Altogether, TQC represents material entity John who lives between 2000 and 2085 as having a unique history, the sum of all processes occurring in the spatiotemporal region John occupies, which, we may say without loss of generality, has phase occurrent parts, such as John2000, John2001, John2002, etc., that are the phases of a Tqc John2000-2085, itself the tqcOf John.
TQC Strategy Applied
Adding these commitments to BFO-OWL permits the following characterizations of varieties of relatedness involving continuant parthood (we include characterizations in BFO-FOL for comparison):
(TQC-2) ∀(x)∃(y)∃(z)∃(w)∃(u)(hasHistory(x,y) & hasOccurrentPart(y,z) & phaseOf(z,w) & hasContinuantPart∀(w,u))
(TQC-3) ∀(x)∃(y)∃(z)∃(w)∃(u)(hasPhase(x,y) & hasOccurrentPart(y,z) & phaseOf(z,w) & continuantPartOf∀(w,u))
Recall, our motivating example for Permanent Specific Relatedness, reflected in BFO-FOL as (FOL-1), is every brain is always continuant part of the same host. Observe, (TQC-1) captures this relationship by appealing only to the binary tensed universal continuant parthood relation. Turning to Permanent Generic Relatedness, reflected in BFO-FOL with the inverse of continuantPartOfAt, named hasContinuantPartOfAt, as (FOL-2), our motivating example was every tissue has some cell as part at all times. Observe, with (TQC-2) this becomes every tissue has a history which has a phase as occurrent part, which is the phase of some temporally qualified continuant that has some continuant cell part at all times. Turning finally to Temporary Relatedness, reflected in BFO-FOL as (FOL-3), our motivating example was organisms having wing parts at some, but not all, portions of their development. Observe with (TQC-3) this becomes every organism has a history with a phase as occurrent part that is the phase of some temporally qualified continuant which is itself continuant part of some continuant wing.
In short, proponents of TQC claim the binary universal tensed relation is adequate for (1), and distinguish (2) and (3) in terms of whether a given temporally qualified continuant has a continuant part or is a continuant part of some relevant continuant. The TQC strategy thus appears to address one aspect of our particular challenge.
Relationship to BFO-FOL
However, our particular challenge requires an adequate characterization of varieties of relatedness in BFO-OWL be translatable into BFO-FOL. With respect to BFO-FOL, the path to (ii) can be understood as a series of relation parametrizations resulting in additions to the domain. Roughly, to parametrize a relation is to define a lower arity relation with satisfaction conditions dependent on the higher arity relation. To illustrate, consider the sentence “John’s hand is part of John on Tuesday” characterized with the BFO-FOL continuantPartOfAt relation:
(I) continuantPartOfAt(John’s Hand, John, Tuesday)
Which, assuming standard first-order semantics, is satisfied (roughly) iff the ordered triple <John’s Hand, John, Tuesday> is a member of a subset of DxDxD, where “D” denotes the domain. This ternary expression can be parametrized by introducing a binary relation with satisfaction conditions tied to the ternary relation. Parametrizing with respect to time (and replacing ‘at’ in the name for readability) we have:
(II) continuantPartOfTuesday(John’s Hand, John)
Satisfied iff the ordered pair <John’s Hand, John> is a member of a subset of DxD. We might continue parametrizing, this time with respect to John, resulting in:
(III) continuantPartOfTuesdayJohn(John’s Hand)
Satisfied iff <John’s Hand> is a member of a subset of D. Finally, we might complete the parametrization by introducing an individual:
Satisfied iff the individual is a member of D. Parametrization resulting in an individual is called total. Otherwise, the parametrization is called partial. (II) and (III) are thus partial parametrizations, and (IV) a total parametrization.
Our choice of example was only illustrative, of course, since rather than parametrizing ternary BFO-FOL relations, TQC replaces them. Nevertheless, temporally qualified continuants may be understood as the result of parametrizing the binary existsAt relation of BFO-FOL, with the domain understood as restricted to continuants (rather than applying to all entities), and the range restricted to temporal regions (and so unchanged). For example, in BFO-FOL the sentence “John exists during 2000 and 2085” might be characterized as (where “2000-2085” names a temporal region):
Satisfied iff <John,2000-2085> is a member of CxT, where C and T are, respectively, continuant and temporal sorts of D. Total parametrization results in the individual:
Satisfied iff the individual is a member of D. If we assume the existsAt relation is implicit for any individual, then we can, without loss, drop this portion of the name, resulting in the familiar:
More generally we might represent individuals such as “John2000-2085” with the variable notation reflecting continuants existing at temporal regions as: xt, yt, zt, with “t” subscripts reflecting temporal indices. These observations suggest some reason to think individuals postulated by the TQC strategy may be translated into BFO-FOL.
Additionally, commitment to (i) has a clear relationship with BFO-FOL, as straightforward translations for tensed binary relations in BFO-OWL follow the pattern:
R∀(x,y) =def ∀(t)(existsAt(x,t) & existsAt(y,t) & R(x,y,t))
R∃(x,y) =def ∃(t)(existsAt(x,t) & existsAt(y,t) & R(x,y,t))
In light of our observations concerning (ii), we might then characterize TQC’s commitment to (i) as accepting only the first definitional pattern, replacing the existsAt conjuncts with our temporally qualified continuant notation:
R∀(x,y) =def ∀(t)(R(xt,yt,t))
Again, these observations provide some reason to think were BFO-OWL to accept (i) and (ii), the resulting implementation would be translatable into BFO-FOL.
The same cannot be said for commitment to (iii). To be sure, some proposed TQC classes and relations have apparently straightforward translations into BFO-FOL, e.g. the class Phase appears only terminologically distinct from a certain BFO-FOL class discussed below. Others, however, have neither obvious parallels in nor translations to BFO-FOL, e.g. the class Tqc, the binary relation tqcOf, etc. This is perhaps to be expected for computational artifacts, but it is a noteworthy cost that TQC commitment to (iii) appears to conflict with a major design principle adopted by BFO developers. Proponents of TQC will likely reply these computational artifacts are needed to address the particular challenge, and that the native machinery of BFO-FOL is inadequate for the job. Hence, they might continue, the cost is worth paying.
But such a response exaggerates the need for these computational artifacts. We demonstrate why in a follow-up post, where we introduce an alternative strategy with a clear translation into BFO-FOL.