Boğaziçi University Workshop on Paradox Session 1A: to the Arché Research Centre for Logic, Language, Metaphysics and Epistemology University of St Andrews, Scotland 5 April 2012 to the Theses Proof The legacy of Aristotle logica vetus (c. 1100): Porphyry s Isagoge, Aristotle s Categories and De Interpretatione, various logical works of Boethius logica nova (by c. 1200): the rest of the Organon: Prior Analytics, Posterior Analytics, Topics, De Sophisticis Elenchis The medievals contribution: logica modernorum (from c. 1150) theory of properties of terms (signification, supposition, ampliation, restriction, copulation, relation, etc.) theory of consequences theory of insolubles theory of obligations stimulated by the theory of fallacy, following recovery of De Sophisticis Elenchis around 1140 reached fulfilment in the 14th century, the most productive century for logic before the 19th. to the Theses Proof 1 / 20 2 / 20 are logical paradoxes, like the Liar paradox. Suppose Socrates says Socrates says something false (call it σ) and nothing else: First, suppose σ is true. Then things are as it says they are, so σ is false, and not true; so by reductio ad absurdum, σ is not true, and so by Bivalence, σ is false. But, secondly, given that σ is false (as we have just proved), then things are indeed as σ says they are, so σ is true. Contradiction. Bivalence Every proposition is either true or false. Contravalence No proposition is both true and false. Upwards T-inference A proposition is true if things are as it says they are. Downwards T-inference If a proposition is true then things are as it says they are. to the Theses Proof More Suppose that in some share-out all and only those who do not utter a falsehood will receive a penny, and suppose Socrates only says: I will not receive a penny. Suppose Socrates only says: Plato speaks truly, and Plato only says: Socrates speaks falsely. (The yes - no, or postcard, paradox) Suppose Socrates only says: Plato speaks falsely, and Plato only says: Socrates speaks falsely. (The no - no paradox) Suppose there is only one disjunction in the world: A man is an ass or some disjunction is false. or consider the conjunction: God exists and this conjunction is false, supposing this is the only conjunction (God has destroyed all others). or the conditional: If this conditional is true then God does not exist. (Curry s paradox) or the consequence: God exists; therefore, this consequence is invalid. (Pseudo-Scotus paradox) or the proposition: This proposition is not known by you. (The Knower paradox) to the Theses Proof 3 / 20 4 / 20
Born Hartfield, Sussex, around 1300 Regent master, Oxford (Balliol) 1321 Insolubilia, early 1320s Merton College 1323, first of the Oxford Calculators De Proportionibus, 1328 Member of Richard de Bury s circle, from 1335 Chancellor of St Paul s Cathedral, London, from 1337 De Causa Dei contra Pelagium, published 1344 Consecrated Archbishop of Canterbury, Avignon July 1349 Died at Lambeth from the Black Death, August 1349. to the Theses Proof Insolubilia Preserved in thirteen MSS, eight complete Every passage preserved in at least ten, at most eleven, MSS Edited from just two MSS by Marie-Louise Roure in 1970 New edition from all MSS with English translation by 2010 First half (chs. 1-5): a refutation of eight earlier views, especially that of the restricters (restringentes) Second half (chs. 6-12): own theory to the Theses Proof Madrid MS: Expliciunt insolubilia magistri thome de bradwardyn de anglia regentis Oxonie. Amen Ralph Strode (1360s): then arose that prince of modern natural philosophers, namely master, who was the first to come upon something worthwhile concerning insolubles. 5 / 20 6 / 20 : London Royal 12 F XIX f. 149 r Solvere non est ignorantis vinculum, 3 o Metaphysice, capitulo primo. Qui ergo insolubilium vinculi sunt ignari nodum illorum ambiguum nequeunt aperire, sed huiusmodi vinculo ut bruta funiculo [in] demum adducuntur. To untie a knot is not a job for the nitwit. (Aristotle, Metaphysics B1) Aristotle means that those who are unacquainted with the tangle of the insolubles are unable to release their Janus-faced grip, but are sure in the end to be brought to heel by a knot of this kind like an animal on a short leash. to the Theses Proof Classification of s 1. the restricters (restringentes), of which there are two groups: 1.1 term-restricters are so called because they do not permit terms to supposit for all their instances. Of these there are three sorts. 1.1.1 for some of them solve insolubles by relative and absolute, that is, the part cannot supposit for the whole (e.g., the predicate of the Liar proposition cannot supposit for the Liar itself) 1.1.2 others solve them by a figure of speech, that is, the supposition changes from one proposition to another (e.g., when tries to infer σ from σ is false ) 1.1.3 yet others by false cause, that is, for much the same reason: the fact that σ is false is no reason (or cause ) to believe σ 1.2 others restrict the time, that is, supposition is restricted to some time before the present (so the utterance of the Liar must refer to an earlier utterance) to the Theses Proof 7 / 20 8 / 20
Classification, continued to the Assumptions Definitions to the 2. the nullifiers (cassantes), of which there are again two types: 2.1 for some nullify the potency, that is, deny that the Liar can be uttered 2.2 some nullify the act, that is, deny that anything has been said by uttering the Liar 3. others propose a middle way (mediantes), that is, that the Liar is neither true nor false 4. others make distinctions (distinguentes), e.g., between the Liar in thought and the Liar as uttered 5. lastly, there is the solution of Aristotle and his followers, that is, himself. Theses Proof 1. A true proposition is an utterance signifying only as things are. 2. A false proposition is an utterance signifying other than things are. Postulates 1. Every proposition is true or false (and not both). 2. Every proposition signifies or means as a matter of fact or absolutely everything which follows from it as a matter of fact or absolutely. 3. The part can supposit for its whole and for its opposite and for what is equivalent to them. 4. Conjunctions and disjunctions with mutually contradictory parts contradict each other. 5. From any disjunction together with the opposite of one of its parts the other part may be inferred. 6. If a conjunction is true each part is true and conversely; and if it is false, one of its parts is false and conversely. And if a disjunction is true, one of its parts is true and conversely; and if it is false, each part is false and conversely. Theses Proof 9 / 20 10 / 20 The interpretation of to the Theses to the Paul Spade takes at face value: Theses BP A [proposition] signifies or denotes whatever follows from it. However, BP is not sufficient for argument. So Spade conjectures: CBP whatever a [proposition] signifies follows from it. But CBP leads to paradox. Arguably, BP follows from (P2), and would agree to it. But never mentions CBP. My interpretation: a proposition signifies whatever follows from what it signifies. is closed under consequence. This is how uniformly and repeatedly uses (P2), even though not strictly how he states it. Theses Proof 1. Every proposition signifies or means affirmation or denial for the supposita of its subject or predicate. 2. If a proposition signifies itself not to be true or itself to be false, it (also) signifies itself to be true and is false. 3. If a proposition only signifies itself to be unknown to someone, or if in addition it only signifies some thing or things known to him, then it signifies that it is unknown to him that it is unknown to him. E.g., take σ: Socrates says something false, where this is all Socrates says. Then σ signifies that σ is itself false, by Thesis 1 and Postulate (P2). So by Thesis 2, σ also signifies that σ is true. So σ does not signify only as things are (for σ cannot be both true and false), so by Definition (D2), σ is false. Theses Proof 11 / 20 12 / 20
The Liar strikes back: revenge to the Formalizing Postulates to the But if says that what Socrates says is false, then surely Socrates was right when he said the same thing? No, says. What Socrates said was σ, what said was, let s say, σ. Then σ and σ seem similar. But they are not. For σ is self-referential, and says of itself that it is false (and so, by Thesis 2, also says of itself that it is true). σ is not self-referential, but says of σ that it is false. So σ and σ have different truth-conditions, and accordingly, one is false, the other true. Theses Proof (D1) Tr(s) := ( p)sig(s, p) ( p)(sig(s, p) p) (D2) Fa(s) := ( p)(sig(s, p) p) (D0) Prop(s) := ( p)sig(s, p) (P1) Prop(s) (Tr(s) Fa(s)) (P1) embodies both Bivalence (every meaningful utterance is either true or false) and Contravalence (no meaningful utterance is both true and false). Theses Proof 13 / 20 14 / 20 Theory of is a closure postulate: (P2) ( p, q)((p q) (Sig(s, p) Sig(s, q))) main claim, Thesis (T2), is that every utterance which signifies itself not to be true, or to be false, also signifies itself to be true, and is false. Sig(s, Fa(s)) (Sig(s, Tr(s)) Fa(s)) Given (P1) and (P2), this is the same as Sig(s, Tr(s)) (Sig(s, Tr(s)) Fa(s)) to the Theses Proof Proof We can outline the essentials of proof of Thesis 2 as follows: Suppose Sig(s, Fa(s) Q) is all s signifies If Fa(s) then p(sig(s, p) p) (D2) i.e. either Q or Fa(s) (T1) But Sig(s, Fa(s)) so Sig(s, Fa(s) Q)) (P2) i.e., Sig(s, Tr(s) Q)) (P1) Moreover, ((Tr(s) Q) Q) Tr(s) (P5) and Sig(s, Q) So Sig(s, Tr(s)) (P2) Hence Sig(s, Tr(s) Fa(s)) So Fa(s) (P1) and (D2) to the Theses Proof 15 / 20 16 / 20
Upwards and Downwards T-inference On theory, Downwards T-inference (Maudlin s term: Field calls it T-OUT) is unrestrictedly valid: Tr( p ) p, for if Tr( p ) then q(sig( p, q) q), so on the assumption that Sig( p, p), it follows that p. That is, if p is true then everything that p says is the case, in particular, that p. But theory restricts Upwards T-inference (what Field calls T-IN). For p to be true, everything that p says must be the case. So in general, the fact that p is insufficient for p to be true. An immediate counterexample is Socrates utterance (σ). What σ says, namely, that σ is not true, is indeed the case, since σ is false. But that is not enough to infer that σ is true. For σ to be true, everything that σ says would have to be the case, and that s impossible. to the Theses Proof Back to (BP) and (P2) We can now establish the earlier claims: If Sig( p, p) then (P2) entails (BP), that a proposition signifies everything which follows from it: Suppose p q. Then by (P2), ( s)(sig(s, p) Sig(s, q)), so Sig( p, p) Sig( p, q). But Sig( p, p), so Sig( p, q), i.e., ( p, q)((p q) Sig( p, q)) We can represent Spade s (CBP) whatever a proposition signifies follows from it as if Sig( p, q) then p q (BP) (CBP) Counterexample: according to, Sig(σ, Tr(σ)), i.e., Sig( Tr(σ), Tr(σ)). But Tr(σ) is false, so Tr(σ) Tr(σ) is false too. (CBP) leads to contradiction: We have shown that Sig( Tr(σ), Tr(σ)) So by (CBP), Tr(σ) Tr(σ) But by (T2), Tr(σ). So Tr(σ). Contradiction. to the Theses Proof 17 / 20 18 / 20 Medieval logicians extended Aristotle s logical theory in many ways, in their theories of obligations, consequences and insolubles provided a thoroughly original solution to insolubles like the Liar paradox, a solution still of interest and promise today The heart of solution is his Postulate (P2), that signification is closed under consequence, so propositions can signify more than at first appears To be true, everything a proposition signifies must obtain; if anything it signifies fails to obtain, it is false shows that insolubles in general signify contradictory things, and hence are all false We ll see tomorrow how to extend discussion of the Liar paradox to other semantic paradoxes and to epistemic paradoxes, like the Knower Paradox. to the Theses Proof 19 / 20 Albert of Saxony,, tr. in N. Kretzmann and E. Stump, Cambridge Translations of Medieval Philosophical Texts, vol. I, Cambridge UP 1988 (treatise 6 of his Perutilis Logica), Insolubilia, Ed. & Eng. tr. S. Read (Dallas Medieval Texts and Translations 10), Peeters 2010 Catarina Dutilh Novaes and, Insolubilia and the Fallacy Secundum Quid et Simpliciter, Vivarium 46 (2008), 175-91 S. Read, Plural signification and the Liar paradox, Philosophical Studies 145 (2009), 363-75 T. Maudlin, Paradox, Oxford UP 2004 Paul Spade, Insolubilia and theory of signification, Medioevo 7 (1981), 115-34 Paul Spade and,, Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/archives/win2009/entries/insolubles/ to the Theses Proof 20 / 20