159
quantity of an object value of a quantity
⊕
as well, as follows:
• An individual quantity q ref is singled out as a quantity unit (e.g., the metre); q ref
may be defined as the quantity Q[r] of an object r or, being defined in some other
way, may be realized by some object r; in either case r is a measurement standard, and possibly in particular the/a primary standard.
• The individual quantities x q ref (e.g., 2 m) are values of quantities, being by construction the multiples of q ref obtained by means of the concatenation of the chosen unit.
• Working standards r′ can be calibrated against the primary standard r, Q[r′] ≈ Q[r]
(ignoring calibration uncertainty), so that the quantity Q[a] of an object a can be
compared with Q[r′]; hence the inference that from q ref = Q[r], Q[r′] ≈ Q[r], and
Q[a] ≈ x Q[r′] leads by transitivity
19
to Q[a] ≈ x q ref is the simplest case of a
metrological traceability chain (JCGM, 2012: 2.42).
• The relation Q[a] ≈ x q ref for a given x (e.g., L[a] ≈ 2 m) is a Basic Evaluation
Equation, and thanks to this traceability it may be a measurement result for the
measurand Q[a] (ignoring measurement uncertainty).
• Hence the relations
the
is indistinguishable from a multiple
quantity of an object
of the quantity of
another object
(e.g., L[a] ≈ 2 L[r]) and
the
isa
quantity of an object
value of a quantity
(e.g., L[a] ≈ 2 ℓ ref (or L[a] ≈ 2 m))
refer to the same empirical situation, the difference being in the way the two relations convey the information about the individual quantities involved.
The conclusion is then obvious: a value of a quantity is an individual quantity identified as a multiple of a given reference quantity, designated as the unit.
20
The analysis in Sect. 5.3.2, which led us to interpret a relation such as Q[a i ] ≈ Q[a j ]
as including expressions with different senses but possibly the same individual
length as their referent, can be now straightforwardly extended to scale transformations and Basic Evaluation Equations:
• In the scale transformation q ref = k q ref* the expressions “q ref ” and “k q ref* ” have
different senses but the same individual length as referent.
21
19 In Sect. 5.2.6 we pointed out that indistinguishability is generally not transitive: how traceability
chains can be constructed in spite of this obstacle is discussed by Mari and Sartori (2007).
20 As explained in Footnote 12 of Chap. 5, we use the concept in a broad
sense, admitting also non-integer multiples.
21 As a consequence, we can provide a simple answer to a question such as whether, e.g., 1.2345
metres and 48.602 inches are the same value or not: “1.2345 m” and “48.602 in” have the same
6.3 Constructing values of quantities
quantity of an object value of a quantity
⊕
as well, as follows:
• An individual quantity q ref is singled out as a quantity unit (e.g., the metre); q ref
may be defined as the quantity Q[r] of an object r or, being defined in some other
way, may be realized by some object r; in either case r is a measurement standard, and possibly in particular the/a primary standard.
• The individual quantities x q ref (e.g., 2 m) are values of quantities, being by construction the multiples of q ref obtained by means of the concatenation of the chosen unit.
• Working standards r′ can be calibrated against the primary standard r, Q[r′] ≈ Q[r]
(ignoring calibration uncertainty), so that the quantity Q[a] of an object a can be
compared with Q[r′]; hence the inference that from q ref = Q[r], Q[r′] ≈ Q[r], and
Q[a] ≈ x Q[r′] leads by transitivity
19
to Q[a] ≈ x q ref is the simplest case of a
metrological traceability chain (JCGM, 2012: 2.42).
• The relation Q[a] ≈ x q ref for a given x (e.g., L[a] ≈ 2 m) is a Basic Evaluation
Equation, and thanks to this traceability it may be a measurement result for the
measurand Q[a] (ignoring measurement uncertainty).
• Hence the relations
the
is indistinguishable from a multiple
quantity of an object
of the quantity of
another object
(e.g., L[a] ≈ 2 L[r]) and
the
isa
quantity of an object
value of a quantity
(e.g., L[a] ≈ 2 ℓ ref (or L[a] ≈ 2 m))
refer to the same empirical situation, the difference being in the way the two relations convey the information about the individual quantities involved.
The conclusion is then obvious: a value of a quantity is an individual quantity identified as a multiple of a given reference quantity, designated as the unit.
20
The analysis in Sect. 5.3.2, which led us to interpret a relation such as Q[a i ] ≈ Q[a j ]
as including expressions with different senses but possibly the same individual
length as their referent, can be now straightforwardly extended to scale transformations and Basic Evaluation Equations:
• In the scale transformation q ref = k q ref* the expressions “q ref ” and “k q ref* ” have
different senses but the same individual length as referent.
21
19 In Sect. 5.2.6 we pointed out that indistinguishability is generally not transitive: how traceability
chains can be constructed in spite of this obstacle is discussed by Mari and Sartori (2007).
20 As explained in Footnote 12 of Chap. 5, we use the concept
sense, admitting also non-integer multiples.
21 As a consequence, we can provide a simple answer to a question such as whether, e.g., 1.2345
metres and 48.602 inches are the same value or not: “1.2345 m” and “48.602 in” have the same
6.3 Constructing values of quantities