Many-sorted logic Quine’s Conjecture on Many-Sorted Logic Thomas Barrett and Hans Halvorson Dominik Ehrenfels St Cross College March 2, 2019 1 / 41
Many-sorted logic Many-sorted logic - Syntax Non-logical vocabulary: 1. non-empty set of sort symbols σ 1 ,σ 2 ,... 2. variables x ( σ 1 ) , x ( σ 1 ) ,... x ( σ 2 ) , x ( σ 2 ) ,... , indexed with a sort 1 2 1 2 symbol 3. constant symbols c ( σ ) , indexed with a sort symbol 4. predicate symbols P i of arity σ i 1 × ... × σ i n 5. function symbols f j of arity σ j 1 × ... × σ j n → σ j n + 1 Logical vocabulary: 1. connectives: ¬ , ∨ ,... 2. quantifiers ∀ σ x , ∃ σ x for each sort symbol σ 3. equality sign = 2 / 41
Many-sorted logic Semantics Structure M : 1. A non-empty domain σ M for each sort symbol σ . The domains of the sort symbols are pairwise disjoint ( σ M ∩ σ M = ∅ for i ≠ j ) i j 2. For any constant symbol c of sort σ , c M ∈ σ M 3. For any predicate P of arity σ i 1 × ... × σ i n , P M ⊆ σ M i 1 × ... × σ M i n 4. For function symbols analogous M ⊧ ∀ σ x φ ( x ) iff M ⊧ φ [ a ] for all a ∈ σ M Note that there are no quantifiers that range over multiple domains 3 / 41
Many-sorted logic Quine’s conjecture Quine’s conjecture Every many-sorted theory is equivalent to a single-sorted theory What is the precise sense of ’equivalent’ here? A standard notion of equivalence in the context of (single-sorted) FOL is Definitional Equivalence 4 / 41
Many-sorted logic Definitional Equivalence Explicit definitions Let Σ be a signature and let P / ∈ Σ, where P is some predicate symbol. An explicit definition of P in terms of Σ is a sentence ∀ ¯ x ( P ¯ x ↔ δ ( ¯ x )) where δ ( ¯ x ) is a Σ-formula Constant symbols and function symbols can also be explicitly defined in a straightforward manner. 5 / 41
Many-sorted logic Definitional extensions Let T be a theory in signature Σ, and let ψ i , i ∈ I be explicit definitions in terms of Σ of symbols not in Σ. Then T ∪ { ψ i ∣ i ∈ I } is a definitional extension of T . Definitional equivalence Two theories T 1 and T 2 of signature Σ 1 and Σ 2 , respectively, are definitionally equivalent if they possess logically equivalent definitional extensions T + 1 and T + 2 of signature Σ 1 ∪ Σ 2 6 / 41
Many-sorted logic An example Let Σ 1 = { P } and Σ 2 = { Q } be signatures. Let T 1 be a Σ 1 -theory and T 2 a Σ 2 -theory: T 1 = {∀ xPx } T 2 = {∀ x ¬ Qx } Let δ ≡ ∀ x ( Qx ↔ ¬ Px ) Let δ ′ ≡ ∀ x ( Px ↔ ¬ Qx ) T 1 ∪ { δ } and T 2 ∪ { δ ′ } are definitional extensions of T 1 and T 2 , respectively. They are furthermore logically equivalent. T 1 and T 2 are therefore definitionally equivalent 7 / 41
Many-sorted logic Generalising definitional equivalence - defining new sorts So far, we have defined new constant symbols, predicate symbols, and function symbols via definitional extensions. But in the setting of many-sorted logic, we also encounter new sort symbols. How should we define these? New sorts are definable from old sorts via four constructions: We can introduce product sorts, coproduct sorts, subsorts, and quotient sorts 8 / 41
Many-sorted logic Product sort A product sort can be thought of as the Cartesian product of two sorts Example: Let σ M = { a , b } and σ M = { c } 1 2 Then σ M + = {⟨ a , c ⟩ , ⟨ b , c ⟩} P 9 / 41
Many-sorted logic Formally: The product sort σ of sorts σ 1 and σ 2 is defined by ∀ σ 1 x ∀ σ 2 y ∃ = 1 σ z ( π 1 ( z ) = x ∧ π 2 ( z ) = y ) Here, the π i are new function symbols of arity σ → σ i . Think of them as projections. 10 / 41
Many-sorted logic Coproduct sort A coproduct sort can be thought of as the disjoint union of two sorts Example: Let σ M = { a , b } and σ M = { α,β } 1 2 Then σ M + = {⟨ a , 1 ⟩ , ⟨ b , 1 ⟩ , ⟨ α, 2 ⟩ , ⟨ β, 2 ⟩} C 11 / 41
Many-sorted logic Formally: The coproduct sort σ of sorts σ 1 and σ 2 is defined by ∀ σ z (∃ = 1 σ 1 x ( ρ 1 ( x ) = z )∨∃ = 1 σ 2 y ( ρ 2 ( y ) = z ))∧∀ σ 1 x ∀ σ 2 y ( ρ 1 ( x ) ≠ ρ 2 ( y )) Here, the ρ i are new function symbols of arity σ i → σ . Think of them as equipping each element of σ i with an index i 12 / 41
Many-sorted logic Subsort Think of a subsort of σ as a copy of a definable subset of σ M Example: Let σ M = { a , b , c } and let P M = { a , b } . We can then define a subsort σ S of σ that is a copy of P M , i.e. σ M = { a ′ , b ′ } S 13 / 41
Many-sorted logic Formally: A subsort σ of a sort σ 1 is defined by ∀ σ 1 x ( φ ( x ) ↔ ∃ σ z ( h ( z ) = x )) ∧ ∀ σ y ∀ σ z ( h ( y ) = h ( z ) → y = z ) Here, φ ( x ) is an old formula which defines the subset of σ 1 we want to copy. h is a new function symbol of arity σ → σ 1 . Think of h as a bijection between σ and its copy. Note that we cannot allow the domain of σ to be empty. ∃ σ 1 x φ ( x ) must therefore hold. This is called the admissibility condition for the subsort σ 14 / 41
Many-sorted logic Quotient sort The elements of a quotient sort σ Q of σ are the equivalence classes of elements of σ with respect to some equivalence relation φ ( x 1 , x 2 ) on σ Example: Let σ M = { Mark , John , Rachel , Mary } Let φ ( x 1 , x 2 ) describe the equivalence relation ′ x 1 is the same gender as x ′ 2 [ Mark ] φ = { Mark , John } [ Rachel ] φ = { Rachel , Mary } σ M + = {[ Mark ] φ , [ Rachel ] φ } Q 15 / 41
Many-sorted logic Formally: A quotient sort σ of a sort σ 1 is defined by ∀ σ 1 x ∀ σ 1 y ( ǫ ( x ) = ǫ ( y ) ↔ φ ( x , y )) ∧ ∀ σ z ∃ σ 1 x ( ǫ ( x ) = z ) Here, ǫ is a new function symbol of arity σ 1 → σ . ǫ maps every element of σ 1 to its equivalence class. Once again, there is an admissibility condition: φ ( x , y ) must be an equivalence relation, i.e. reflexive, symmetric, transitive 16 / 41
Many-sorted logic Morita equivalence We are now in a position to define our new notion of Generalised Definitional Equivalence, or Morita Equivalence Morita extensions Let Σ ⊂ Σ + be signatures and T a Σ-theory. A Morita extension T + of T is a Σ + -theory T ∪ { δ s ∣ s ∈ Σ + − Σ } For which it holds that 1. δ s is an explicit definition of s 2. If α s is an admissibility condition for s , then T ⊧ α s 17 / 41
Many-sorted logic Morita equivalence Let T 1 be a Σ 1 -theory and T 2 a Σ 2 -theory. T 1 and T 2 are Morita equivalent if there are theories T 1 1 ,... T m 1 and T 1 2 ,... T n 2 such that 1. T i + 1 is a Morita extension of T i i for 0 ≤ i ≤ m − 1 1 2. T i + 1 is a Morita extension of T i 1 for 0 ≤ i ≤ n − 1 2 3. T m 1 and T n 2 are logically equivalent Why are multiple steps upwards needed (unlike for definitional equivalence)? Answer: We can construct new sorts from complex sorts, which in turn are constructed from more basic sorts 18 / 41
Many-sorted logic Example The following two theories are Morita equivalent: T 1 = {∃ = 1 σ 1 x ( x = x )} and T 2 = {∃ = 2 σ 2 y ( y = y )} , ∃ = 1 σ 2 yPy } To show this, we need to define the symbols of Σ 1 = { σ 1 } in terms of the symbols of Σ 2 = { σ 2 , P } , and vice versa. Then we can build a common Morita extension T + of T 1 and T 2 The domain of σ 1 in any model M of T 1 has exactly one element, e.g. σ M = { a } . To construct a domain for σ 2 out of this, we need 1 to turn this one element into two. ⇒ σ 2 must be defined as the coproduct of σ 1 with itself σ M + = {⟨ a , 1 ⟩ , ⟨ a , 2 ⟩} 2 19 / 41
Many-sorted logic We also need to define P ∈ Σ 2 in terms of Σ 1 . For this, we can just define P M + = {⟨ a , 1 ⟩} , i.e. the first element of the σ M + we just 2 constructed Now we need to define σ 1 in terms of Σ 2 . σ M + needs to have 1 exactly one element. We just saw that P M + has exactly one element. So let’s define σ 1 as a copy of P M + , i.e. as a subsort of σ 2 . For instance, σ M + = { a } 1 Now let’s do all of this in the syntax! 20 / 41
Many-sorted logic The signature Σ + of our common Morita extension will be Σ + = { σ 1 ,σ 2 , P ,ρ 1 ,ρ 2 } . ρ 1 and ρ 2 are function symbols of arity σ 1 → σ 2 which we need for the definitions of the product sort and the subsort. Let’s define σ 2 and P : δ σ 2 ≡∀ σ 2 z (∃ = 1 σ 1 x ( ρ 1 ( x ) = z ) ∨ ∃ = 1 σ 1 x ( ρ 2 ( x ) = z )) ∧ ∀ σ 1 x ∀ σ 1 y ( ρ 1 ( x ) ≠ ρ 2 ( y )) δ P ≡∀ σ 2 z ( Pz ↔ ∃ σ 1 x ( z = ρ 1 ( x ))) And let’s define σ 1 : δ σ 1 ≡∀ σ 2 z ( Pz ↔ ∃ σ 1 x ( z = ρ 1 ( x ))) ∧ ∀ σ 1 x ∀ σ 1 y ( ρ 1 ( x ) = ρ 1 ( y ) → x = y ) 21 / 41
Many-sorted logic T 1 ∪ { δ σ 2 ,δ P } is a theory in the target signature Σ + = { σ 1 ,σ 2 , P ,ρ 1 ,ρ 2 } . We have reached our common Morita extension, starting from T 1 But T 2 ∪ { δ σ 1 } is in the signature Σ ′ = { σ 1 ,σ 2 , P ,ρ 1 } . We have not yet reached the common Morita extension from T 2 since we have not yet defined ρ 2 . We need to extend T 2 ∪ { δ σ 1 } once more to reach the common Morita extension. Let’s add δ ρ 2 ≡ ∀ σ 1 x ∀ σ 2 y ( ρ 2 ( x ) = y ↔ ρ 1 ( x ) ≠ y ) 22 / 41
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