Abstract
Analogical reasoning projects a property taken to hold of something or things (the source) to something else (the target) on the basis of just those similarities premised in the analogy. Standard similarity-based accounts of analogical reasoning face the question: Under what conditions does a collection of similarities sufficiently warrant analogical projection? One answer is: When a thing’s having the premised similarities somehow determines its having the projected property. Standardly, this answer has been interpreted as claiming that a formally defined determination relation exists between the variables of which the determining properties are values and the variable of which the determined property is a value. This paper supplies another answer: Analogical projection is warranted when an item’s having the projected property is grounded in its having the premised similarities. Drawing on the metaphysics of grounding, we propose a model of grounded analogy on which analogical reasoning is valid under metaphysical necessity. Our model thus provides one answer to John Stuart Mill’s question of how analogical reasoning based on limited information about a single source case can provide an indefeasible justification for analogical projection. Once a relation of total grounding is ascertained to hold between some similarities and the projected property, no other properties bear on the projectability of the projected property from the source to the target.
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Notes
Rather than by argument form, Mill distinguished induction from analogy according to the connection premised to hold between the similarities and the projected property: “In the case of a complete induction, it has previously been shown, by due comparison of instances, that there is an invariable conjunction between the former property or properties and the latter property, but in what is called analogical reasoning no such conjunction has been made out” (Logic, III.xx.2 VII: 555).
Mary Hesse (1966: p. 73) called this the logical problem of analogy.
As most answers to Mill’s Problem do, ours relies on identifying an “invariable conjunction” between the similarities and the projected property of an analogy, which is then cited as a premise in analogical argument. On Mill’s categorization, then, such an argument would be inductive rather than analogical. According to Mill the connection strength of distinctively analogical argument is a function of three factors: the positive, negative, and neutral analogies (as explained later in Sect. 2.1).
The value of an analogical argument[,] inferring one resemblance from other resemblances without any antecedent evidence of a connexion between them, depends on the extent of ascertained resemblance, compared first with the amount of ascertained difference, and next with the extent of the unexplored region of unascertained properties … The cases in which analogical evidence affords in itself any very high degree of probability are … only those in which the resemblance is very close and extensive … (Logic, III.xx.3 VII: p. 559)
Argumentation theorists among our readers will be familiar with a Toulminean sense of ‘grounding,’ which should be distinguished from the metaphysical sense we use here. Toulmin generally used ‘grounds’ to speak of the support an argument offers a claim, and he spoke of the quality of that support in terms of how well a claim is grounded ([1958] 2003). By contrast, when we speak of a grounded analogy, we mean ‘grounding’ in its metaphysical sense, according to which something’s having the projected property of the analogy is metaphysically grounded in its having the premised similarities. With these two senses in hand, we may say that, in some analogical reasoning, the metaphysical grounding relation Toulmin-grounds the warrant: that the analogy is groundedm groundsT our reasoning via it.
The literal analogies relied upon in reasoning by analogy should be distinguished from figurative analogies that can have an illustrative or heuristic, but not a justificatory, function (Cf. Waller 2001).
Sometimes, theorists of analogy and analogical reasoning use talk of different source and target domains to indicate some putatively salient difference in the kinds of items or cases being compared in the analogy from source to target (e.g., Weitzenfeld 1984). For example, the source might be an artificial or mathematical model, and the target the natural world. On the analysis of analogy set forth here, ‘domain/s’ is not used with this connotation. Any difficulty in, or possibility of, analogizing across such “qualitatively different” domains, and hence reasoning by analogy across them, is left to the possibility and construction of the mapping function, φ, that defines the analogy.
For simplicity, the table given in Fig. 1 represents an analogy with only single source and target cases. Were there to be several source or target cases, the entries in the left-hand and/or right-hand columns would be repeated for each case accordingly.
It also roughly matches the schematization appearing in Walton et al. (2008) “User’s Compendium” in Argumentation Schemes, which omits claim [1] (Walton et al. 2008: p. 315; cf. 55–60 especially their “version 1” scheme). Our similarity-based scheme is actually a hybrid of two schemes distinguished by Walton (2014: pp. 24ff., 28ff.). His “first scheme” omits claim [1] from our standardization, while his “second scheme” omits claim [2]. (The (2008) “User’s Compendium” lists Walton’s (2014) “first scheme.”).
Woods and Hudak (1989) present analogical argument as having the structure of a second-order, by parity of reasoning meta-argument about the similarities of two arguments.
Mill claimed: “There can be no doubt that every such resemblance which can be pointed out between B and A, affords some degree of probability, beyond that which would otherwise exist, in favor of the conclusion [that some further similarity between B and A obtains] drawn from it” (Logic, III.xx.2 VII: p. 556).
Russell (1988: p. 256ff.) argues that, in the absence of prior knowledge of a determination relation or criteria for the salience of (dis)similarities, “it still seems plausible that the most similar source is the best analogue; certainly in the absence of any other information, it seems perverse to choose an analogue that is demonstrably less similar.”
Bartha (2010: p. 19; cf. 2019), for example, identifies the following list:
(G1)
The more similarities (between the two domains), the stronger the analogy
(G2)
The more differences, the weaker the analogy
(G3)
The greater the extent of our ignorance about the two domains, the weaker the analogy
(G4)
The weaker the conclusion, the more plausible the analogy
(G5)
Analogies involving causal relations are more plausible than those not involving causal relations
(G6)
Structural analogies are stronger than those based on superficial analogies
(G7)
The relevance of the similarities and differences to the conclusion … must be taken into account
(G8)
Multiple analogies [e.g., source domains, or cases within a domain] supporting the same conclusion make the argument stronger
Cf. Walton’s (2014: p. 29) survey of Hurley’s text A Concise Introduction to Logic (2003: pp. 469–470) and Copi, Cohen, and Flage’s text Essentials of Logic, 2nd ed., (2007: p. 334ff., list begins on 339).
For example, Walton et al. (2008: p. 62, 315, notation adapted) prescribe several critical questions as aiding the evaluation of analogical arguments.
- CQ1:
-
Is Pt true (false) in a?
- CQ2:
-
Are a and t similar in the respects cited?
- CQ3:
-
Are there important dissimilarities between a and t? Are there respects in which a and t are different that would tend to undermine the force of the similarity cited?
- CQ4:
-
Is there some other case, c, that is also similar to a but in which Pt is false (true)
Like fallacies, critical questions are properly used to “stress test” arguments of a given schematic type for failures to which they are stereotypically susceptible. Well designed sets of critical questions should do just this. For example, when stress testing an appeal to authority (e.g., expert testimony) one would do well to check whether it is a fallacious ad verecundiam by asking whether the authority appealed to is properly authoritative concerning the claim concluded and in the discursive context in which their authoritativeness is invoked as a reason to accept the claim.
Of course, inferring [1*] inductively, where Pa and Pb are premises in the induction, makes the argument question-begging (viciously premise circular). Inferring [1*] inductively without including Pa and Pb in the induction makes premise [1*] at least as uncertain as the conclusion [4*], thereby making the argument incapable of offering epistemic reasons. There are, of course, other, unproblematic, ways of securing the conclusion-independent acceptability of the identity claim [2*] as a premise. For example, it could be given by stipulation, or by prior knowledge (e.g., taking it on authority).
Importantly, Mill seems to have been aware of the nonredundancy problem, as it appears to be the point by which he distinguished between analogical and inductive reasoning:
For if we have the slightest reason to suppose any real connexion between the two properties A [the similarities] and B [the projected property], the argument is no longer one of analogy. If it had been ascertained … that there was a connexion by causation between the fact of … [the similarities] and … [the obtaining of the projected property] … we should then infer from the fact of the ascertained or presumed law of causation, and not from the analogy… (Logic, V.v.6 VIII: p. 794)
A similar example can be found in Weitzenfeld (1984: p. 145; cf. Hitchcock 2017: p. 209ff.) about realtors predicting, and lenders appraising, the fair market value of a house by comparison with recent sale prices of comparable houses where the features shared between the houses (e.g., neighborhood, age, construction, size, updates, etc.) are taken to determine their value. As Weitzenfeld writes:
Realtors know what variables influence the price of a house and assume in the absence of knowledge of the details of the structure, that together they determine the “market value” of the house. This assumption justifies the evaluation of houses by comparison with recent known sales despite the fact that no two houses are identical. (p. 145)
In this respect, the illustrative examples of Determining trade-in value and predicting a house price by consulting the recent closing prices on comparable houses (discussed in the previous footnote), are unrealistic (i.e., implausible) as “complete” determination relations. Rather, they are best understood as instances of “partial” determination relations, where other features of the items in question, or the surrounding circumstances, also bear on the projectability of the projected property at issue. It is for precisely this reason that mortgage lenders consider a variety of “comps” when appraising the fair market value of a house. Realistically, it would be prudent for Bob to search for other trade-ins of cars sharing the same “determining features” as his, before accepting an offer that matched Sue’s.
Here, we limit, by stipulation, our consideration to those determination relations that are “complete” in this respect, for they alone make true Davies’s claim that consulting a single case will conclusively settle the matter. Importantly, expanding our conception to include “partial” determination relations would return us to the situation we found ourselves in with similarity-based analogical arguments—one where we lack an answer to Mill’s Problem, where any solution to the justification problem relies on considerations beyond the partial determination relation itself, and where analogical projection is, at best, inconclusive.
While we adopt Davie's notation for determination relations, we depart from his analysis of them.
We offer our thanks to an anonymous reviewer for suggesting this formulation, and identifying problems with an earlier formulation we had given.
We use ‘∃!’ to express the unique quantifier: ∃!x Px ⇔ ∃x∀y (Py ↔ x = y). As a second-order quantifier we have the wff: ∃!P Px ⇔ ∃P ∀Q (Qx ↔ Q = P). We use ‘⇔’ to mean ‘is logically equivalent to.’
This is offered as a simplest, or single-case definition, which follows our earlier assumption, made to simplify the discussion, that there is only a single determining variable, Φ, and a single determined variable, Ψ. We have already introduced the expressive means to accommodate more complicated cases, and these may be incorporated into our abbreviated, single-case definition as necessary.
In offering this analysis of determination generalization we again depart from Davies (1988), who suggests that determination rules express “the relation of one predicate deciding the truth of another,” thus giving determination rules of the form:
[D*] (∀x Px ⊃ Qx) ∨ (∀x Px ⊃ ~ Qx) (234).
To better express the thought that “Px decides whether or not Qx” Davies introduces a polar variable, i, where ‘iPx’ means ‘whether or not Px.’ Using this notation, Davies expresses determination rules as having the general form:
[D] i1Px > i2Qx,
which reads “whether or not Px determines whether or not Qx” (Davies 1988: p. 236; notation adapted).
We grant that each of [D*] and [D] (when properly interpreted) yield valid analogical argument. Yet, on our view, neither interpretation of determination satisfactorily captures Weitzenfeld’s key thought that knowledge of a determining structure expresses knowledge of a non-accidental relation among variables of which the specific properties cited in the analogy are values (1984: pp. 142–143)—a thought that we take to inform the logical structure of determination-based analogical reasoning. While Davies’s example of Determining trade-in value captures this thought nicely, his formalizations, in our judgment, do not.
Nevertheless, the macrostructure of determination-based analogical reasoning can be more perspicuously displayed using Hesse’s tabular model. Determining relations express a connection between the similarities cited in the analogy and the projected property. Often, these connections are conveniently displayed as vertical relations in the tabular model, and premise [2d] asserts that such a vertical, determination relation holds. Hesse thought that they were necessary: “An argument from analogy requires a certain kind of similarity relation between the horizontal terms of the analogy, and also a certain kind of vertical relation” (1966: p. 77). While Hesse conceived of this vertical relation as causal (the cited similarities cause the projected property), other determining relations, e.g., supervenience, are possible.
Our determination-based model of analogical reasoning bears some resemblances to Bartha’s (2010, 2019) articulation model. Specifically, determining relations provide prior associations that support generalization. Following Hesse (1966), on Bartha’s (2010: p. 25, ch.4) articulation model, a suitable vertical relation, called the prior association, must obtain between the similarities and projected property in the source domain. The second requirement of good analogical reasoning is a potential for generalization whereby the prior association ascertained to hold in the source domain may be generalized to extend to the target domain. The prior association selects for those elements of the analogy that prima facie warrant the analogical projection, while the potential for generalization seeks to ensure that there are no undermining or overriding considerations.
While determining relations seem entirely suitable as prior associations supporting generalization, Bartha (2010: p. 46ff.) rejects the determination-based model of analogy as a general model for analogical reasoning.
Do determination rules give us a solution to the problem of providing a justification for analogical arguments? In general: no. Analogies are commonly applied to problems … where we are not even aware of all relevant factors, let alone in possession of a determination rule. (2019)
We do not take this point to pose an objection to our model of reasoning by grounded analogy, or for the corrective account we offer of determination-based analogical reasoning, since we do not take either to provide a general model for analogical argument. Nor do we seek to provide such a model. While we take our model of reasoning by grounded analogy to satisfy Bartha’s conditions of prior association and generalization, here we seek only to show that it solves the problems of justification and nonredundancy.
We offer our thanks to an anonymous reviewer for calling our attention to the points discussed in this section.
Previously, Trudy Govier (1989: p. 144) had considered the idea that (some) a priori analogies might be schematized as follows:
[1G]
A has x,y,x
[2G]
B has x,y,z
[3G]
A is W
[4G′]
It is in virtue of x,y,z, that A is W
[5G]
Therefore, B is W
where 4G′ is added as an unstated premise to a more traditionally schematized argument from analogy. Her reasoning for reconstructions drawn along these lines is found in an earlier (1985) paper where she, like Hitchcock, considers the reasoning involved in constructing parity of reasoning analogies. “We indicate our sense of what an argument depends on,” Govier writes (1985: p. 30), “when we construct a parallel argument in which the central features of the original are preserved while its incidental features may be varied.”
Govier ultimately rejects this augmented schematization as representing the macrostructure of analogical reasoning, and the proper grounds of analogical projection, on the grounds that it fails the nonredundancy requirement. In Govier’s view (1989: p. 144): “from [4G′] it is a very short step to a universal statement [4G*] of the type:
[4G*]. All things which have x,y,z are W.”
Yet, she observes, while adding [4G*] to the argument satisfies the justification requirement by making the argument valid, it also makes the premises, [1G] and [3G], about the analog redundant and inert in the derivation. Thus, the reconstruction fails the nonredundancy requirement. Information about the source need not be relied on in the reasoning, which, thereby, ceases to be analogical. Govier’s assessment of such reconstructions has been adopted as an orthodoxy within argumentation theory by theorists like Guarini (2004: pp. 161–162) and André Juthe (2005: p. 19) who explicitly follow Govier in their own assessments. (See Waller (2001) for a dissenting opinion.)
Nevertheless, Juthe (2005: pp. 19–20) seeks to save such analogical reasoning from redundancy by claiming that even if a claim like [4G*] holds, our reasoning need not proceed via it. Rather, he claims, we can base the conclusions of our analogical arguments on the analogy, rather than the generalization.
That there is a true universal generalization of all relevant features is one thing; the claim that one must know it and have it explicit so the argument reduces to a deductive or inductive argument is another claim. It is the truth of the latter claim that is necessary to make the reference to analogy redundant. I think that it is more plausible that we often do not know what the completely specified generalization of all relations between particular cases is. But that does not prevent us from perceiving analogical relations between particular cases.
Also, Juthe (2005: p. 11, 2009: p. 139) recognizes a distinctive form of “argument by conclusive analogy,” whose schematization involves a premise to the effect that the source has the projected property in virtue of its having the similarities.
Thus, we adopt the predicational view of the grounding relation as developed by, for example, Audi (2012), rather than the operational view of grounding that authors such as Fine (2001) take up. This is because some of the substantive presuppositions we will make have been helpfully presented in previous work on grounding using the predicational idiom, e.g. in Rosen (2010). For a nice discussion of some of the differences between the predicational and operational views of grounding, see Correia and Schneider (2012: pp. 10–12).
We will assume throughout that causation and grounding are distinct relations. For an alternative approach, see Wilson (2018).
Note that we do not endorse this particular grounding claim; it is merely illustrative. For extensive discussion of the importance of grounding to moral philosophy, see Berker (2018).
These examples should also illustrate that, although the epistemology of grounding claims is somewhat unclear, we do rely on grounding in our reasoning quite often. Some grounding claims may strike us as tenuous, but in other cases they would be hard to give up. Since our interest in this article is only to show that grounding provides the basis for valid arguments from analogy drawn from a single source case, the epistemology of grounding is something we largely set to the side in what follows.
For example, in the case of the debate between internalist and externalist accounts of mental content, both sides will typically agree that mental content is partly grounded in facts about one’s brain (and body). At issue is whether some further grounds are needed to explain mental content.
Rosen (2010) labels this property non-monotonicity, but we prefer to call it anti-monotonicity as a way of drawing attention to how much stronger it is than the mere failure of monotonicity. Grounding is non-monotonic because, for some p, q, and Γ, [p] ← Γ but not [p] ← Γ, [q]. But the property we are interested in is universal: for any p, q, and Γ, when [p] ← Γ, we are guaranteed that it is false that [p] ← Γ, [q] (if Γ, [q]≠ Γ).
The example comes from Trogdon (2013b), who denies that this neural correlate could be a total ground of the visual experience of motion. On the neural correlate of the visual experience of motion, see Block (2005).
Here Rosen treats entailment as strict implication. One may worry that, if entailment is characterized in terms of strict implication, our model of grounding will be subject to the paradoxes of strict implication. Since ◻(∧Γ ⊃ p) appears in the consequent of the Entailment principle, that principle will sometimes be trivially satisfied (in cases where p is a tautology, for instance). However, the Entailment principle is not a biconditional. Therefore these paradoxical trivial instances of the Entailment principle do not in turn imply the existence of any grounding relation. For this reason, the paradoxes of strict implication do not pose a problem either for the model of grounding introduced here or for the account of reasoning by grounded analogy developed later in this section.
The necessity of grounding has been the subject of recent debate that is worth flagging here. For example, Skiles (2015) argues that there are at least some cases in which Entailment fails due to Ship of Theseus-style counterexamples: cases in which the existence of an object is grounded in the arrangement of its parts, but the facts about that arrangement are held fixed in spite of a series of replacements of parts that culminates in the original object ceasing to exist. While it is not possible here to give such objections the detailed reply they deserve, there are several ways that they might be addressed. Most promisingly, one might take such cases to show only this: if an object is such that a series of replacements of its parts could result in its ceasing to exist, then its existence is not fully grounded in the arrangement of its parts. (After all, if an object’s existence were so grounded, how could the replacement of its parts have any impact on its existence?) However, given that our aim is not primarily to justify the various elements of the standard view of grounding, we will leave the issue at that. In what follows, we simply assume the Entailment principle.
The grounding relationship between a’s having the properties S and its having the projected property Pt therefore secures the relevance of S as a collection of similarities between the source and target case. However, it is also clear that not all relevant similarities need to be a total ground of the projected property. In general, similarities and differences may be relevant to the projected property without grounding the instantiation of that property; however, if something’s having a collection of similarities suffices to ground its instantiating the projected property, that is a guarantee of relevance. (Indeed, as shown below, it is a guarantee of validity.) The framework of Macagno (2017) provides a helpful way of thinking through our account of the validity of grounded analogy. Macagno treats relevance as a relation between (1) an abstracted functional genus of the similarities between source and target, and (2) the projected property. The inference rule of Ground Generalization that we introduce here can be understood as way of generating an appropriate functional genus in cases where something’s having the projected property is grounded in its having the premised similarities.
We do not here undertake to supply an exhaustive list of such applications as such a project is well beyond the scope of this paper. Rather, we seek only to demonstrate “proof of concept” by showing that the model of grounded analogy can successfully represent some ordinary analogical arguments.
Thanks to an anonymous reviewer for raising this objection.
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Acknowledgements
We consider this joint work; our names are listed in alphabetical order. We offer our thanks to Matt McKeon for his constructive and insightful discussion of this topic as we developed the argument of the paper. We also extend our thanks to the anonymous referees, whose thorough and incisive comments contributed significantly to improving the paper in substantive ways.
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Godden, D., Grey, J. Reasoning by grounded analogy. Synthese 199, 5419–5453 (2021). https://doi.org/10.1007/s11229-020-02974-9
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DOI: https://doi.org/10.1007/s11229-020-02974-9