(This gem is pulled up from the vasty deeps of the ComBox to where it may shine in a more fitting setting. Minor editing, bolding, and comments in blue by BV.)
1). Let us say that a *real* contradiction is a sentence which comes out false according to every possible model (M): i.e., M = language-plus-domain-plus-interpretation, where an ‘interpretation’ is a complete and systematic assignment of extensions to the non-logical terms of the language (L). We assume that L is a well developed natural language such as English and we have a sufficiently rich domain that includes whatever entities are required to implement an interpretation that will suffice for theological purposes.
1.1) Note: We are assuming throughout classical logic in two sense: (a) the logical constants are interpreted classically; (b) there are no *real* true contradictions.
1.2) Sentence S is a *real* contradiction just in case there is no *normal model* M in which it comes out true. A normal model in this context is one which features an interpretation that assigns extensions to the non-logical terms in the usual way prior to resolving any potential ambiguities. On a realist conception of truth, S [if contradictory] has no truth-maker (T-maker) in any normal model or possible world.
2) Let us now define at least one sense of an *apparent contradiction* in model theoretic terms. Let S be a sentence expressible in L and suppose S comes out false in every normal model M. S appears to be a contradiction. Is it really a contradiction? Prof. Anderson maintains that there are sentences which are contradictory in every normal model, but are non-contradictory in some other models of L. How can that be? [Shouldn't Peter have 'false' for contradictory and 'true' for non-contradictory in the preceding sentence? After all, in (1) we are told in effect that contradictoriness is falsehood in every model, which implies that noncontradictoriness is truth in some model. 'Contradictory in every model' is a pleonastic expression.]
2.1) Well, we can make initial sense of this claim as follows. Suppose that one or more terms in S are equivocal or ambiguous. According to one such meaning of these terms, every normal interpretation assigns these terms an extension that renders S contradictory. But according to the other meaning, there is a normal interpretation of these terms according to which S is non-contradictory: i.e., comes out true in some normal models.
Example: 'Bill's end was the finish line, but he continued to vex his enemies and delight his friends after crossing it.' This is contradictory if 'end' means cessation. But it is not if 'end' means goal.
2.2) Cases such as these can be accommodated without a significant conceptual revision of the normal-model picture above. We simply need to assume that L is rich enough to express all the meanings of the infected terms. We then introduce into the language two or more new terms to replace the ambiguous term, each of which is assigned an unequivocal meaning, and make appropriate adjustments throughout. Next we assign suitable extensions to the new terms according to their assigned meanings. We then reinterpret S by introducing new sentences S’ and S* to replace S. The result is that at least one of the new sentences, S’ or S*, has a normal-model and hence is true in such a model. The original sentence S, then, is only *apparently* contradictory. Obviously, similar adjustments will have to be made to all sentences expressible in L in which the infected terms occur. Let us call such models *adjusted-normal-models* of S.
And so, on Peter's analysis, the example I gave is only apparently contradictory.
2.3) As we have seen, as long as we assume that L is rich enough to express the equivocation or ambiguity, it is possible to devise an adjusted-normal-model for S so that its apparent contradictoriness is removed.
3) However, I suspect that the above way of removing the apparent contradictoriness of S is not what Prof. Anderson calls MACRUEs [merely apparent contradictions resulting from unarticulated contradictions] and it is not what Prof. Anderson thinks is the case with the apparent contradictoriness of the Trinitarian sentence.
3.1) MACRUEs, according to Prof. Anderson, share with the above cases the fact that MACRUE sentences are only apparently contradictory due to the fact that they include equivocal or ambiguous terms. However, the ambiguity involved may be *inscrutable* to us in the sense that we may not be able to find an adjusted-normal-model for a MACRUE sentence. Let us call such sentences *inscrutable-sentences*.
3.2) Now, I wish here to distinguish two cases of such inscrutability:
(A) Epistemic Inscrutability. Cases of epistemic-inscrutability feature the following properties:
(i) S is apparently contradictory;
(ii) Some term(s) in S is equivocal or ambiguous;
(iii) L is conceptually rich enough to express the equivocation or ambiguity;
(iv) There is an adjusted-normal-model Mi in which at least one interpretation of S is true;
(v) We are currently or (perhaps) permanently unable to find the adjusted-normal-model Mi for S according to which some version of S comes out true because of our own cognitive limitations and consequent inability to correctly disambiguate the infected terms.
(B) Conceptual Inscrutability. These cases feature the following properties:
(i) S is apparently contradictory;
(ii) Some term(s) in S is equivocal or ambiguous;
(iii) L is not rich enough to express the equivocation or ambiguity;
(iv) There is no adjusted-normal-model Mi for S such that according to it S comes out true;
(v) Nevertheless, there is a language L* in which the affected terms could be disambiguated and there is an abnormal-model Ma for L* according to which a revised version of S would come out true;
(iv) We are currently or permanently unable to have cognitive access to L* and to Ma in order to ascertain that S has a model in which it is true.
3.3) It is important to see that in the case in which conceptual-inscrutability holds, then the abnormal-model Ma of S is “invisible” from the point of view of the set of models for L. Whatever abnormal-models make S true, these models cannot even be formulated with respect to L: they are not detectable. If they were, then they would become models of L.
4) I think that Prof. Anderson’s MACRUE’s are more akin to conceptually-inscrutable sentences rather than epistemically inscrutable sentences. And Prof. Anderson maintains, I believe, that the Trinitarian-sentence is conceptually-inscrutable. [Seems right.]
4.1) It is important to appreciate some general ramifications of these forms of inscrutability. On both forms of inscrutability, the ambiguity or equivocation is contagious in the sense that the ambiguity infects *all* sentences in which the infected term(s) occurs. This means that if one or more terms of the Trinitarian-sentence are alleged to be ambiguous, then all occurrences of the said terms are infected with the said ambiguity throughout all theological discourse. Thus, if one maintains that the apparent contradictoriness of the Trinitarian-sentence is due to an ambiguity in the term ‘person’, the term ‘God’, or the term ‘divine’, or all of them, then whenever these terms occur throughout all theological discourse, they must be viewed as infected with ambiguity.
4.2) Thus, to the extent that we insist that we cannot discern the ambiguity in the case of the Trinitarian-sentence, we are thereby committed to plead ignorance regarding all other cases as well. This means that even in the case of the weaker epistemic-inscrutability, we must admit that we cannot understand the meaning of all the theological sentences in which the infected terms occur, even though ex hypothesi our language is rich enough to disambiguate the alleged equivocation and, therefore, the contradictoriness is merely apparent. Thus, in the case of epistemic-inscrutability, these sentences do have a clear and distinct meaning in the language, but we are unable to discern this meaning and to that extent fail to understand any of the sentences in which the infected terms occur.
4.3) We cannot underestimate the far-reaching consequences of these results, even in the case of the weaker epistemic-inscrutability. For instance, if one maintains that the apparent contradictoriness of the Trinitarian-sentence [Peter should give an example. Perhaps this will serve: "There is one God in three divine Persons."] is due to an ambiguity of the term ‘God’, for instance, then the sentence ‘God created the world’ becomes an inscrutable sentence. We are committed to maintain that while the theological language contains the resources to assign an unambiguous meaning to this sentence and that therefore there is an adjusted-normal-model according to which it is true, we simply do not know what this model is.
Here a detailed example is needed. I'll take a stab at providing one. To explain how there can be one God in three divine Persons, one might say that 'God,' which is ambiguous as between individual divine nature (substantial essence) and individual nature + whatever makes this essence a full-fledged individual, refers to the individual divine nature, and that this nature is the nature of three numerically distinct supposita, the Father, the Son, and the Holy Ghost. But on this disambiguation of 'God,' it cannot be the case that God created the world. For it cannot be the case that a mere substantial essence created the world: only a full-fledged individual substance possessing causal powers could do such a thing.
Peter's point, I take it, is that if the meaning of 'God' in a Trinitarian sentence is epistemically inscrutable, then it will also be epistemically inscrutable in non-Trinitarian sentences such as 'God created the world.'
But isn't it open to a critic to argue that condition (v) of epistemic inscrutability is not satisfied in this example?
4.4) We are now in a position of epistemic-ignorance with respect to every theological sentence in which the infected terms occur. Hence, we fail to understand all of them.
Question: Why can't theological terms shift their sense as we move from one area of theological discourse to another?
5) The situation becomes intolerably worse if we interpret the situation of the Trinitarian-sentences along the lines of conceptual-inscrutability. For in this case we not only fail to know the actual adjusted-normal-model for the Trinitarian-sentence, but we must admit that there is no such model for our theological language as a whole. The situation is rather that while in our own theological language the alleged ambiguity is inexpressible, there is another language to which we have no cognitive access in which the ambiguity is expressible. The result is that we cannot even conceive of a model for the Trinitarian-sentence, although we hold that such a model exists (albeit it is an abnormal-model relative to our own language). Hence, we cannot claim to even possibly understand the Trinitarian-sentence, for we do not even have the linguistic resources to express its real meaning in our own language.
Well, we cannot SPECIFY such a model, but perhaps we can conceive of the existence of one analogously as a person who has no inkling of irrational numbers (numbers that cannot be expressed as simple fractions) cannot SPECIFY the length of the hypotenuse of a right triangle whose other sides are 1 unit in length but can nevertheless refer to that length in a general sort of way. Peter is placing a very stringent constraint on the nature of understanding, one that Anderson could reasonably resist.
5.1) Once again the above result becomes contagious. Every sentence in which the infected terms occur becomes completely inscrutable to us. Except this time the inscrutability is so complete that we cannot even hope to approximate the real meaning of these sentences, for their real meaning is beyond the conceptual resources available to us in our own language. I shall call this *hyper-inscrutability*.
5.2) Hyper-inscrutability renders virtually all important sentences of theology completely beyond our comprehension. We are simply unable to tell or claim to understand what they mean. We are even unable to tell whether or not this or that sentence in which some of the infected terms occur is even meaningful. It is difficult to see how one can delineate any criteria for what is orthodoxy and what is not under these sort of conditions.
6) Finally, I wish to draw attention to several questions:
6.1) How do we tell whether a given case is a case of merely epistemic-inscrutability or that it belongs to the more severe case of conceptual-inscrutability, which entails hyper-inscrutability with its completely unacceptable consequences?
6.2) What reasons do we have to believe that there are cases of conceptual-inscrutability?
Here I suppose Anderson would invoke divine revelation: The Trinity is a revealed truth. Peter will presumably demand the criteria whereby putative revelation can be known to be genuine revelation, and then we open another can of worms. The modern mind won't allow in any Transcendence that cannot be certified, whose epistemological certificate of authenticity cannot be produced on demand, to put it vaguely but suggestively.
6.3) Are we willing to accept the consequence that the Trinitarian-sentence and therefore our whole theological language is hyper-inscrutable in order to admit the requisite distinction between apparent and real contradictoriness of the severe kind entailed by conceptual-inscrutability?