A Medieval A Medieval Solution to the Medieval Logic Solution to the Bo˘gazi¸ciUniversity Liar Paradox Liar Paradox Stephen Read I The legacy of Stephen Read

Workshop on Truth and Paradox I logica vetus (c. 1100): ’s , Aristotle’s Medieval Logic Medieval Logic Insolubles and , various logical works Insolubles of ’ Session 1A: Thomas Thomas Bradwardine Bradwardine I logica nova (by c. 1200): the rest of the : Prior Bradwardine’s Bradwardine’s A Medieval Solution to the Liar Paradox Insolubles Analytics, , , De Sophisticis Insolubles Earlier Solutions Earlier Solutions Elenchis Bradwardine’s Bradwardine’s Solution I The medievals’ contribution: logica modernorum (from Solution Stephen Read Postulate 2 Postulate 2 Bradwardine’s Theses c. 1150) Bradwardine’s Theses Bradwardine’s Proof Bradwardine’s Proof Truth and I theory of properties of terms (signification, supposition, Truth and Arch´eResearch Centre for Logic, Language, Metaphysics and Signification ampliation, restriction, copulation, relation, etc.) Signification Epistemology Summary Summary University of St Andrews, Scotland References I theory of consequences References I theory of insolubles I theory of obligations I stimulated by the theory of fallacy, following recovery of De Sophisticis Elenchis around 1140 5 April 2012 I reached fulfilment in the 14th century, the most productive century for logic before the 19th.

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A Medieval A Medieval Insolubles Solution to the More Insolubles Solution to the Liar Paradox Liar Paradox Insolubles are logical paradoxes, like the Liar paradox. Suppose Stephen Read I Suppose that in some share-out all and only those who do not Stephen Read Socrates says ‘Socrates says something false’ (call it σ) and nothing utter a falsehood will receive a penny, and suppose Socrates else: Medieval Logic Medieval Logic Insolubles only says: ‘I will not receive a penny’. Insolubles I First, suppose σ is true. Then things are as it says they are, so Thomas I Suppose Socrates only says: ‘Plato speaks truly’, and Plato Thomas σ is false, and not true; so by reductio ad absurdum, σ is not Bradwardine only says: ‘Socrates speaks falsely’. (The ‘yes’-‘no’, or Bradwardine Bradwardine’s Bradwardine’s true, and so by Bivalence, σ is false. Insolubles postcard, paradox) Insolubles Earlier Solutions Earlier Solutions I But, secondly, given that σ is false (as we have just proved), I Suppose Socrates only says: ‘Plato speaks falsely’, and Plato Bradwardine’s Bradwardine’s then things are indeed as σ says they are, so σ is true. Solution only says: ‘Socrates speaks falsely’. (The ‘no’-‘no’ paradox) Solution Postulate 2 Postulate 2 Contradiction. Bradwardine’s Theses I Suppose there is only one disjunction in the world: ‘A man is Bradwardine’s Theses Bradwardine’s Proof an ass or some disjunction is false’. Bradwardine’s Proof Bivalence Every proposition is either true or false. Truth and Truth and Signification I or consider the conjunction: ‘God exists and this conjunction is Signification Contravalence No proposition is both true and false. Summary false’, supposing this is the only conjunction (God has Summary References References Upwards T-inference A proposition is true if things are as it says destroyed all others). they are. I or the conditional: ‘If this conditional is true then God does Downwards T-inference If a proposition is true then things are as it not exist’. (Curry’s paradox) says they are. I or the consequence: ‘God exists; therefore, this consequence is invalid’. (Pseudo-Scotus’ paradox) I or the proposition: ‘This proposition is not known by you’. (The Knower paradox)

3 / 20 4 / 20 A Medieval A Medieval Thomas Bradwardine Solution to the Bradwardine’s Insolubilia Solution to the Liar Paradox Liar Paradox I Born Hartfield, Sussex, around 1300 Stephen Read Stephen Read I Regent master, Oxford (Balliol) 1321 I Preserved in thirteen MSS, eight complete Medieval Logic Medieval Logic I Insolubilia, early 1320s Insolubles I Every passage preserved in at least ten, at most eleven, Insolubles I Merton College 1323, first of the Oxford Calculators Thomas MSS Thomas I De Proportionibus, 1328 Bradwardine Bradwardine Bradwardine’s Bradwardine’s I Member of Richard de Bury’s circle, from 1335 Insolubles I Edited from just two MSS by Marie-Louise Roure in Insolubles Earlier Solutions Earlier Solutions I Chancellor of St Paul’s Cathedral, London, from 1337 1970 Bradwardine’s Bradwardine’s I De Causa Dei contra Pelagium, published 1344 Solution I New edition from all MSS with English translation by Solution I Consecrated Archbishop of Canterbury, Avignon July 1349 Postulate 2 Postulate 2 Bradwardine’s Theses Stephen Read 2010 Bradwardine’s Theses I Died at Lambeth from the Black Death, August 1349. Bradwardine’s Proof Bradwardine’s Proof Truth and Truth and Signification I First half (chs. 1-5): a refutation of eight earlier views, Signification Summary especially that of the restricters (restringentes) Summary References References I Second half (chs. 6-12): Bradwardine’s own theory Ralph Strode (1360s): “then arose that prince of Madrid MS: ‘Expliciunt insolubilia magistri thome de bradwardyn de I anglia regentis Oxonie. Amen’ modern natural philosophers, namely master Thomas Bradwardine, who was the first to come upon something worthwhile concerning insolubles.”

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A Medieval A Medieval Bradwardine’s Insolubles: London Royal 12 F XIX Solution to the Bradwardine’s Classification of Solutions Solution to the r Liar Paradox Liar Paradox f. 149 Stephen Read Stephen Read 1. the restricters (restringentes), of which there are two Solvere non est ignorantis vinculum, Medieval Logic groups: Medieval Logic o 3 Metaphysice, capitulo primo. Insolubles 1.1 term-restricters are so called because they do not Insolubles Thomas Thomas Qui ergo insolubilium vinculi sunt Bradwardine permit terms to supposit for all their instances. Of Bradwardine ignari nodum illorum ambiguum Bradwardine’s these there are three sorts. Bradwardine’s Insolubles Insolubles nequeunt aperire, sed huiusmodi Earlier Solutions 1.1.1 for some of them solve insolubles by “relative and Earlier Solutions vinculo ut bruta funiculo [in] Bradwardine’s absolute”, that is, the part cannot supposit for the Bradwardine’s Solution Solution demum adducuntur. Postulate 2 whole (e.g., the predicate of the Liar proposition Postulate 2 Bradwardine’s Theses cannot supposit for the Liar itself) Bradwardine’s Theses ‘To untie a knot is not a job for the Bradwardine’s Proof Bradwardine’s Proof Truth and 1.1.2 others solve them by a figure of speech, that is, the Truth and nitwit.’ (Aristotle, Metaphysics Signification Signification supposition changes from one proposition to another B1) Aristotle means that those Summary Summary (e.g., when tries to infer σ from ‘σ is false’) who are unacquainted with the References References 1.1.3 yet others by “false cause”, that is, for much the same tangle of the insolubles are unable reason: the fact that σ is false is no reason (or to release their Janus-faced grip, “cause”) to believe σ but are sure in the end to be 1.2 others restrict the time, that is, supposition is restricted brought to heel by a knot of this to some time before the present (so the utterance of the kind like an animal on a short Liar must refer to an earlier utterance) leash.

7 / 20 8 / 20 A Medieval A Medieval Bradwardine’s Classification, continued Solution to the Bradwardine’s Assumptions Solution to the Liar Paradox Liar Paradox

Stephen Read Definitions Stephen Read 1. A true proposition is an utterance signifying only as things are. Medieval Logic Medieval Logic 2. the nullifiers (cassantes), of which there are again two Insolubles 2. A false proposition is an utterance signifying other than things are. Insolubles types: Thomas Thomas Bradwardine Bradwardine 2.1 for some nullify the potency, that is, deny that the Liar Bradwardine’s Postulates Bradwardine’s Insolubles Insolubles can be uttered Earlier Solutions 1. Every proposition is true or false (and not both). Earlier Solutions 2.2 some nullify the act, that is, deny that anything has Bradwardine’s 2. Every proposition signifies or means as a matter of fact or Bradwardine’s been said by uttering the Liar Solution Solution Postulate 2 absolutely everything which follows from it as a matter of fact or Postulate 2 Bradwardine’s Theses absolutely. Bradwardine’s Theses 3. others propose a middle way (mediantes), that is, that Bradwardine’s Proof Bradwardine’s Proof Truth and 3. The part can supposit for its whole and for its opposite and for Truth and the Liar is neither true nor false Signification Signification what is equivalent to them. Summary Summary 4. others make distinctions (distinguentes), e.g., between References 4. Conjunctions and disjunctions with mutually contradictory parts References the Liar in thought and the Liar as uttered contradict each other. 5. From any disjunction together with the opposite of one of its parts 5. lastly, there is the solution of Aristotle and his followers, the other part may be inferred. that is, Bradwardine himself. 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.

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A Medieval A Medieval The interpretation of Postulate 2 Solution to the Bradwardine’s Theses Solution to the Liar Paradox Liar Paradox Paul Spade takes Postulate 2 at face value: Stephen Read Theses Stephen Read BP “A [proposition] signifies or denotes whatever Medieval Logic Medieval Logic Insolubles 1. Every proposition signifies or means affirmation or Insolubles follows from it.” Thomas denial for the supposita of its subject or predicate. Thomas Bradwardine Bradwardine However, BP is not sufficient for Bradwardine’s argument. Bradwardine’s Bradwardine’s Insolubles 2. If a proposition signifies itself not to be true or itself to Insolubles So Spade conjectures: Earlier Solutions be false, it (also) signifies itself to be true and is false. Earlier Solutions Bradwardine’s Bradwardine’s CBP “whatever a [proposition] signifies follows from Solution 3. If a proposition only signifies itself to be unknown to Solution it.” Postulate 2 Postulate 2 Bradwardine’s Theses someone, or if in addition it only signifies some thing or Bradwardine’s Theses Bradwardine’s Proof Bradwardine’s Proof But CBP leads to paradox. Arguably, BP follows from (P2), Truth and Truth and Signification things known to him, then it signifies that it is unknown Signification and Bradwardine would agree to it. But Bradwardine never Summary to him that it is unknown to him. Summary mentions CBP. References References My interpretation: a proposition signifies whatever follows E.g., take σ: ‘Socrates says something false’, where this is from what it signifies. Signification is closed under all Socrates says. Then σ signifies that σ is itself false, by consequence. Thesis 1 and Postulate (P2). So by Thesis 2, σ also signifies This is how Bradwardine uniformly and repeatedly uses (P2), that σ is true. So σ does not signify only as things are (for σ even though not strictly how he states it. cannot be both true and false), so by Definition (D2), σ is false.

11 / 20 12 / 20 A Medieval A Medieval The Liar strikes back: revenge Solution to the Formalizing Bradwardine’s Postulates Solution to the Liar Paradox Liar Paradox

Stephen Read Stephen Read

Medieval Logic (D1) Tr(s) := (∃p)Sig(s, p) ∧ (∀p)(Sig(s, p) → p) Medieval Logic I But if Bradwardine says that what Socrates says is Insolubles Insolubles false, then surely Socrates was right when he said the Thomas Thomas Bradwardine (D2) Fa(s) := (∃p)(Sig(s, p) ∧ ¬p) Bradwardine same thing? Bradwardine’s Bradwardine’s Insolubles Insolubles Earlier Solutions Earlier Solutions I No, says Bradwardine. What Socrates said was σ, what 0 Bradwardine’s Bradwardine’s Bradwardine said was, let’s say, σ . Solution (D0) Prop(s) := (∃p)Sig(s, p) Solution Postulate 2 Postulate 2 0 I Then σ and σ seem similar. But they are not. Bradwardine’s Theses Bradwardine’s Theses Bradwardine’s Proof Bradwardine’s Proof Truth and (P1) Prop(s) → (Tr(s) ∨ Fa(s)) Truth and I For σ is self-referential, and says of itself that it is false Signification Signification (and so, by Thesis 2, also says of itself that it is true). Summary Summary References (P1) embodies both Bivalence (every meaningful utterance References 0 I σ is not self-referential, but says of σ that it is false. is either true or false) and Contravalence (no meaningful 0 utterance is both true and false). I So σ and σ have different truth-conditions, and accordingly, one is false, the other true.

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A Medieval A Medieval Bradwardine’s Theory of Signification Solution to the Bradwardine’s Proof Solution to the Liar Paradox Liar Paradox I Postulate 2 is a closure postulate: Stephen Read We can outline the essentials of Bradwardine’s proof of Thesis 2 Stephen Read

Medieval Logic as follows: Medieval Logic (P2) (∀p, q)((p ⇒ q) → (Sig(s, p) → Sig(s, q))) Insolubles Insolubles Thomas Suppose Sig(s, Fa(s) ∧ Q) is all s signifies Thomas I Bradwardine’s main claim, Thesis (T2), is that every Bradwardine Bradwardine Bradwardine’s If Fa(s) then ∃p(Sig(s, p) ∧ ¬p) (D2) Bradwardine’s utterance which signifies itself not to be true, or to be Insolubles Insolubles Earlier Solutions i.e. either ¬Q or ¬Fa(s) (T1) Earlier Solutions false, also signifies itself to be true, and is false. Bradwardine’s Bradwardine’s Solution But Sig(s, Fa(s)) Solution Postulate 2 so Sig(s, ¬Fa(s) ∨ ¬Q)) (P2) Postulate 2 Sig(s, Fa(s)) → (Sig(s, Tr(s)) ∧ Fa(s)) Bradwardine’s Theses Bradwardine’s Theses Bradwardine’s Proof i.e., Sig(s, Tr(s) ∨ ¬Q)) (P1) Bradwardine’s Proof Truth and Truth and Signification Signification I Given (P1) and (P2), this is the same as Moreover, ((Tr(s) ∨ ¬Q) ∧ Q) ⇒ Tr(s) (P5) Summary and Sig(s, Q) So Sig(s, Tr(s)) (P2) Summary References References Sig(s, ¬Tr(s)) → (Sig(s, Tr(s)) ∧ Fa(s)) Hence Sig(s, Tr(s) ∧ Fa(s)) So Fa(s) (P1) and (D2)

15 / 20 16 / 20 A Medieval A Medieval Upwards and Downwards T-inference Solution to the Back to (BP) and (P2) Solution to the Liar Paradox Liar Paradox We can now establish the earlier claims: I On Bradwardine’s theory, Downwards T-inference (Maudlin’s Stephen Read Stephen Read If Sig(hpi, p) then (P2) entails (BP), that a proposition term: Field calls it T-OUT) is unrestrictedly valid: I Medieval Logic signifies everything which follows from it: Medieval Logic Insolubles Insolubles I Suppose p ⇒ q. Then by (P2), (∀s)(Sig(s, p) → Sig(s, q)), so Tr(hpi) ⇒ p, Thomas Sig(hpi, p) → Sig(hpi, q). But Sig(hpi, p), so Sig(hpi, q), i.e., Thomas Bradwardine Bradwardine Bradwardine’s Bradwardine’s for if Tr(hpi) then ∀q(Sig(hpi, q) → q), so on the assumption Insolubles (∀p, q)((p ⇒ q) → Sig(hpi, q)) (BP) Insolubles that Sig(hpi, p), it follows that p. Earlier Solutions Earlier Solutions Bradwardine’s Bradwardine’s I That is, if hpi is true then everything that hpi says is the case, Solution Solution We can represent Spade’s (CBP)—whatever a proposition in particular, that p. Postulate 2 I Postulate 2 Bradwardine’s Theses signifies follows from it—as Bradwardine’s Theses I But Bradwardine’s theory restricts Upwards T-inference (what Bradwardine’s Proof Bradwardine’s Proof Truth and Truth and Field calls T-IN). For hpi to be true, everything that hpi says Signification if Sig(hpi, q) then p ⇒ q (CBP) Signification must be the case. So in general, the fact that p is insufficient Summary Summary References References for hpi to be true. I An immediate counterexample is Socrates’ utterance (σ). I Counterexample: according to Bradwardine, Sig(σ, Tr(σ)), i.e., What σ says, namely, that σ is not true, is indeed the case, Sig(h¬Tr(σ)i, Tr(σ)). But Tr(σ) is false, so ¬Tr(σ) ⇒ Tr(σ) is false too. since σ is false. But that is not enough to infer that σ is true. (CBP) leads to contradiction: For σ to be true, everything that σ says would have to be the I I We have shown that Sig(h¬Tr(σ)i, Tr(σ)) case, and that’s impossible. I So by (CBP), ¬Tr(σ) ⇒ Tr(σ) I But by (T2), ¬Tr(σ). So Tr(σ). Contradiction. 17 / 20 18 / 20

A Medieval A Medieval Summary Solution to the References Solution to the Liar Paradox Liar Paradox I Medieval logicians extended Aristotle’s logical theory in Stephen Read I Albert of Saxony, ‘Insolubles’, tr. in N. Kretzmann and E. Stephen Read many ways, in their theories of obligations, Stump, Cambridge Translations of Medieval Philosophical Medieval Logic Medieval Logic consequences and insolubles Insolubles Texts, vol. I, Cambridge UP 1988 (treatise 6 of his Perutilis Insolubles I Thomas Bradwardine provided a thoroughly original Thomas Logica) Thomas Bradwardine Bradwardine solution to insolubles like the Liar paradox, a solution Bradwardine’s I Thomas Bradwardine, Insolubilia, Ed. & Eng. tr. S. Read Bradwardine’s Insolubles Insolubles still of interest and promise today Earlier Solutions (Dallas Medieval Texts and Translations 10), Peeters 2010 Earlier Solutions Bradwardine’s Catarina Dutilh Novaes and Stephen Read, “Insolubilia and Bradwardine’s I The heart of Bradwardine’s solution is his Postulate Solution I Solution Postulate 2 the Fallacy Secundum Quid et Simpliciter’, Vivarium 46 Postulate 2 (P2), that signification is closed under consequence, so Bradwardine’s Theses Bradwardine’s Theses Bradwardine’s Proof (2008), 175-91 Bradwardine’s Proof propositions can signify more than at first appears Truth and Truth and Signification I S. Read, ‘Plural signification and the Liar paradox’, Signification I To be true, everything a proposition signifies must Summary Philosophical Studies 145 (2009), 363-75 Summary obtain; if anything it signifies fails to obtain, it is false References References T. Maudlin, Truth and Paradox, Oxford UP 2004 Bradwardine shows that insolubles in general signify I I Paul Spade, ‘Insolubilia and Bradwardine’s theory of contradictory things, and hence are all false I signification’, Medioevo 7 (1981), 115-34 I We’ll see tomorrow how to extend Bradwardine’s I Paul Spade and Stephen Read, ‘Insolubles’, Stanford discussion of the Liar paradox to other semantic Encyclopedia of Philosophy: paradoxes and to epistemic paradoxes, like the Knower http://plato.stanford.edu/archives/win2009/entries/insolubles/ Paradox. 19 / 20 20 / 20