatibility with the potential affective evaluation.

To sum up, affective representations support a certain degree of flexibility of thought and reasoning by being able to represent absent entities and relations, by having recombinable components, by being applicable to an infinite variety of potential objects, and by being context sensitive and revisable in light of new evidence. However, unlike LOT representations, flexibility of ARs is constrained by the experiential and material limitations, so that even though all potential object-evaluation combinations are possible in principle, they need to be given in experience, and.

even then they require a degree of compatibility between the valences of the object and the evaluation. The affective representational system can be understood as capable of ‘mid-range’ flexibility: not fully systematically recombinable and stimulus independent, but flexible enough to be suitable not for all, but for some kinds of inferences.

5 Conclusion

The main aim of the paper was to offer an account of the success of non-human animals in the transitive inference task. To create a framework for such an account, I first needed to clear the way from the unjustified constraints of the mainstream view on the kinds of elements and processes to be involved in inferences. I argued against the view that linguistic capability is necessary for making inferences. As I discussed in Sect. 2, even though a lack of human language does not automatically imply a lack of structured representations, one obstacle for going further with the idea that animals can make inferences was that no viable alternatives to linguistic vehicles have been proposed. The burden of proof was on those who argue against such a constraint to provide a viable alternative. The strategy of this paper was to present affective representations as a candidate for such an alternative and show how it is possible to explain the processes underlying inferential reasoning that conform to the rules of logic and rationality, but with affective states as constituents. This account opens the space for an inferential procedure operating over ARs, but also for a different kind of inference than the one assumed by the mainstream views. Applying affective representations to explain the case of animal transitive inference fits with the experimental data and does not give rise to the problems facing the existing approaches. It seems that this grants some explanatory value to introducing such representations. Their value in other theoretical roles would need to be further assessed, and there is room to discuss the further details and commitments of the account. For now, I take it as sufficient to offer a framework in which we can think about mental representation and inference more broadly, and open up new ways of explaining the intelligent behaviors already observed and documented in many animal species. Importantly, even if ARs are not fully spelled out, this proposal opens the way to introducing intermediate representations that are less demanding than LOT representations, but more demanding than associations, which seems to be needed in interpreting many findings in comparative cognition.