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Appendix: the Use of Symbolic Notation in Descriptive Logic Appendix: The Use of Symbolic Notation in Descriptive Logic One well-known introductory textbook on normative logic happens to use, as an example, certain propositions from Patristic theology,1 with the implication that Christians would do well to accept the correct- ness of the comparatively little remembered Patripassian heresy. In this case, the point can be adequately stated using the ordinary notation employed in normative predicate logic:2 PATRIPASSIANISM Px. x is God. Qx. x suffered and died on the Cross. a. the Father. b. the Son. 1. ∃∀xyPy() ↔ x = y 2. Pa 3. Pb ∴= 4. ab 5. Qb ∴ 6. Qa In fact, however, Christians in all periods have usually found the idea of the Father’s having suffered and died to be a repugnant one; and the Church of the fourth century was disturbed much less by Patripassianism than by another doctrine, one that can be stated using the same vocabulary: ARIANISM 1. ∃∀xyPy() ↔ x = y 1 Hodges, Logic, pp. 138, 262. 2 Since normative logicians have not achieved a complete standardization of the notation they use, it is probably worth specifying how I intend the symbols employed in the example sequences to be read. They are as follows: ∃x : ‘There exists some x such that ... ’; ∀x : ‘For all x ... ’; PQ↔ : ‘P if and only if Q’ (i.e., P and Q are either both true or both false); xy= : ‘x equals y’; ∴: ‘therefore’; ¬P : ‘not P’ (i.e., P is false); xy≠ : ‘x does not equal y’; PQ∧ : ‘P and Q’. 155 156 Appendix 2. Pa 3. ¬Qa 4. Qb 5. ∴≠ab 6. ∴¬Pb This, too, has conclusions that some Christians will have found unpal- atable: if the younger Pliny’s informants can be believed, worshipping ‘Christ as God’ was central to Christianity as early as the reign of Trajan.3 The difficulty, as presented, can be resolved – but at the price of another substantial derogation from the faith of the Apostles: DOCETISM 1. ∃∀xyPy() ↔ x = y 2. Pa 3. Pb ∴= 4. ab ¬ 5. Qa ∴¬ 6. Qb I do not assert, nor do I even think it especially likely, that this repre- sents an accurate reconstruction of any specific Docetist belief system; it will be recalled that Islam takes a Docetic view of the Crucifixion, holding that Jesus only seemed to die on the Cross (Koran 4.157), without regarding Jesus as the Son of God at all. But the situation is still worse if we try to write out Consubstantialism, the viewpoint that was ultimately accepted as orthodox, in the same form: CONSUBSTANTIALISM (1) Px. x is God. Qx. x suffered and died on the Cross. Rxy. x and y are the same person. Sxy. x and y are of the same substance. a. the Father. b. the Son. 1. ∃∀xyPy() ↔ Sxy 2. ∃∃x y() Px ∧ Py ∧¬ Rxy 3 Pliny the Younger, Complete Letters X.96, p. 279. Appendix 157 3. Pa 4. Pb 5. Qb 6. ¬Qa 7. ∴∧¬Sab Rab This sequence fails entirely, as a descriptive reconstruction of Consubstantialism. The ideas of ‘person’ and ‘substance’ are simply imported as content, rather than arising within the belief system as part of a reasoning process. What we have is not a reconstruction of believers’ actual logic: it is merely a static declaration of an answer, and the meaning of ‘person’ and ‘substance’ within the belief system has not been established. To understand the teaching of the Nicene Fathers correctly, we must return to stating the original problem – and then show, as plausibly as we can, how a sequence like the one given in the last paragraph comes to answer it. Some of the steps, however, are difficult to express using the existing symbolic notation; and it is more convenient to include some propositions in verbal form. CONSUBSTANTIALISM (2) Px. x is God. Qx. x suffered and died on the Cross. a. the Father. b. the Son. 1. ∃∀xyPy() ↔ x = y 2. Pa 3. Pb 4. Qb 5. ¬Qa 6. The question ‘ab= ?’ cannot be unequivocally answered. 7. Therefore, there is a difficulty with ‘=’: which means that 8. expressions involving ‘=’ need to be rewritten to avoid the difficulty. = 9. This can be done by distinguishing two kinds of identity, ‘ 1 ’ and = ‘ 2 ’. ∃∀ ↔ = 10. xyPy() x2 y ∃∃ ∧ ∧ ≠ 11. x y() Px Py x1 y 12. Pa 158 Appendix 13. Pb 14. Qb 15. ¬Qa =∧≠ 16. ()()abab21 = = ‘17. ‘17. 1’ and ‘ 2’ can be described using the relations Rxy and Sxy. Rxy. x and y are the same person. Sxy. x and y are of the same substance. 18. ∃∀xyPy() ↔ Sxy 19. ∃∃x y() Px ∧ Py ∧¬ Rxy 20. Pa 21. Pb∧∧¬ Qb Qa 22. Sab∧¬ Rab The unkind will be likely to regard this sequence as an argument having the form 1. ab= 2. ab≠ 3. ∴‘=’ ≠ ‘=’ But we do, in any case, reason this way all the time: confronted by two propositions, neither of which we feel able to deny, but which strike us as contradictory, we find a way of reconciling them. ‘It depends on what the meaning of “is” is.’ It is more interesting to observe that this reconstruction allows us to draw two significant conclusions. Firstly, it would be quite wrong to claim that the Fathers of the Church only developed their doctrines on ‘person’ and ‘substance’ because they were Hellenistic philosophers and these were the terms available to them. It is true (step 17) that the particular terminology was borrowed; but any Christian who wants to believe that there is only one God, and that the Father is God, and the Son is God, and the Son died on the Cross, and the Father did not die on the Cross is already thinking implicitly in terms of two distinct kinds of identity relation. In one way the Father is the same as the Son; in another way they are different; yet there is only one God. Whether or not our intellectual and educa- tional background predisposes us to express the point by talking about πρόσωπον and οὐσία is comparatively inessential; we will in any event need to think about God using two distinct kinds of identity, neither confounding the first nor dividing the second. It is interesting in this Appendix 159 connection to note that some analogous disputes appear to have trou- bled nineteenth-century Christadelphianism. Bryan R. Wilson, a writer in the sociological tradition, mentions that the movement experienced heresies asserting, among other things, that Christ was a mere man; [ ... ] that Christ was not a son of Adam and not a sharer in human mortality [ ... ]; that Christ in the days of his flesh was partly of ‘divine substance’; [ ... ] that Christ was a passive medium with no will of his own.4 Wilson does not go into further detail, and I have not myself had the opportunity for the extensive reading in Christadelphian literature that would be needed before one could venture a reconstruction of what these ideas actually were or meant. In particular, Wilson provides no reference for the phrase ‘divine substance’, although he places it in quotation marks: and he gives his reader no clue as to whether the expression was borrowed from the ecclesiastical tradition, or whether heterodox Christadelphians adopted it for themselves. If it could be demonstrated that Christadelphianism had independently (or semi- independently) replicated some of the same theological debates that occupied the Church in the fourth and fifth centuries, without the participants’ being heavily immersed either in Patristics or in Hellenistic philosophy, that would be an exceptionally important piece of compar- ative evidence. It would do for the study of Patristics what the discovery of life on other planets might do for genetics; but, then, the hope of discovering extraterrestrial life has so far always been disappointed. The matter will only be resolved when some competent descriptive logician feels inspired to start looking through back numbers of The Christadelphian. Secondly, this reconstruction makes it easy to see the nullity of the purported compromise represented by the Homoiousion, the doctrine that the Father and the Son were ‘of like substance’. Even if we do decide to augment our vocabulary with the relation Txy. x and y are of like substance, we shall still have to decide what to do with it. It is clear from steps 10 and 18 of the reconstruction of Consubstantialism that the Homoousion is here doing the logical work hitherto done by identity in formulating 4 Bryan R. Wilson, Sects and Society, p. 244. 160 Appendix the belief that there is only one God. If the Homoiousion is simply inserted in its place, we have: THE HOMOIOUSION (1: ‘only a diphthong’) 1. ∃∀xyPy() ↔ Txy 2. ∃∃x y() Px ∧ Py ∧¬ Rxy 3. Pa 4. Pb∧∧¬ Qb Qa 5. Tab∧¬ Rab which differs from steps 18–22 of the Consubstantialism reconstruction only in an insignificant matter of terminology. The ‘like substance’ rela- tion is still functioning as a kind of identity relation; we are still distin- guishing two kinds of identity whenever we talk about God. If, on the other hand, we take the point of ‘like substance’ as being that it cannot be treated as an identity relation, we shall need to fall back on ordinary identity in our expression of monotheism. We will then have: THE HOMOIOUSION (2: ‘Semi-Arianism’) 1. ∃∀xyPy() ↔ x = y 2. Pa 3. ¬Qa 4. Qb 5. ∴≠ab 6. ∴¬Pb 7. Tab – which I have labelled ‘Semi-Arianism’, in line with tradition, but which is really pure Arianism. The Homoiousion has simply been tacked on at the end, with no close logical relation to the rest of the argument and, therefore, with no real meaning.
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