Four-Valued First-Order Semantics for RW Shay Logan North Carolina - PowerPoint PPT Presentation

Four-Valued First-Order Semantics for RW Shay Logan North Carolina State University Department of Philosophy These slides: https://tinyurl.com/ShaysMelbourneTalk October 19, 2017

The plan: I’m going to spell out a semantic theory for you. I’m going to tell you why you should like it. Then I’m going to go to bed. Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 2 / 46

More detail on the first two points: Step 1: I’m going to remind you who RWQ is. Step 2: I’ll then give you a semantic theory for the zero-order fragment of RWQ. Step 3: Then I’ll give a varying-domain stratified semantic theory for full RWQ. Step 4: I’ll raise the standard objections to the stratified semantics. Step 5: I’ll respond to the objections, along the way tossing out a constant-domain stratified semantic theory. Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 3 / 46

Note: At no point will we go through soundness or completeness proofs. They’re just too messy for this sort of talk. But send me an email if you want to read the results! When the paper is in a readable state I’ll pass it on. Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 4 / 46

The Languages Zero-Order Polyadic sentential language. Predicates (of any arity) Names Connectives: ∧ , ¬ , → . α ∨ β = def ¬ ( ¬ α ∧ ¬ β ). First-Order Add ∀ and variables. ∃ x = def ¬∀ x ¬ Define wff and sentence in the usual ways. Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 5 / 46

The Logic (part I) A1 α → α A2 ( α ∧ β ) → α A3 ( α ∧ β ) → β A4 (( α → β ) ∧ ( α → γ )) → ( α → ( β ∧ γ )) A5 ( α ∧ ( β ∨ γ )) → (( α ∧ β ) ∨ ( α ∧ γ )) A6 ¬¬ α → α A7 ( α → ¬ β ) → ( β → ¬ α ) A8 ( α → β ) → (( β → γ ) → ( α → γ )) A9 α → (( α → β ) → β ) Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 6 / 46

The Logic (part II) R1 α, α → β β α, β R2 α ∧ β Zero-order RW is the logic generated by A1-A9, R1-R2, restricted to the zero-order language. Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 7 / 46

The Logic (part III) A10 ∀ νφ → φ ( τ/ν ) ( τ free for ν in φ ). A11 ∀ ν ( φ → ψ ) → ( φ → ∀ νψ ) ( ν not free in φ ). A12 ∀ ν ( φ ∨ ψ ) → ( φ ∨ ∀ νψ ) ( ν not free in φ ). φ R3 ∀ νφ Logic generated by all A1-A12 and R1-R3 is RWQ. (This is Ross Brady’s axiomatization) Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 8 / 46

Zero-Order Semantics (part I) A zero-order premodel is a 7-tuple � D , S , N , R , δ, E + , E − � where D is a set (the domain ). S is a set ( setups ). N ⊆ S ( normal setups). R ⊆ S 3 (compatibility). δ is a denotation function . E + ( P , a ) is the extension of P at a . E − ( P , a ) is the antiextension of P at a . We say a premodel is reduced when N = { g } . Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 9 / 46

Defintions Rabcd = def for some x Rabx and Rxcd Ra ( bc ) d = def for some x , Rbcx and Raxd . x ≤ y = def for some n ∈ N , Rnxy . Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 10 / 46

Zero-Order Semantics (part II) A zero-order model is a zero-order premodel such that Ordering For all s , t , and u in S s ≤ s . If s ≤ t and t ≤ u , then s ≤ u . If s ≤ t and t ≤ s , then s = t . Monotonicity For all a , b , c , and x in S , If a ≤ x and Rxbc , then Rabc . If b ≤ x and Raxc , then Rabc . If x ≤ c and Rabx , then Rabc . Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 11 / 46

Closure If n ∈ N and n ≤ m , then m ∈ N . Rearranging If Rabcd , then B : Ra ( bc ) d B’: Rb ( ac ) d C : Racbd Horizontal Atomic Heredity If P is an i -ary predicate and a ≤ b , then If � d 1 , . . . , d i � ∈ E ± ( P , a ), then � d 1 , . . . , d i � ∈ E ± ( P , b ). Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 12 / 46

Truth values are in {∅ , { 1 } , { 0 } , { 1 , 0 }} : 1 ∈ M a ( P α 1 . . . α n ) iff � δ ( α 1 ) , . . . , δ ( α n ) � ∈ E + ( P , a ) 0 ∈ M a ( P α 1 . . . α n ) iff � δ ( α 1 ) , . . . , δ ( α n ) � ∈ E − ( P , a ) 1 ∈ M a ( φ ∧ ψ ) iff 1 ∈ M a ( φ ) and 1 ∈ M a ( ψ ). 0 ∈ M a ( φ ∧ ψ ) iff 0 ∈ M a ( φ ) or 0 ∈ M a ( ψ ). 1 ∈ M a ( ¬ φ ) iff 0 ∈ M a ( φ ) 0 ∈ M a ( ¬ φ ) iff 1 ∈ M a ( φ ) 1 ∈ M a ( φ → ψ ) iff for all b and c , if Rabc then If 1 ∈ M b ( φ ), then 1 ∈ M c ( ψ ), and If 0 ∈ M b ( ψ ), then 0 ∈ M c ( φ ). 0 ∈ M a ( φ → ψ ) iff for some b and c with Rbca , 1 ∈ M b ( φ ) and 0 ∈ M c ( ψ ). Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 13 / 46

φ is zero-order valid when for all models M , if n is normal, then 1 ∈ M n ( φ ) Theorem If φ is a zero-order theorem, then φ is zero-order valid. Theorem If φ is zero-order valid, then φ is a zero-order theorem. Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 14 / 46

Stratified Models: Generatlities Some history: In two papers in the late 1980s, Kit Fine showed (a) Incompleteness wrt the na¨ ıve first-order models. (b) Completeness wrt stratified first-order models. Impetus for stratified models comes from Fine’s prior work on arbitrary objects. Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 15 / 46

Stratified Models: Generalities Here’s a recipe for building a stratified model: Stack up a family of zero-order models. 1 Single out an increasing sequence of sets of 2 ‘arbitrary’ objects. Ensure the arbitrary objects behave like arbitrary 3 objects. Use the arbitrary objects to define truth for 4 quantified sentences. Call the resulting stack of zero-order models a 5 first-order model. Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 16 / 46

“. . . a universal sentence ∀ x ψ ( x ) is true just in case ψ ( x ) is true of an arbitrary or generic individual. But let me not be misunderstood. My saying that ψ ( x ) is true of an arbitrary individual is not a fancy way of saying that ψ ( x ) is true of every individual. I mean to be taken literally; for the universal sentence ∀ x ψ ( x ) to be true, there must actually be an arbitrary individual of which the condition ψ ( x ) is true” (Fine, 1988) Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 17 / 46

Varying-Domain RWQ Premodels In our case, here’s what this looks like: Varying-domain RWQ-premodels are 5-tuples � D , Ω , δ, M , ⇓� D is a set (the base domain) Ω = { ω i } ∞ i =1 is a set that is disjoint from D . (arbitrary objects) δ is a denotation function. M is a function mapping each finite set X of natural numbers to a zero-order-model M X . ⇓ is a family of restriction functions Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 18 / 46

X , E − Requirements: if M X = � D X , S X , N X , R X , δ, E + X � , then D X = D ∪ { ω i } i ∈ X . If X ⊇ Y , there is a function ↓ X Y : S X → S Y in ⇓ . (Write ↓ X Y with postfix notation) For a ∈ S X , a ↓ X Y ↓ Y Z = a ↓ X Z ↓ X X = id S X Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 19 / 46

Terminology: Let a ∈ S X and m , n ∈ D X . We say that a is symmetric in m and n exactly when a does not extensionally distinguish m from n ; that is, when � d 1 , . . . , m , . . . , d i � ∈ E ± X ( P , a ) iff � d 1 , . . . , n , . . . , d i � ∈ E ± X ( P , a ) Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 20 / 46

Varying-Domain RWQ Models Varying-domain RWQ-models are varying-domain RWQ-premodels that satisfy the following six conditions: Heredity If a ↓ X Y = b , then E ± Y = E ± X ( P , a ) ∩ D i Y ( P , b ). Normality a ↓ X Y ∈ N Y iff a ∈ N X . Lifting If a ∈ S X , b ∈ S Y , and a ↓ X X ∩ Y = b ↓ Y X ∩ Y then for some c ∈ S X ∪ Y , a = c ↓ X ∪ Y and X b ≤ c ↓ X ∪ Y . Y Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 21 / 46

Homomorphism If a , b , and c are in S X and R X abc , then R Y a ↓ X Y b ↓ X Y c ↓ X Y . Extension If a , b , and c are in S Y and R Y abc , then if d ↓ X Y = a then there are e and f such that e ↓ X Y = b , f ↓ X Y = c and R X def ; and if f ↓ X Y = c then there are d and e such that d ↓ X Y = a , e ↓ X Y = b and R X def . Symmetry If a ∈ S Y , X ⊇ Y , m ∈ D Y and n ∈ D X − D Y , then there is a b that is symmetric in m and n such that b ↓ X Y = a . Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 22 / 46

Variable Assignments If ν is a variable and X is a finite set of numbers then a variable assignment maps the pair � ν, X � to an element of D X . va is X - coherent when for all ν and Y ⊇ X , va( ν, X ) = va( ν, Y ). Notice if va is X -coherent, then va is Y -coherent when Y ⊇ X . Shay Logan Four-Valued First-Order Semantics for RW October 19, 2017 23 / 46