Falsity, measures, and repelling assignments Giovanni Panti Department of Mathematics and Computer Science University of Udine
Falsum-free product logic Consider the propositional language · , → , 1 and the set of axioms: commutative monoid identities for · , 1 , x → x = 1 , z → ( y → x ) = ( z · y ) → x, x · ( x → y ) = y · ( y → x ) , y → ( x · y ) = x.
Falsum-free product logic Consider the propositional language · , → , 1 and the set of axioms: commutative monoid identities for · , 1 , x → x = 1 , z → ( y → x ) = ( z · y ) → x, x · ( x → y ) = y · ( y → x ) , y → ( x · y ) = x. They define a rather basic propositional logic, which has as canonical algebraic semantics the half-open real unit interval (0 , 1] with conjunction · interpreted as ordinary product and → as the corresponding residuum (explicitely, x → y = min(1 , y/x )).
Truncated subtraction It is remarkable that the above axioms capture precisely the equational theory of truncated subtraction in the nonnegative reals.
Truncated subtraction It is remarkable that the above axioms capture precisely the equational theory of truncated subtraction in the nonnegative reals. Indeed, fixing and arbitrary real number 0 < c < 1, the exponential function in base c provides an order-reversing isomorphism ([0 , + ∞ ) , + , · − , 0) → ((0 , 1] , · , → , 1) ,
Truncated subtraction It is remarkable that the above axioms capture precisely the equational theory of truncated subtraction in the nonnegative reals. Indeed, fixing and arbitrary real number 0 < c < 1, the exponential function in base c provides an order-reversing isomorphism ([0 , + ∞ ) , + , · − , 0) → ((0 , 1] , · , → , 1) , and it is a —not really trivial— fact that an identity is true in ( R ≥ 0 , + , · − , 0) iff it can be deduced from the axioms, which now read commutative monoid identities for + , 0 , x · − x = 0 , ( x · − y ) · − z = x · − ( y + z ) , x + ( y · − x ) = y + ( x · − y ) , ( x + y ) · − y = x.
Cancellative hoops Let us stick to the + , · − , 0 language. The models of the axioms are called [cancellative] hoops . All hoops we consider will be separating subhoops of C ( X ) ≥ 0 , for X some compact Hausdorff space; they are just positive cones of lattice-ordered abelian groups.
Cancellative hoops Let us stick to the + , · − , 0 language. The models of the axioms are called [cancellative] hoops . All hoops we consider will be separating subhoops of C ( X ) ≥ 0 , for X some compact Hausdorff space; they are just positive cones of lattice-ordered abelian groups. (P., LNAI 2007). The free n -generated hoop is the set of all functions from [0 , 1] n − 1 to R ≥ 0 which are generated by x 1 , . . . , x n − 1 ,1 l − ( x 1 ∨ · · · ∨ x n − 1 ) under pointwise operations. These functions are continuous and piecewise-linear with integer coefficients, and all such functions are thus obtainable.
Measuring things The prototypical cancellative hoop is R ≥ 0 . Nonnegative real numbers are excellent for measuring
Measuring things The prototypical cancellative hoop is R ≥ 0 . Nonnegative real numbers are excellent for measuring but
Measuring things The prototypical cancellative hoop is R ≥ 0 . Nonnegative real numbers are excellent for measuring but we cannot measure until we fix a gauge scale.
Measuring things The prototypical cancellative hoop is R ≥ 0 . Nonnegative real numbers are excellent for measuring but we cannot measure until we fix a gauge scale. We usually fix 1, but in the real numbers the choice is irrelevant: since the automorphism group of R ≥ 0 acts transitively on positive reals, all gauge scales are created equal.
Belying our politician Gauge scales must have the archimedean property. Therefore, we define a unit u in the cancellative hoop H to be an archimedean element: ∀ h ∈ H ∃ n ≥ 1 h ≤ nu.
Belying our politician Gauge scales must have the archimedean property. Therefore, we define a unit u in the cancellative hoop H to be an archimedean element: ∀ h ∈ H ∃ n ≥ 1 h ≤ nu. For the class of hoops we are considering (the subhoops of some C ( X ) ≥ 0 ), the units are precisely the functions which are never zero.
Belying our politician Gauge scales must have the archimedean property. Therefore, we define a unit u in the cancellative hoop H to be an archimedean element: ∀ h ∈ H ∃ n ≥ 1 h ≤ nu. For the class of hoops we are considering (the subhoops of some C ( X ) ≥ 0 ), the units are precisely the functions which are never zero. Upon defining x ⊕ y = ( x + y ) ∧ u and ¬ x = u · − x , the interval [0 , u ] = { x ∈ H : x ≤ u } becomes an MV-algebra (the algebraic conterparts of � Lukasiewicz many-valued logic), with u denoting “absolute falsity”.
Belying our politician Gauge scales must have the archimedean property. Therefore, we define a unit u in the cancellative hoop H to be an archimedean element: ∀ h ∈ H ∃ n ≥ 1 h ≤ nu. For the class of hoops we are considering (the subhoops of some C ( X ) ≥ 0 ), the units are precisely the functions which are never zero. Upon defining x ⊕ y = ( x + y ) ∧ u and ¬ x = u · − x , the interval [0 , u ] = { x ∈ H : x ≤ u } becomes an MV-algebra (the algebraic conterparts of � Lukasiewicz many-valued logic), with u denoting “absolute falsity”. Note that in free hoops not all units are created equal: the automorphism group of Free n CH does not act transitively. Different units will usually give rise to nonisomorphic MV-algebras.
Average truth-values Fix your favorite finitely presented cancellative hoop H , and your favorite unit u in it.
Average truth-values Fix your favorite finitely presented cancellative hoop H , and your favorite unit u in it. I’m interested in the following question:
Average truth-values Fix your favorite finitely presented cancellative hoop H , and your favorite unit u in it. I’m interested in the following question: • can we assign an “average truth value” to all elements of H in a consistent way?
Average truth-values Fix your favorite finitely presented cancellative hoop H , and your favorite unit u in it. I’m interested in the following question: • can we assign an “average truth value” to all elements of H in a consistent way? More formally, is there a monoid homomorphism (which usually is not a hoop homomorphism) s : H → R ≥ 0 that maps u to 1 and is automorphism-invariant (i.e., s ( σh ) = s ( h ) for every h and every automorphism σ of H that fixes u )?
Average truth-values Fix your favorite finitely presented cancellative hoop H , and your favorite unit u in it. I’m interested in the following question: • can we assign an “average truth value” to all elements of H in a consistent way? More formally, is there a monoid homomorphism (which usually is not a hoop homomorphism) s : H → R ≥ 0 that maps u to 1 and is automorphism-invariant (i.e., s ( σh ) = s ( h ) for every h and every automorphism σ of H that fixes u )? s ( h ) must be thought of as the average truth-value of h . The set of all such s ’s (usually called the state space ) is a compact convex subset of R H ≥ 0 , whose boundary is precisely the set of hoop homomorphism.
Basic stuff • Kroupa, P. (independently, 2005). States correspond 1-1 to finite measures on the dual space of H (which is the —unique up to homeo— space X such that H is embeddable in C ( X ) ≥ 0 ).
Basic stuff • Kroupa, P. (independently, 2005). States correspond 1-1 to finite measures on the dual space of H (which is the —unique up to homeo— space X such that H is embeddable in C ( X ) ≥ 0 ). • P. (2007) The duals of automorphisms of Free n CH are precisely the piecewise-fractional GL n Z -homeomorphisms of [0 , 1] n − 1 , p 4 p 3 q 4 q 3 σ ∗ p 6 p 5 q 6 q 5 p 1 p 2 q 1 q 2 and a state is σ -invariant iff the corresponding measure is σ ∗ -invariant.
Basic stuff • Kroupa, P. (independently, 2005). States correspond 1-1 to finite measures on the dual space of H (which is the —unique up to homeo— space X such that H is embeddable in C ( X ) ≥ 0 ). • P. (2007) The duals of automorphisms of Free n CH are precisely the piecewise-fractional GL n Z -homeomorphisms of [0 , 1] n − 1 , p 4 p 3 q 4 q 3 σ ∗ p 6 p 5 q 6 q 5 p 1 p 2 q 1 q 2 and a state is σ -invariant iff the corresponding measure is σ ∗ -invariant. • Mundici (1995) The Lebesgue measure on [0 , 1] n − 1 is invariant under all the [duals of the] elements in Stab(1 l) < Aut Free n CH .
Actually, the latter is a characterization P. (Comm. in Alg. 2008), Marra (J. Group Th. 2009). The Lebesgue measure on [0 , 1] n − 1 is the only nonatomic measure that is invariant under Stab(1 l).
Actually, the latter is a characterization P. (Comm. in Alg. 2008), Marra (J. Group Th. 2009). The Lebesgue measure on [0 , 1] n − 1 is the only nonatomic measure that is invariant under Stab(1 l). This really means that once you fixed the constant 1 l as your gauge scale for falsity, then your policician’s statements have precisely one nonambiguous average truth-value: their integrals w.r.t. Lebesgue measure.
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