The Mother of All Paradoxes Volker Halbach Truth and Intensionality Amsterdam 3rd December 2016
A theory of expressions The symbols of L are: 1. infinitely many variable symbols v 0, v 1, v 2, v 3,... 2. predicate symbols = and, 3. function symbols q, and sub, 4. the connectives, and the quantifier symbol, 5. auxiliary symbols ( and ), 6. possibly finitely many further function and predicate symbols of arbitrary arity but at least the sentence letter p, and 7. for each string e of symbols there is exactly one constant in the language L. The constant for e is called the quotation constant of e. We write e for the quotation constant of e. I write x for v 0, y for v 1, and z for v 2 to make the notation less cluttered.
A theory of expressions I distinguish three kinds of nonlogical vocabulary (beyond the logical vocabulary): the syntactic (quotations constants, sub etc) and the contingent vocabulary (p etc), and.
A theory of expressions All instances of the following schemata and rules are axioms of the theory A: a.1. all axioms and rules of first-order predicate logic including the identity axioms. a.2. a b = ab, where a and b are arbitrary strings of symbols. a.3. q(a) = a a.4. sub(a, b, c) = d, where a and c are arbitrary strings of symbols, b is a symbol (or, equivalently, a string of symbols of length 1), and d is the string of symbols obtained from a by replacing all occurrences of the symbol b with c. a.5. x y z((x y) z) = (x (y z)) a.6. x y(x y = 0 x = 0 y = 0) a.7. x y(x y = x y = 0) x y(y x = x y = 0) a.8. v 3 v 4 y z(sub(v 3, y, z) sub(v 4, y, z)) = sub(v 3 v 4, y, z) a.9. a = b, if a and b are distinct expressions.
A theory of expressions A standard model for the language L without interprets the syntactic vocabulary in the intended way and assigns some interpretations to the other vocabulary, in particular to p. The domain of a standard model is always the set of all L-expressions (finite strings of symbols). Let S be a set of expressions. A sentence φ of L holds in E, S formally E, S φ if and only if φ holds when all nonlogical symbols of L except are interpreted according to the model E and is interpreted by S. S is called the extension of in the model E, S. We have: E, S e iff e S
A theory of expressions A standard model for the language L without interprets the syntactic vocabulary in the intended way and assigns some interpretations to the other vocabulary, in particular to p. The domain of a standard model is always the set of all L-expressions (finite strings of symbols). Let S be a set of expressions. A sentence φ of L holds in E, S formally E, S φ if and only if φ holds when all nonlogical symbols of L except are interpreted according to the model E and is interpreted by S. S is called the extension of in the model E, S. We have: E, S e iff e S
A theory of expressions A standard model for the language L without interprets the syntactic vocabulary in the intended way and assigns some interpretations to the other vocabulary, in particular to p. The domain of a standard model is always the set of all L-expressions (finite strings of symbols). Let S be a set of expressions. A sentence φ of L holds in E, S formally E, S φ if and only if φ holds when all nonlogical symbols of L except are interpreted according to the model E and is interpreted by S. S is called the extension of in the model E, S. We have: E, S e iff e S
A theory of expressions Definition dia(x) = sub(x, x, q(x))
A theory of expressions Lemma Assume φ(x) is a formula not containing bound occurrences of x. Then the following holds: A dia(φ(dia(x))) = φ(dia(φ(dia(x)))) dia(φ(dia(x))) = sub(φ(dia(x)), x, q(φ(dia(x)))) = sub(φ(dia(x)), x, φ(dia(x))) = φ(dia(φ(dia(x))))
A theory of expressions Lemma (strong diagonalization) If φ(x) is a formula of L with no bound occurrences of x, then one can find a term t such that the following holds: A t = φ(t) and therefore A φ(t) φ(φ(t))
Possible worlds semantics Definition A frame is an ordered pair W, R where W is nonempty and R is a binary relation on W.
Possible worlds semantics Definition A valuation V for a frame W, R is a function that assigns to every w W a standard model E. Definition A pw-model is a quadruple W, R, V, B such that W, R is a frame, V is a valuation for W, R and B is a -interpretation for W, R satisfying the following condition: B(w) = {φ L for all u W (if wru, then V(u), B(u) φ)} V(u), B(u) is always a standard model and V(u), B(u) φ means that φ is true in the standard model V(u), B(u) in the usual sense of first-order predicate logic.
Possible worlds semantics Definition A valuation V for a frame W, R is a function that assigns to every w W a standard model E. Definition A pw-model is a quadruple W, R, V, B such that W, R is a frame, V is a valuation for W, R and B is a -interpretation for W, R satisfying the following condition: B(w) = {φ L for all u W (if wru, then V(u), B(u) φ)} V(u), B(u) is always a standard model and V(u), B(u) φ means that φ is true in the standard model V(u), B(u) in the usual sense of first-order predicate logic.
Possible worlds semantics Definition A frame W, R admits a pw-model on every valuation iff for every valuation V on W, R there is a such that B such that W, R, V, B is a pw-model. Definition A frame admits a pw-model iff the frame admits a pw-model on some valuation, that is, iff there is a valuation V such that W, R, V, B is a pw-model model. Strong Characterization problem Which frames admit a pw-model on every valuation?
Possible worlds semantics Definition A frame W, R admits a pw-model on every valuation iff for every valuation V on W, R there is a such that B such that W, R, V, B is a pw-model. Definition A frame admits a pw-model iff the frame admits a pw-model on some valuation, that is, iff there is a valuation V such that W, R, V, B is a pw-model model. Strong Characterization problem Which frames admit a pw-model on every valuation?
Possible worlds semantics Definition A frame W, R admits a pw-model on every valuation iff for every valuation V on W, R there is a such that B such that W, R, V, B is a pw-model. Definition A frame admits a pw-model iff the frame admits a pw-model on some valuation, that is, iff there is a valuation V such that W, R, V, B is a pw-model model. Strong Characterization problem Which frames admit a pw-model on every valuation?
Possible worlds semantics Lemma (normality) Suppose W, R, V, B is a pw-model, w W and φ, ψ sentences of L. Then the following holds: 1. If V(u), B(u) φ for all u W, then V(w), B(w) φ. 2. V(w), B(w) φ ψ ( φ ψ)
Possible worlds semantics Lemma 1. If a frame W, R is transitive and W, R, V, B a pw-model on that frame, we have for all sentences φ in L and worlds w W: V(w), B(w) φ φ 2. If a frame W, R is reflexive and W, R, V, B a pw-model on that frame, we have for all sentences φ in L and worlds w W: V(w), B(w) φ φ
The paradoxes w Theorem (liar paradox) The frame W 1, R 1 does not admit a pw-model.
The paradoxes Example (Montague) If W, R admits a pw-model, then W, R is not reflexive.
The paradoxes Example The frame two worlds see each another displayed above does not admit a pw-model.
The paradoxes The following frame does not admit a pw-model.
The paradoxes Example The frame one world sees itself and one other world does not admit a pw-model. We call the frame W 3, R 3. w 1 w 2
The paradoxes Example The frame ω, Pre does not admit a pw-model. Here ω is the set of all natural numbers and Pre is the successor relation. Hence every world n sees n + 1 but no other world. The frame ω, Pre can be displayed by the following diagram: 0 1 2
The paradoxes Theorem The frame ω, < does not admit a pw-model. Here < is the usual smaller than relation on the natural numbers: The frame ω, < can be displayed by the following diagram: 0 1 2
Frames with possible worlds models Example The frame {w}, Ø admits a pw-model on every valuation.
Frames with possible worlds models By Suc we denote the successor relation { k, n k = n + 1} on the set ω of natural numbers. Example The frame ω, Suc admits a pw-model on every valuation. 2 1 0
Frames with possible worlds models Definition A frame W, R is converse wellfounded (or Noetherian) iff for every non-empty M W there is a w M that is R-maximal in M. Lemma Every converse wellfounded frame W, R admits a pw-model on every valuation.
Frames with possible worlds models Definition A frame W, R is converse wellfounded (or Noetherian) iff for every non-empty M W there is a w M that is R-maximal in M. Lemma Every converse wellfounded frame W, R admits a pw-model on every valuation.
The Strong Characterization Problem Theorem (Strong Characterization theorem) A frame admits a pw-model on every valuation iff it is converse wellfounded.
The Strong Characterization Problem We define x as n..... x. n can be defined properly using the diagonal lemma. corresponds ot the transitive closure of R. Lemma For all sentences φ and ψ und pw-models W, R, V, B the following holds: 1. If V(w), B(w) φ for all w W, then V(w), B(w) φ. 2. V(w), B(w) φ ψ ( φ ψ) 3. V(w), B(w) φ φ 4. V(w), B(w) φ φ φ
The Strong Characterization Problem Lemma The transitive closure R of the accessibility relation R of any pw-model that admits a pw-model on every valuation is converse wellfounded. Lemma A frame W, R is converse wellfounded iff its transitive closure W, R is converse wellfounded. This concludes the proof of the Strong Characterization theorem.
All worlds share the same domain, the set of all expressions. The definition of a standard model can be generalized, so that a standard model also specifies a set of contingent objects. Since some of them may lack names, the unary predicate is replaced with a binary predicate applying to formulae and sequences of objects (variable assignments). This requires a theory sequences of arbitrary objects (symbols or contingent objects). Extensions
Löb s theorem is the mother of all paradoxes as long as the modality is normal. Various predicate modalities aren t normal in this sense, for instance, KF. Conclusion
Thank you. Conclusion
Now for something completely different... Conclusion
Classical Determinate Truth The theory CD is formulated in the language of arithmetic augmented with the new predicate symbols D and T. The axioms below are added to the axioms of PA with the schema of induction is expanded to the language with D and T.
Classical Determinate Truth Determinateness axioms (d1) s t D(s=. t) (d2) x ( Sent(x) (D(. x) Dx)) (d3) x y ( Sent(x. y) (D(x. y) Dx Dy)) (d4) v x ( Sent(. vx) (D(. vx) t Dx[t/v])) (d5) s(dṭs Ds Sent(s ))) Truth axioms (t1) s t(ts=. t s = t ) (t2) s( Sent(s ) Ds (TṬs Ts )) (t3) x ( Sent(x) (T(. x) Tx)) (t4) x y ( Sent(x. y) (T(x. y) Tx Ty)) (t5) v x ( Sent(. vx) (T(. vx) t Tx[t/v]))
Classical Determinate Truth Optional axioms (o1) s(dḍs Ds Sent(s ))) (o1) Can we add s( Sent(s ) Ds TḌs Ds ))