Scott's two challenges

Let us now rehearse Scott's two challenges for religious expressivism in turn. His first challenge for religious expressivists is to explain how a non-truth-evaluable religious sentence and its negation could be logically inconsistent. To illustrate, consider a theist who believes that:

1. (1) God is all-loving.

If religious expressivists are correct, then (1) is just an expression of some non-cognitive attitude; thus, it cannot be judged as true or false. This is well and good. However, consider its negation:

1. (2) God is not all-loving.

(1) and (2) seem inconsistent. To assert both implies asserting a contradiction. The question now for religious expressivists is how to explain this inconsistency, given that religious sentences are not truth-evaluable.

Religious expressivists may answer that (1) and (2) are inconsistent because they express incompatible attitudes, which, in turn, implies that (1) contradicts (2). However, this answer is unavailable for religious expressivists since non-cognitive attitudes do not contradict one another. To illustrate, compare belief states and desire states. If someone believes p and also believes ~p, we know that one of these beliefs must be false because both cannot be true. In contrast, if someone desires q and also desires ~q, these desires may conflict, they may lack clear direction on how to live, but they do not contradict each other (in the logical sense). The non-cognitive attitudes expressed by (1) and (2) are like desire states. They may conflict but are not contradictory. Thus, to explain how a non-truth-evaluable religious sentence and its negation can be inconsistent, religious expressivists ‘must therefore find a different account of negation in religious language’ (Scott (Reference Scott2013), 81–82).

Let us now turn to Scott's second challenge of explaining how non-truth-evaluable religious sentences could figure in logically valid arguments. To illustrate, consider the following argument:

1. (3) God is all-loving.

2. (4) If God is all-loving, God offers us the gift of salvation.

3. (5) Therefore, God offers us the gift of salvation.

The argument is valid since all its premises cannot be true while its conclusion is false. However, religious expressivists are confronted with a dilemma. On the one hand, if they maintain that (3) and (5) are devoid of cognitive meaning, they still have to account for the meaning of (4). Being a conditional sentence, (4) could be asserted without asserting its constituent sentences. If this is so, then (3) and (5) may have a different meaning when they are embedded in (4) as opposed to when they are unembedded. This means that argument (3)–(5) is invalid since it commits the fallacy of equivocation.

On the other hand, if religious expressivists want to maintain the argument's validity, they have to say that (3) and (5) preserve their non-truth-evaluability while being embedded in (4). If this is so, then since (4) does not necessarily express the attitudes expressed by its constituent sentences, it seems to follow that the non-cognitive attitudes do not entirely determine the meanings of (3) and (5). This makes religious expressivism false (Scott (Reference Scott2013), 83). Thus, religious expressivists have a problem in either case.Footnote ²

Religious expressivists may reply that the validity of argument (3)–(5) could be defined in terms of the non-cognitive attitudes of approval and disapproval. Argument (3)–(5) is valid since approving (3) and (4) while disapproving (5) is attitudinally incoherent. There will be some rational pressure to either approve the argument's conclusion or disapprove of at least one of its premises (Schroeder (Reference Schroeder2008), 709–710). This, then, sidesteps Scott's second challenge.

This response might be unavailable for religious expressivists since coherence requires a minimal notion of truth-evaluability. To say that two attitudes cohere with one another implies that they must at least be consistent; they can both be true. However, this goes against the basic tenet of religious expressivism. Moreover, even if coherence does not imply truth-evaluability, there is still the issue of how the meanings of religious sentences are determined by the attitudes they express. As Scott's second challenge has shown, there seems to be no common meaning between asserted and unasserted occurrences of religious sentences, which is a problem for religious expressivists (Scott (Reference Scott2013), 83).

Scott's two challenges amount to the same thing. If religious expressivists are right that religious sentences are not truth-evaluable, then they have to explain the logical behaviour of such sentences. In the case of the first challenge, this implies explaining the inconsistency of a given religious sentence and its negation. In the case of the second challenge, this means explaining how religious sentences could figure in valid arguments. In the following two sections, I argue that religious expressivists could adopt a semantic framework that models non-truth-evaluable expressions to address these challenges. And this framework, I suggest, is the semantic framework of Weak Kleene (WK3).Footnote ³

A Weak Kleene semantics

The language of WK3 consists of a countable set of atomic sentences: p, q, r. . . and a set of familiar logical connectives: ‘~’ (negation), ‘v’ (disjunction), ‘&’ (conjunction), ‘⊃’ (material conditional), and ‘≡’ (material biconditional).Footnote ⁴ The formation rules for well-formed formulas (wffs) are standard. Regarding the metalinguistic variables, A, B, C,. . . refer to wffs, while X, Y, Z refer to sets of wffs.

WK3 is a three-valued semantics, where a valuation function v maps each atomic sentence into a set of valuations, V: {1, e, 0 }. ‘1’ and ‘0’ represent the classical true and false values, respectively, while ‘e’ is the non-classical value for the expressive, non-truth-evaluable value. Given this, compounds will have the truth tables given in Table 1.

The logical connectives in Table 1 behave in a perfectly classical (two-valued) way if their constituents are 1 or 0. They only behave non-classically when at least one of their constituents has value e. This semantic behaviour is expected since WK3 is a sub-classical logic. This leads us to the infectious feature of WK3, and let us define it as follows:

Table 1. WK3 truth tables

Definition 1 (Infectiousness) For any sentence A, if v(A) = e, then for any compound sentence, C, that has A as a constituent, v(C) = e.

Think of the infectious nature of expressive sentences as a kind of sentential virus. Once a sentence gets infected by it, it contaminates other sentences within its immediate vicinity. In the formal picture, the value of a compound sentence is e if at least one of its constituents has value e.Footnote ⁵

Let us now turn to the notion of validity in WK3. WK3-validity, ‘⊨_wk3’, is defined in terms of the familiar notion of truth-in-a-model or truth-in-a-given-valuation, with value ‘1’ as the only designated value.Footnote ⁶ A sentence A is true iff v(A) = 1. A set of sentences X is true iff v(B) = 1 for all B ∈ X. Given this, ⊨_wk3 is defined as:

Definition 2 (WK3-validity) X ⊨_wk3 A iff there is no WK3-valuation where v(B) = 1 for all B ∈ X, but v(A) ≠ 1.

That is, the argument from a set of premises X to a single conclusion A is WK3-valid just in case there is no valuation where all its premises are true, but its conclusion is either false or has value e.

Some philosophical remarks

With the machinery of WK3 at hand, we could now address Scott's two challenges. Before doing this, let me first make a few philosophical remarks. First, notice that the basic tenet of religious expressivists fits in nicely with the WK3 semantics. Arguably, religious expressivists divide atomic sentences into two general types: those that are truth-evaluable and those that are not. Value e plays an important role here since it allows us to semantically distinguish between sentences of the former sort, namely, sentences that have value 1 or 0, from those of the latter sort, namely, sentences that always have value e. If religious expressivists are right, then religious sentences will be of this latter sort.

Second, given Definition 1, a sentence that has value e is infectious and remains so regardless of whether it is in an asserted or unasserted context. This follows from the main features of the WK3 semantics. To illustrate, if v(A) = e, then (i) in an asserted context like ‘~A’, the semantics yields v(~A) = e, and (ii) in an unasserted context like ‘A v B’, the semantics still yields v(A v B) = e, regardless of what the value of B may be. This, then, answers Scott's worry that religious sentences might have different ‘meanings’ in asserted and unasserted contexts since religious sentences are always evaluated as having value e, regardless of their context. This means that embedded and unembedded religious sentences have value e.

Third, given that religious sentences are infections, compounds that have them as constituents will have value e. This means that compound sentences about religious and non-religious matters will be, according to religious expressivists, devoid of cognitive significance. For instance, despite having a true disjunct, a disjunction like ‘God is all-loving or 2 + 2 = 4’ will register value e. On the other hand, despite having a false conjunct, a conjunction like ‘God is all-loving and 2 + 2 = 5’ will likewise register value e. This only means that non-cognitive content always trumps cognitive content, regardless of whether the cognitive content is true.Footnote ⁷

WK3-validity and Scott's second challenge

Let us now address Scott's second challenge of showing how non-truth-evaluable religious sentences could figure in logically valid arguments. Since the WK3 framework provides a semantic interpretation for expressive sentences, it also provides the best model for the logical behaviour of religious sentences. In particular, their behaviour in (WK3-)valid arguments.

Given Definition 2, it is relatively easy to show that the argument forms in Table 2 are WK3-valid.

The argument forms in Table 2 are WK3-valid since there is no valuation where all their premises have value 1, while their conclusion has value 0 or value e. This means that though religious sentences may be devoid of cognitive meaning, they could still be constituents of logically valid arguments. For instance, argument (3)–(5) above is valid since it is an instance of modus ponens. This, then, answers Scott's second challenge.

Table 2. WK3-valid arguments

However, while all of the WK3-valid arguments in Table 2 are classically valid, there are classically valid arguments that are not WK3-valid. A notable example is addition, i.e., A ⊭_wk3 A v B.Footnote ⁸ A counterexample for addition is a valuation where v(A) = 1 and v(B) = e (Beall (Reference Beall and Maclaurin2012)).Footnote ⁹

The Nuances of ‘e’ and Scott's first challenge

Let us now turn to Scott's first challenge of explaining how a religious sentence and its negation could be inconsistent. This challenge seems harder to answer than the first since in WK3, if A is a religious sentence, then A and ~A have the same e value. It seems, then, that WK3 could not account for the inconsistency of a religious sentence and its negation.

This complication could be addressed by a different account of inconsistency. For instance, Definition 2 implies that WK3 is an explosive logic. It permits any arbitrary conclusion B to follow from a contradiction: that is, A & ~A ⊨_wk3 B. This means that if A is a religious sentence, then A and ~A entail any sentence B. Thus, the explosive nature of WK3 already evidences the inconsistency of a religious sentence and its negation.Footnote ¹⁰

However, Scott might insist that despite WK3's explosiveness, religious expressivists who subscribe to it might still fail to account for the inconsistency of a religious sentence and its negation. For religious expressivists, religious sentences express attitudes, and, as was discussed above, while attitudes may conflict, they do not seem to contradict each other in the logical sense. If this is right, then Scott's first challenge remains unresolved.

To answer this, we may extend WK3 to account for the nuanced character of e. After all, e is more expressive than was first approximated. As a true-blooded religious expressivist might argue, there are different non-cognitive attitudes that a person could have towards a religious sentence. These attitudes might range from pure disgust to pure ecstasy. Taken this way, the resulting WK semantics would perhaps be a fuzzy-like one. For simplicity however, let us borrow the ‘yey!’ and ‘boo!’ language of moral expressivists and turn them into a more nuanced appreciation of value e.Footnote ¹¹ This nuanced appreciation implies the dual feature of e that includes a positive feature, ‘e+’ and a negative feature, ‘e−’. Let us stipulate that e+ and e− are mutually exclusive values. e+ represents a strong positive emotional response towards a given religious sentence, while e− represents a strong negative response to it. As such, WK3 becomes a four-valued semantics, WK4, since the more nuanced e+ and e− values replace the e value in the set of valuations. This means that the logical connectives in WK4 behave as in Table 3.

Notice that in Table 3, e+ trumps e− in the case of disjunctions, while e− trumps e+ in the case of conjunctions. For example, if v(A) = e+ and v(B) = e−, then v(A v B) = e+, while v(A & B) = e−. This semantic behaviour is expected if we suppose that the attitude that e+ represents is stronger than what e− represents. This follows the common thought that positive emotions trump negative ones every time.

Table 3. WK4 truth tables

The nuanced values of e+ and e− provide a way to answer Scott's first challenge since the negation of a religious sentence becomes semantically tractable. If v(A) = e+, then v(~A) = e−, and vice versa. Thus, via this addendum, religious expressivists could say that a religious sentence and its negation are inconsistent because, as the semantics shows, they express mutually exclusive attitudes toward a given religious sentence. This mutual exclusivity is not defined in terms of truth and falsity. It is defined in terms of the duality of two non-cognitive attitudinal responses. This, then, answers Scott's first challenge.

One might worry that explaining the inconsistency of religious sentences in terms of the duality of e+ and e− seems to invite the above-mentioned objection raised against religious expressivists who explain the inconsistency in terms of holding incompatible non-cognitive attitudes. Moreover, even if we grant that this duality does not assume truth-evaluability, their mutual exclusivity seems to be determined not by logic but by pragmatic considerations. They behave more like Moorean inconsistencies of the form ‘p, but I don't believe p’ rather than logical inconsistencies epitomized by the form ‘A & ~A’ (Schroeder (Reference Schroeder2008), 710).Footnote ¹²

In response, we must reiterate that in the proposed formal picture, e+ and e− are not non-cognitive attitudes per se. They are semantic values that represent such attitudes. As semantic values, they only serve as nuanced valuations of religious sentences. This means that religious sentences may have an e+ or an e− value for a given valuation. Thus, these values cannot be likened to pragmatic or logical inconsistencies.

But how do these nuanced e values explain the inconsistency? In the proposed WK framework, inconsistent religious sentences of the form ‘A & ~A’ do not yield the value 0. However, they will always register value e− given the WK4 truth tables in Table 3. To illustrate, if v(A) = e+, then v(A & ~A) = e−; on the other hand, if v(A) = e−, then v(A & ~A) = e−. This explanation preserves the religious expressivist's view that attitudes conflict and explains why these attitudes can be logically inconsistent given the semantics. For religious expressivists who think that religious sentences are not truth-evaluable, this accounts for the inconsistency of a religious sentence and its negation.Footnote ¹³

Finally, note that adding e+ and e− to the set of valuations does not rule out some of the established WK3-valid arguments, nor does it rule in invalid ones. After all, WK3 and WK4 are logically equivalent. All valid and invalid arguments in the former are also valid and invalid arguments in the latter. This is not surprising since WK4-validity is just WK3-validity (see Definition 2), with the caveat that value e includes two more nuanced values: e+ and e−.