133
prevent its ideal realization. Hence, even when two properties are found to be indistinguishable, a more specific comparison might reveal that they are not equal, but
only similar.
Like any generic similarity, indistinguishability is
• reflexive (any property is indistinguishable from itself: P[a] ≈ P[a]) and
• symmetric (if two properties are indistinguishable, then the order in which they
are considered is immaterial: P[a i ] ≈ P[a j ] if and only if P[a j ] ≈ P[a i ]), but
• not transitive (given three properties, from the facts that the first and the second
are indistinguishable and that the second and the third are indistinguishable, the
conclusion that also the first and the third are indistinguishable does not follow:
P[a i ] ≈ P[a j ] and P[a j ] ≈ P[a k ] do not imply that P[a i ] ≈ P[a k ]).
The non-transitivity of indistinguishability has at least one critical consequence:
properties of objects could not be consistently represented, or even named, in any
sufficiently simple form. Indeed, the observation that P[a i ] and P[a j ] are indistinguishable would lead one to represent them with the same symbol, and the observation that P[a j ] and P[a k ] are also indistinguishable would lead one to represent P[a k ]
with the same symbol as P[a j ], and therefore as P[a i ]; but since P[a i ] and P[a k ] could
be instead distinguishable, this would lead to the situation in which distinguishable
properties are represented by the same symbol, a case of homonymy and therefore
information loss in representation. Furthermore, since the comparison can be iterated, there might be a sequence of objects a 1 , a 2 , …, a n such that P[a 1 ] ≈ P[a 2 ] and
P[a 2 ] ≈ P[a 3 ] and … P[a n–1 ] ≈ P[a n ] even though, for each i, P[a i ] ≉ P[a i+2 ], with the
consequence that all comparable but mostly distinguishable properties of objects
are represented by the same symbol, and therefore that the representation conveys
no information at all.
31
This issue does not seem to have a general solution better than provisionally
assuming the transitivity of indistinguishability, and therefore modeling the comparison of properties of objects as an equivalence relation, which could be then
discovered to be not such by means of further and more refined comparisons. As we
discuss below, providing information on properties of objects in terms of traceable
values is the specific solution to this problem adopted in measurement science (see
also Mari & Sartori, 2007).
31 This is the paradox known as sorites, a term which derives from the Greek word “soros”, meaning (see Hyde & Raffman, 2018). The classical way to present it is in terms of logical
properties, for example as follows. Let a n be a set of n grains of wheat. Of course a 0 is not a heap,
i.e., is_heap(a 0 ) = false. Moreover, a n and a n+1 are indistinguishable in their being heaps, in the
sense that if a set is not a heap, adding a grain to it does not make it a heap, i.e., if is_heap(a n ) = false
then is_heap(a n+1 ) = false. Then starting from the first clause and by the repeated application of the
second clause the conclusion is reached that is_heap(a n ) = false no matter how large n is.
5.2 Some clarifications about properties
prevent its ideal realization. Hence, even when two properties are found to be indistinguishable, a more specific comparison might reveal that they are not equal, but
only similar.
Like any generic similarity, indistinguishability is
• reflexive (any property is indistinguishable from itself: P[a] ≈ P[a]) and
• symmetric (if two properties are indistinguishable, then the order in which they
are considered is immaterial: P[a i ] ≈ P[a j ] if and only if P[a j ] ≈ P[a i ]), but
• not transitive (given three properties, from the facts that the first and the second
are indistinguishable and that the second and the third are indistinguishable, the
conclusion that also the first and the third are indistinguishable does not follow:
P[a i ] ≈ P[a j ] and P[a j ] ≈ P[a k ] do not imply that P[a i ] ≈ P[a k ]).
The non-transitivity of indistinguishability has at least one critical consequence:
properties of objects could not be consistently represented, or even named, in any
sufficiently simple form. Indeed, the observation that P[a i ] and P[a j ] are indistinguishable would lead one to represent them with the same symbol, and the observation that P[a j ] and P[a k ] are also indistinguishable would lead one to represent P[a k ]
with the same symbol as P[a j ], and therefore as P[a i ]; but since P[a i ] and P[a k ] could
be instead distinguishable, this would lead to the situation in which distinguishable
properties are represented by the same symbol, a case of homonymy and therefore
information loss in representation. Furthermore, since the comparison can be iterated, there might be a sequence of objects a 1 , a 2 , …, a n such that P[a 1 ] ≈ P[a 2 ] and
P[a 2 ] ≈ P[a 3 ] and … P[a n–1 ] ≈ P[a n ] even though, for each i, P[a i ] ≉ P[a i+2 ], with the
consequence that all comparable but mostly distinguishable properties of objects
are represented by the same symbol, and therefore that the representation conveys
no information at all.
31
This issue does not seem to have a general solution better than provisionally
assuming the transitivity of indistinguishability, and therefore modeling the comparison of properties of objects as an equivalence relation, which could be then
discovered to be not such by means of further and more refined comparisons. As we
discuss below, providing information on properties of objects in terms of traceable
values is the specific solution to this problem adopted in measurement science (see
also Mari & Sartori, 2007).
31 This is the paradox known as sorites, a term which derives from the Greek word “soros”, meaning
properties, for example as follows. Let a n be a set of n grains of wheat. Of course a 0 is not a heap,
i.e., is_heap(a 0 ) = false. Moreover, a n and a n+1 are indistinguishable in their being heaps, in the
sense that if a set is not a heap, adding a grain to it does not make it a heap, i.e., if is_heap(a n ) = false
then is_heap(a n+1 ) = false. Then starting from the first clause and by the repeated application of the
second clause the conclusion is reached that is_heap(a n ) = false no matter how large n is.
5.2 Some clarifications about properties