, Volume 11, Issue 2, pp 153–162

The Metaphysics of Mixed Inferences: Problems with Functionalist Accounts of Alethic Pluralism


    • Department of Philosophy, College of Arts and SciencesUniversity of Massachusetts Dartmouth

DOI: 10.1007/s12133-010-0065-z

Cite this article as:
Nulty, T.J. Int Ontology Metaphysics (2010) 11: 153. doi:10.1007/s12133-010-0065-z


Alethic pluralists argue truth is a metaphysically robust higher-order property that is multiply realized by a set of diverse and domain-specific subvening alethic properties. The higher-order truth property legitimizes mixed inferences and accounts for a univocal truth predicate. Absent of this higher-order property, pluralists lack an account of the validity of mixed inferences and an adequate semantics for the truth predicate and thereby appear forced to abandon the central tenets of alethic pluralism. I argue the use of many-valued logics to support pluralism fails to address the pluralist’s metaphysical problem regarding mixed inferences and mixed truth functional connectives. The high degree of heterogeneity of the alethic realizers (unlike the realizers for pain) challenges the plausibility of a single higher-order functional property. A functional property with such a heterogeneous base cannot be projectable at a theoretically significant level. The problem with mixed inferences and truth functions is but one symptom of the deeper projectability problem.


TruthAlethic functionalismAlethic pluralismMixed inferences

1 The Argument for Supervenient Alethic Functionalism

Michael Lynch has argued recently (2000, 2001, 2004, 2005, 2006) that truth is a multiply realized supervenient property. Despite its supervenient status, truth is an objective, higher-order kind which has explanatory power not reducible to its disjunctive base properties. Supervenient functionalism about truth is motivated by an attempt to avoid problems associated with strong alethic pluralism (SAP). SAP states the concept of truth affords more than one robust account; the robust accounts are pluralistic because they vary with domain e.g., the moral domain, the mathematical domain, and the domain of truths about the physical world. Truth is a metaphysically robust property, but how that property is realized depends on the context of the proposition.

Generally speaking, SAP makes the truth predicate unacceptably vague. If the predicate expresses different truth properties in different domains, then the predicate is polysemous; it expresses a different concept depending on the context of use. There are two main problems with a polysemous truth predicate. First, such a view fails to accord with speaker blind generalizations such as “Everything Bill said is true.” Generalizations of this sort attribute truth uniformly, not in one sense in one domain and another sense in a different domain. Nothing about the behavior of speakers suggests they view the truth predicate as varying in meaning or conceptual content across domains. The second main problem involves truth-preserving (i.e., valid) inferences. If our truth predicate expresses a different concept by picking out different properties in different domains, we now lack an explanation of seemingly valid inferences across domains. Valid inference is normally taken to preserve a single property.

Lynch intends to respond to these problems while preserving the central tenet of pluralism: propositions can be true (in a metaphysically robust sense) in different ways. Recall that the problem for SAP arises because SAP entails that the truth predicate expresses multiple truth concepts. Lynch proposes that the truth predicate expresses a single, uniform supervenient property. Truth is a higher-order supervenient property.

ST: For any proposition, if it is true, then it has some property F such that, necessarily, if a proposition has F, then it has the property of truth.

Lynch points out a similar move has been made in the philosophy of mind. Our network of folk beliefs about pain defines our concept of “pain.” The functional state of pain, as defined by the folk network, can be realized in a variety of different physical systems. The functional role of truth, or truth-role, is defined by the platitudes first advanced by Crispin Wright (1992, 1996, 1999, 2001). In addition to the minimalist platitudes advanced by Wright, Lynch expands these platitudes to include non-semantic properties and states connected to truth (e.g., beliefs and other thought attitudes).

Lynch defines the functional truth-role using the Ramsey–Lewis method for defining theoretical terms. Following Lynch’s notion, let A be the conjunction of our folk platitudes, then we replace occurrences of “true” in the platitudes with a variable allowing for existential quantification. We are now in a position to define the functional property identified as the supervenient property in ST.

x has a property that plays the truth-role if and only if ∃t1 [A(t1…O1…On) & x has t1].

Thus, the conditions under which a proposition can be true are easily stated:

FT: The proposition that p is true if and only if it has the property that plays the truth role.

The adequacy of Lynch’s attempt to avoid the problems confronting SAP depends on whether truth as a supervenient property is minimally objective. In other words, the truth predicate must rigidly designate the supervenient property in all domains, not merely the domain specific first-order properties. We must avoid reducing the supervenient property to simply having one member of the disjunctive set of realizer properties. The realizer properties cannot be grouped merely by our application of the truth concept, if the concept is to pick out an objective property. The reductive worry is what partially motivated Kim’s causal exclusion problem. The second-order property of having mental state M is not explanatorily necessary since all of the causal work occurs with the subvening physical property P (Kim 1998). In what way does the supervenient property of truth avoid the reductive worries developed by Kim?

The truth predicate rigidly designates a supervenient property when two conditions are met: (1) the members of the kind share a property in virtue of which they are members of that kind; and (2) their having that property is not a mere projection of, or construction out of, our concept of that kind. Lynch contends a proposition is true when it has the supervenient property of truth, that is, the property of having the property that realizes the truth-role (Lynch 2006).

2 The Mixed Inferences Objection

Before discussing the adequacy of Lynch’s functionalist approach to truth, we first need to understand the force of the mixed inferences objection. Additionally, we need to understand why one response to the objection, advanced by JC Beall, fails to be adequate.

Christine Tappolet (1997) presents the following kind of argument:
  1. 1.

    Wet cats are funny.

  2. 2.

    This cat is wet.

  3. 3.

    Therefore, this cat is funny.


The argument certainly appears valid, but there is a problem for the pluralist. The pluralist is committed to claiming that the first premise and the second premise are true in different ways; the first sentence is made true by social conventions about humor, while the second is made true by the physical fact of water on the cat’s fur. Truths about which things are funny are different than truths about the physical world. Validity requires the preservation of a single property from premises to conclusion. Pluralism appears to deny a single truth property, and so, this argument cannot be valid. The pluralist could also try to maintain validity by denying the common view that validity consists of truth preservation.

JC Beall (2000) has offered an interesting reply to Tappolet’s criticism. Beall argues the pluralist can accept that claims are true in different ways and also maintain that validity consists of truth preservation. The response consists of moving from a classical bivalent logic to some type of many-valued logic. Consider a standard many-valued logic that has two ways of being true, 1/2 and 1, respectively. Both of the previous values are designated in this logic. Validity is then defined as an argument in which it is impossible for all premises to be designated and the conclusion not to be designated. Thus, we can assign the two premises in the above argument different values (i.e., 1/2 and 1) and have a designated conclusion, thereby preserving truth. Beall concludes that by adopting a many-valued logic, pluralists can accept the validity of mixed inferences and maintain that the premises can be true in different ways.

Tappolet (2000) offered a reply to Beall. Contrary to Beall’s assessment of Tappolet’s argument as a dilemma, Tappolet presents her argument as a trilemma. Pluralists must accept one of the following: (1) deny the validity of mixed inferences; (2) admit that in addition to the multiple individual truth predicates, there is one unique truth predicate that applies to all the premises; or (3) deny that validity consists of truth preservation from premises to conclusion (Tappolet 2000: 383).

Tappolet argues that Beall’s talk of each premise having different “ways of being true” is actually a not-so-hidden reference to a common truth predicate. In the example above, assume premise one has one kind of truth T1 and the second premise has another type T2, both types are designated, which, according to Tappolet, is just a general truth predicate denied by pluralists; there is a kind of truth (i.e., designation) both premises have in common. If we have this general truth predicate that is preserved in valid inference, Tappolet sees no reason for the additional multiplicity of truth predicates, which is tantamount to a rejection of pluralism.

Tappolet then makes a further argument involving the notion of mixed conjunction. Consider the sentences: “The cat is wet and funny” which can certainly be true. If the first conjunct is true T1 because of natural fact, and the second conjunct is true T2 because of social norms, the kind of truth predicate that applies to the conjunction is mysterious. The conjunction is true as a result of neither just a physical fact nor just a social norm (Tappolet 2000: 384). Once again, it seems the pluralist is forced to posit a common truth predicate to handle conjunction analogous to the one needed to make sense of mixed inference.

3 Assessment of the Mixed Inference and Mixed Conjunction Objections

As Devitt (2001) and others have pointed out, the truth literature is rife with equivocation between talking about the truth predicate and talking about truth as a property. Indeed, one of Devitt’s criticisms of deflationists is that they infer metaphysical claims (claims about truth as a nonexistence property) from linguistic claims (claims about how we use the truth predicate). The inferences are not justified according to Devitt because they turn on an equivocation between the truth predicate and the truth property. A similar ambiguity appears in both Beall’s paper and Tappolet’s paper.

Alethic pluralism is a metaphysically robust theory; it is decidedly not deflationary. Truth-bearers can and do have a variety of distinct but metaphysically robust properties that realize the functional truth role. The alethic realizers can be robust in two main ways: realist and antirealist. For example, the correspondence property is a metaphysically robust property of truth-bearers that is non-epistemic, and so qualifies as a form of alethic realism. Verificationist, assertability theorists, and pragmatist also claim truth is a metaphysically robust property (i.e., not deflationary), but it is a property that depends on us in some way. The realizers of the higher order truth property will be a mix of realist and antirealist subvening properties (I will say more about this later).

Beall, on the other hand, is a deflationist. When Beall claims there are different “ways of being true,” this cannot be interpreted as a metaphysical claim. The different ways of being true are purely formal, or linguistic/conceptual. When the pluralists claim there are “different ways of being true,” the claim is a metaphysical one. What we have then is an equivocation on the expression “different ways of being true.” The claim that premises can by true in formally different ways (i.e., 1/2 or 1 as values) does not begin to address the metaphysically different ways in which premises can be true. The real question for pluralists, and those who challenge their position, is what metaphysical property is preserved in valid inference, and what metaphysical property a conjunction has when its conjuncts have two metaphysically different truth realizers.

Both Beall and Tappolet talk about predicates, not properties. Predicates are relatively easy to come by—we just introduce them into our language. The real question, requiring an answer from pluralists, is whether or not those second-order predicates correspond to metaphysically robust second-order functional properties. Following Pedersen’s (2006) formalization, we can express the generalized truth predicate quite easily:
$$ \left( {\hbox{TG}} \right)\left( {\forall {\hbox{p}}} \right)\left( {{\hbox{TG}}\left( {\hbox{p}} \right) \leftrightarrow T1\left( {\hbox{p}} \right){\hbox{v}} \ldots {\hbox{v}}\,{\hbox{Tn}}\left( {\hbox{p}} \right)} \right) $$

We can get a generalized truth predicate simply by quantifying over the plurality of individual truth predicates. We can apply the general predicate to any proposition to which we apply one of the specific truth predicates. Such a move may be satisfying for deflationists, but it cannot satisfy the pluralist because we have to move beyond just talking about language; we are not doing metaphysics by talking only about predicates.

The pluralist needs to show that if each first-order predicate picks out a first-order property, then the second-order predicate also picks out a metaphysically robust second-order property. Again, following Pedersen, we express this requirement as follows:

Let φ(x) be a complex disjunctive predicate, i.e., φ(x) ↔ ψ1(x) ∨ … ∨ ψn(x). If each ψ1(x) …ψn(x) denotes a property, then so too does φ(x).

The mystery for the pluralist is not primarily how to characterize formally mixed inference and mixed conjunction. The real mystery is how, given a mixed inference or mixed conjunction, the different properties denoted by ψ1(x) …ψn(x) interact with each other as premises and conclusion, and as truth functional connectives.

4 The Metaphysics of Truth and the Logic of the Truth Predicate

For Beall and other deflationists, the relationship between the metaphysics of truth (i.e., truth properties) and the logic of truth is a non-existent one; there are no metaphysical truth properties that govern the logic we use. However, robust theorists of both realist and antirealist character have focused on the relationship between the metaphysics of truth and the logical systems we use. Dummett (1978, 1991) has famously argued in a number of places that if we reject the notion of truth as verification transcendent (e.g., rejecting the correspondence theory of truth) we must accordingly reject bivalence. Graham Priest (2000) devoted an entire paper to exploring six different theories of truth and their relationship to the possibility of accepting paraconsistent logic. Intuitively, it is clear that if validity is about the preservation of some kind of metaphysically robust truth property—which it must be for non-deflationists—then the nature of truth will affect the allowable rules of inference and the logical behavior of truth-functional connectives. Conversely, the system of logic we accept might impose some restrictions on an acceptable metaphysical theory of truth. Regardless of which of the two approaches we make theoretically primary, both are committed to a relationship between the logic we use and how we think about truth metaphysically.

For Lynch, using the notation above, the second-order property denoted by φ(x) will be the supervenient functional property that is multiply realized by one of the various properties denoted by one of the following disjuncts: ψ1(x) ∨ … ∨ ψn(x). It is this allegedly robust supervenient property that all true propositions will have in addition to their subvening realizer properties. This higher-order property is the source of a univocal truth predicate at the linguistic level. The pluralist’s case rest not on providing just a predicate, as was shown above; rather the case rests on an acceptable metaphysical theory of this higher-order functional property.

Lynch recognizes the importance of this metaphysical task. He explicitly rejects the notion of second-order concepts forward by Kim (1998). Kim draws a distinction between functional (second-order) concepts and functional (second-order) properties. Kim argues plausibly that we cannot alter our ontology by “mere logical operations on our notations” (Kim 1998: 103). If the second-order property is something more than just its realizer in a given instance, it must have distinct causal powers. Talking about second-order “properties” is best replaced by talking about second-order “descriptions” or “concepts” (Kim 1998: 104–105).

Adapting Kim’s discussion (1998: 107), consider the claim that a proposition has the second-order property denoted by φ(x) because it has the property denoted by ψ1(x) or the property denoted by ψ2(x)…or the property denoted by ψn(x). In this sense of having a “second-order property,” a proposition has the property denoted by φ(x) only in the trivial sense that it has one of the properties denoted by some ψn(x). The stronger claim, and the one needed by the alethic pluralist, is that a proposition has the second-order property φ(x) because it has a disjunctive property that is something distinct from just having one of the disjuncts. Also, note that the first characterization of having a second-order property does not entail the second characterization: having one subvening property is different than having a disjunctive property comprised of all subvening properties.

Returning to how we might introduce a second-order general truth predicate (∀p) (TG(p) ↔ T1(p) ∨ … ∨ Tn(p)), the semantics of TG(p) depends on whether we can make sense on metaphysical grounds of accepting the latter rather than the former characterization of having the second-order property explained in the above paragraph. Pluralists need to explain the logical behavior of the predicate φ(x) in such a way as to relate that behavior to the metaphysical status of the property allegedly denoted by the predicate. In the following section, I argue there are strong considerations against treating φ(x) as denoting a functional property rather than merely expressing a functional concept.

5 The Metaphysics of Alethic Multiple Realizability

One way of seeing the inadequacy of the metaphysical story told by pluralist thus far is to consider how the truth realizers differ from the realizers mentioned by way of analogy (e.g., pain realizers). Any discussion of higher-order properties must consider whether those properties are projectible so that we can make lawlike generalizations. Consider pain as a functional property. Presumably all of the pain realizers are objective, mind-independent physical entities. The realizers are ontologically homogeneous to a significant degree, and so pain will likely be a projectible property. Now consider the ways in which the functional property of truth is different. Note first that the projections of the higher-order truth property ought to involve lawlike generalization in logic, semantics, epistemology, etc. As Lynch and other pluralists repeatedly point out, truth might be realized in terms of correspondence in one domain, provability in another, and superassertability in another. The disjunction of truth realizers is a heterogeneous mix of realist and antirealist truth types.

The heterogeneous disjunction of truth realizers creates a number of significant problems. First, if having the higher-order functional property amounts to having something more than just one of the realizers (see the discussion of Kim in the previous section), we lack an explanation of whether this higher-order truth is realist or antirealist. If we treat having the higher-order functional property as simply having one the realizers, then the higher-order property will be consistent with alethic realism in one case and alethic antirealism in another. Now, we have the absurd result that what is supposed to be a single higher-order property belongs to two mutually exclusive categories (alethic realism and alethic antirealism).

Second, the individual truth realizers have different relationships to our notions of validity, justification, doxastic normativity and meaning. How we think about these latter notions is related to our metaphysical views about truth. The kinds of projections we will make about the realizer truth properties will be quite different, especially between realist and antirealist realizers. The pluralist needs to explain how such a higher-order property will be projectible given the vastly different behaviors of the realizers. Someone might respond that we can make lawlike generalizations about the higher-order property by using instances of the equivalence schema. However, appealing to the equivalence schema and similar expressions will not by itself help solve the projectibility problem. Which background logic we use to express the equivalence schema will be affected by which metaphysical account of truth we accept. Additionally, the function of role of truth involves more than just logic.

The problem of projectibility allows us to make more sense of Tappolet’s challenges. Consider a conjunction A∧B. A will be correspondence true, while B will be verificationist true. How we assign designated values to A and B will depend on which logic we adopt. Likewise, how we assign a value to the conjunction A∧B will depend on which logic we adopt. Knowing the alethic realizers of A and B might help us determine which logic to use to represent A and B individually. For example, if A or B contained vague predicates we might want to use a fuzzy logic to assign values. However, knowing that the conjunction has the alleged higher-order property, or even knowing that each conjunct has the higher-order property does not tell us anything about which logic would be most acceptable. The higher-order truth property is logically epiphenomenal; the realizer properties perform all of the logical work.

Similar problems emerge for the other notions associated with truth such as doxastic normativity and justification. We might know when we are justified in asserting A on one account of truth, and we might think that whether we are justified in asserting B is irrelevant because B has or lacks some correspondence property of truth. We are left wondering what to say about which epistemic stance we should take to the conjunction of A and B since each conjunct is subject to different epistemic norms given different conceptions of truth.

Differing metaphysical views of truth result in radically different theories of meaning. Both Donald Davidson and Michael Dummett believe a theory of meaning will model competent speaker knowledge of some language L. The axioms for L will vary depending on whether the model uses realist truth conditions or antirealist truth conditions (e.g., justification). Realist models will likely be bivalent, while as Dummett argues, the antirealist models will abandon bivalence. The proposed higher-order truth property has no role to play in a theory of meaning; the first-order truth properties perform all the semantic work.

The fact that the realizers are associated in various ways to different conceptions of logical consequence and truth functionality, suggest a further inadequacy in Beall’s response. Beall’s response assumes the use of a single logic is justified, but that assumes the logical behavior of the truth concept applied to the different sentences will be the same. The assumption of a single logic that captures the different ways in which sentences are true correlates in metaphysical terms to an unjustified assumption that the higher-order functional predicate is projectible. The pluralist needs to explain how. The problem with mixed inferences and mixed conjunctions presented by Tappolet is metaphysically grounded in a lack of projectibility of the higher-order truth property because of the heterogeneous disjunctive base.

The heterogeneity of the base properties leads to a third problem. While the previously discussed problems challenge the projectibility of the higher-order property, assuming the supervenience of a multiply realized property, the third problem challenges whether truth can be multiply realized. The basic form of this argument comes from Lawrence Shapiro (2000).

Some features of the base properties are relevant to their functional role, while other features are not. The color of a particular corkscrew is irrelevant to its function as a corkscrew; irrelevant also, according to Shapiro, is the fact that the corkscrew is made of aluminum or steel. Since neither the color nor the material (i.e., steel or aluminum) is relevant to fulfilling corkscrew function, we should not claim the corkscrew function is multiply realized in differently colored tokens or tokens comprised of different materials. We need an argument to show the same function is realized in genuinely and relevantly different ways.

However, the demand for genuinely different realizer properties presents the functionalist with a dilemma. Alethic pluralists claim the higher-order truth property performs “the same function” and that is how we get a univocal truth concept. Either the individual truth realizers perform this “same function” in the same way or they do not. Accepting the former disjunct entails that we do not really have a case of multiple realizeability; whatever differences might exist among the truth realizers is irrelevant to fulfilling the common functional role. This result is analogous to the case in which corkscrews come in different colors but are not genuine cases of multiple realization. This result is tantamount to denying alethic pluralism. Accepting the latter disjunct, in which the truth realizers perform a similar function in radically different ways, results in distinct kinds, not a single higher-order kind.

I think it is clear alethic pluralists are committed to the second horn of dilemma. Since the pluralist allows for radically different truth realizers (realist and antirealist), they must deny that the realizers satisfy the truth function in the same way. There are genuine and relevant differences among the truth realizers. But, now the second horn of the dilemma presents a serious challenge. As I argued above in my discussion of projectibility, the individual truth realizers are playing distinct roles in logic, semantics, and epistemology. While those distinct roles may be “similar” at a suitable level of abstraction, they are not identical roles, and the lack of projectibility suggests the highly abstracted similarities are not theoretical significant or useful. Shapiro (2000) characterizes such generalization as “numbingly dull” (2000: 649). Similar considerations motivate Kim (1998) to reject talk of second-order properties in favor of second-order concepts.

6 Conclusions

Alethic pluralism is an interesting addition to the literature on truth, but more work needs to be done. Specifically, pluralists need to explain, at a sufficiently fine-grained level, the functions of the individual truth realizers. Such an explanation needs to account for the way in which the proposed higher-order truth property differs from other higher-order functional properties such as pain. The property of pain has a primarily causal function. Truth, on the other hand, not only has perhaps a causal function, but plays a role in logic, semantics, and epistemology that has no clear analog to the pain case. Given the different ways the individual truth realizers fulfill these truth roles (logical, semantic, epistemological) in different domains, we need an argument for why we can legitimately consider them instances of the same functional kind. Moreover, the argument for a single higher-order kind must avoid being so abstract and generalized that interesting projections are not possible. That is, the pluralist must show we have a real functional property not merely a functional concept. The pluralist’s case has yet to be made convincing.

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