G Rodighiero
email: giulia.rodighiero@unipd.it

E Daddi

I Baronchelli

A Cimatti

A Renzini

H Aussel

P Popesso

D Lutz

P Andreani

S Berta

A Cava

D Elbaz

A Feltre

A Fontana

N M Förster Schreiber

A Franceschini

R Genzel

A Grazian

C Gruppioni

O Ilbert

E Le Floch

G Magdis

M Magliocchetti

B Magnelli

R Maiolino

H Mccracken

R Nordon

A Poglitsch

P Santini

F Pozzi

L Riguccini

L J Tacconi

S Wuyts

G Zamorani

THE LESSER ROLE OF STARBURSTSFOR STAR FORMATION AT z = 2

Keywords: galaxies, evolution -galaxies, interactions -galaxies, nuclei -galaxies, starburst

Two main modes of star formation are know to control the growth of galaxies: a relatively steady one in disk-like galaxies, defining a tight star formation rate (SFR)-stellar mass sequence, and a starburst mode in outliers to such a sequence which is generally interpreted as driven by merging. Such starburst galaxies are rare but have much higher SFRs, and it is of interest to establish the relative importance of these two modes. PACS/Herschel observations over the whole COSMOS and GOODS-South fields, in conjunction with previous optical/near-IR data, have allowed us to accurately quantify for the first time the relative contribution of the two modes to the global SFR density in the redshift interval 1.5 < z < 2.5, i.e., at the cosmic peak of the star formation activity. The logarithmic distributions of galaxy SFRs at fixed stellar mass are well described by Gaussians, with starburst galaxies representing only a relatively minor deviation that becomes apparent for SFRs more than 4 times higher than on the main sequence. Such starburst galaxies represent only 2% of mass-selected star forming galaxies and account for only 10% of the cosmic SFR density at z ∼ 2. Only when limited to SFR> 1000M ⊙ /yr, off-sequence sources significantly contribute to the SFR density (46 ± 20%). We conclude that merger-driven starbursts play a relatively minor role for the formation of stars in galaxies, whereas they may represent a critical phase towards the quenching of star formation and morphological transformation in galaxies.

1. INTRODUCTION Star forming galaxies follow a tight correlation between their stellar mass (M * ) and star formation rate (SFR), defining a main sequence (MS) that has been recognized in the local Universe (Brinchmann et al. 2004;Salim et al. 2007;Peng et al. 2010), as well as at intermediate redshifts 0.5 < z < 3 (Noeske et al. 2007;Elbaz et al. 2007;Daddi et al. 2007;Pannella et al. 2009;Rodighiero et al. 2010a, Karim et al. 2011), and beyond (Daddi et al. 2009;Gonzalez et al. 2010). With SFR ∝ M α * , the slope α can differ substantially depending on sample selection and the procedures for measuring SFR and M * , with values in the above literature ranging from ∼ 0.6 to ∼ 1. Moreover, its normalization rapidly rises from z = 0 to z ∼ 2-2.5 as (1 + z) ∼3.5 , then flattening all the way to the highest redshifts (Daddi et al. 2007, Rodighiero et al. 2010a;Karim et al. 2011). Slope and normalization of the SFR-M * relation play a crucial role in the growth of galaxies and in the evolution of their mass function (Renzini 2009;Peng et al. 2010Peng et al. , 2011)). Observations of the CO molecular gas content of MS galaxies indicate that their star formation efficiency does not depend strongly on cosmic epoch to z ∼ 2, with the SFR increase being due to higher molecular gas fractions (Daddi et al. 2008;2010a;Tacconi et al. 2010;Geach et al. 2011).

On the other hand, outliers are known to exist, with very high specific SFR (SSFR) compared to normal MS galaxies, such as local ultra-luminous infrared galaxies (ULIRG, Sanders & Mirabel 1996, Elbaz et al. 2007), and at least some of the submm-selected galaxies (SMG) at 1 < z < 4 (Tacconi et al. 2008;Daddi et al. 2007;2009;Takagi et al. 2008). What does differentiate such active behemoths from the dominant MS population? Outliers are galaxies in which the SSFR has been boosted by some event, possibly a major merger (e.g., Mihos & Hernquist 1996;di Matteo et al. 2008;Martig et al. 2010;Bournaud et al. 2011), as suggested by local ULIRGs and SMGs being dominated by gas-rich major mergers (e.g. Sanders et al. 1988). Indeed, MS galaxies and outliers appear to be in different star formation regimes: a quasi-steady, long-lasting mode for disks and a more rapid, starburst mode in major mergers or in the densest SF regions (Daddi et al. 2010b;Genzel et al. 2010).

So far it has been unclear by which of these two modes most of the stars in galaxies were formed. While MS galaxies are optically thin in the UV (Daddi et al. 2005;2007), MS outliers are generally optically thick (Goldader et al. 2002, Chapman et al. 2005) and far-IR observations are required to reliably derive their SFRs. The PACS camera (Poglitsch et al. 2010) onboard Herschel (Pilbratt et al. 2010) now allows for the first time to obtain deep far-IR observation probing SFRs down to MS levels for typical galaxies with M * ∼ 10 10 M ⊙ at z ∼ 2, i.e., at the peak epoch of the cosmic SFR density and of the space density of SMGs (Chapman et al. 2005). In this letter we combine wide area PACS observations of the COSMOS field with deeper data in the GOODS field, both taken as a part of the PEP survey (Lutz et al. 2011), and obtain a first accurate estimate of the relative role of MS and outlier galaxies on the formation of stars at z ∼ 2. Red and dead (passive) galaxies, though, exist at these cosmic epochs, and form a separate sequence below the MS of star-forming galaxies. Their contribution will be ignored in this Letter. We use a Salpeter IMF and a standard WMAP cosmology.

SAMPLE SELECTION AND MEASUREMENTS

In order to obtain a meaningful census of star-forming galaxies on and off the MS, one has to cover the SFR-M * plane down to low SFR and M * levels, and do so over a large area to include rare objects with the highest SFRs. We reach this goal by combining far-IR-selected (i.e., SFR-selected) and near-IR-selected (i.e., M * -selected) star-forming samples in the COSMOS and in GOODS-South fields, having both UV-and far IR-based SFR determinations. We first describe the datasets used, sample selections and SFR and M * measurements. We consider only galaxies within the redshift range of 1.5 < z < 2.5, based either on spectroscopic or photometric redshifts. The datasets include in total 576 and 122 PACS-detected galaxies in the COSMOS and GOODS-South fields, respectively, and 18,981 and 586 BzK-selected (cf. Daddi et al. 2004) sources in the COSMOS and GOODS-South fields.

SFR-selected Herschel samples

Herschel-PACS observations cover 2.04 square degrees over the COSMOS field down to a 5σ limit of 8 and 17 mJy at 100 and 160 µm, respectively, well above confusion limits (Lutz et al. 2011). Blending is not a major issue in the PACS data, and photometry was carried out by PSF-fitting at 24µm prior positions. The detection limits correspond to ∼ 200 M ⊙ yr -1 at z ∼ 2 (rising to 300M ⊙ yr -1 at z=2.5). We cross-matched over a common area of 1.73 square degrees the PACS detections with the IRAC-selected catalog of Ilbert et al. (2010), so to obtain UV-to-8µm photometry, accurate photometric redshifts and stellar masses by SED fits as described in Rodighiero et al. (2010b). At z ∼ 2 the mass completeness is granted above ∼ 10 10 M ⊙ (Ilbert et al. 2010). IR luminosities (hence SFRs) are derived from PACS fluxes using a set of empirical templates of local objects (Polletta et al. 2007;Gruppioni et al. 2010) as described in Rodighiero et al. (2010a). If, instead, we adopt the templates from Chary & Elbaz (2001), consistent SFR estimates are obtained with no bias and a scatter of ∼ 0.15 dex (that represents the typical error associated to our SFRs) . The PACS data in GOODS are deeper than those in COSMOS, reaching 5σ detection levels of 1.8, 2.0 and 4.0 mJy at 70, 100 and 160 µm, respectively (Berta et al. 2011). The typical SFR limit at z ∼ 2 is ∼ 50M ⊙ yr -1 . For PACS data reduction, photometry, far-IR based SFRs, and M * estimates we proceeded as for the COSMOS field.

Mass-selected BzK samples

For the GOODS-S field we used the K-band selected sample of star-forming BzK galaxies from Daddi et al. (2007), which reaches ∼ M * ∼ 10 10 M ⊙ over a 120 arcmin 2 field, with SFR estimated from the UV restframe luminosity corrected for dust reddening and taking advantage of a substantial fraction of spectroscopic redshifts. The deep BzK GOODS sample was complemented with the larger statistics (in particular for the more massive sources) provided by the BzK sample over the COSMOS field (McCracken et al. 2010), where here masses and SFRs have been computed using the same procedure as in Daddi et al. (2007), with typical errors of a factor ∼ 2.

Use of UV-based SFRs allows one to reach down to a few M ⊙ yr -1 at z ∼ 2. Daddi et al. (2007) performed detailed comparisons of UV-based SFRs with 70µm, 850µm, radio and X-ray based SFRs from stacking, finding good agreement. Pannella et al. (2009) used more than 30,000 BzK-selected galaxies in COS-MOS, finding that UV-based SFRs were slightly underestimated compared to radio. More recently Nordon et al. (2010) suggest that UV-based SFRs might actually be overestimated up to a factor of 1.5-2 at z ∼ 2, using Herschel observations (although see Wuyts et al. 2011a for a discussion of how this conclusion depends on sample selections). This relatively low level of uncertainty does not affect the main conclusion of this paper.

3. RESULTS Fig. 1 shows the SFRs versus stellar masses for the four 1.5 < z < 2.5 galaxy samples in this study. At first sight the PACS-based SFR-M * relation for the COSMOS sample (red symbols) bears little resemblance to the corresponding UV-based relation (black points). PACS-based SFRs run almost flat with mass, as opposed to the UV-based SFRs that increase almost linearly with mass. We maintain that this different behavior is due to the different selection of the two samples, one being SFR limited, the other being mass limited. It is immediately apparent from Fig. 1 that the vast majority of BzK-selected galaxies of a few to several 10 10 M ⊙ are not detected by PACS, implying that their SFR is lower than 200-300 M ⊙ yr -1 . Hence the intrinsic SFR-M * relation must be steeper than suggested by the PACS data alone. That this must be the case is indicated by the deeper PACS data over the GOODS field (cyan points), which start populating to lower stellar masses the MS defined Figure 1. Stellar mass -Star Formation Rate relation at 1.5 < z < 2.5. We use four main samples: the "shallow" PACS-COSMOS sources (red filled circles), the deeper PACS-GOODS South (cyan squares), the BzK-GOODS sample (black filled circles) and the BzK-COSMOS sources (black dots). The solid black line indicates the main sequence (MS) for star-forming galaxies at z ∼ 2 defined by Daddi et al. (2007), while the dotted and dashed lines mark the loci 10 and 4 times above the MS (along the SFR axis), respectively. The star indicates the PACS source detected by Aztec at 1.1mm in the COSMOS field. In the smaller inset, we show the same information as in the main panel, however here the stellar mass is presented as a function of the SSFR.

by the BzK-selected sample with UV-based SFRs. Still, the PACS SFR-selection cut is visible at ∼ 60 M ⊙ /yr, and most BzK galaxies remain undetected below this limit.

The PACS-GOODS sample also allows to populate the region with excess SFRs above the MS, something harder to do with UV-based SFRs due to obscuration. The combination of such datasets is clearly ideal to obtain a statistical census of high-SFR galaxies as well as high SSFR (see the insert of Fig. 1). Indeed, half of PACS-detected sources over the COSMOS field, either do not have a BzK counterpart, or their SFR is a factor ∼ 4 or more higher than their UV-based SFR. We interpret this as evidence that most of the SF activity in these galaxies is heavily dust-obscured in the UV.

In order to investigate the frequency and relative role of galaxies on and off the MS, we have derived the num-ber density distribution of galaxies in four stellar mass bins as a function of their SSFR (see Fig. 2). We assume no redshift incompleteness for the BzK galaxies over the 1.5 < z < 2.5 range (see, e.g., McCracken et al. 2010), whereas for PACS galaxies the flux limits imply different SFR limits as a function of redshifts. Therefore, 1/V max were computed with the same templates used to derive the IR luminosities. This procedure implicitly assumes no strong evolution of the number density of the population in the probed redshift range. This is confirmed by the average V /V max being 0.504 ± 0.023 and 0.56 ± 0.05, for the COSMOS-and GOODS-PACS samples, respectively. The SSFR distributions from the four samples agree within the errors (Poisson) in the regions of overlap, providing an important cross-check of the solidity of our approach.

In absence of a clear bimodality in Fig. 1, the distri- bution functions shown in Fig. 2 allow us to objectively define MS outliers, hence starburst galaxies. To this purpose, Gaussians functions are fitted to the BzK distributions in the four mass bins, resulting in a nearly constant σ = 0.24 dex, slightly lower than reported in Daddi et al. (2007). Deviations from the Gaussian distributions start to be clearly detected at SSFRs 0.6 dex above the average (Fig. 2; or some 2.5σ), and we adopt this threshold to define on-sequence and off-sequence galaxies. Such deviations are less obvious in the highest mass bin. We also note that the peak positions of the Gaussians shift as a function of the bin central mass, with a slope of 0.79 ± 0.04 in the log(SFR)-log(M * ) relation 15 (slightly shallower than the 0.9 slope reported by Daddi et al. 2007). It is worth emphasizing the good agreement of the four independent samples in the common SSFR bins, in particular for M * > 10 11 M ⊙ , where PACS-GOODS fully samples the MS distribution around its peak. In this case the UV and IR SFR tracers define exactly the same Gaussian distribution.

From the histograms in Fig. 2 we have estimated the relative contribution of on-sequence and off-sequence galaxies to the total comoving number density and SFR density, in absolute (Fig. 3) as well as relative (Fig. 4) 15 The MS best-fit is log(SFR)=-6.42+log(mass)*0.79 terms. The space density of galaxies in the highest mass bin is getting lower, as we are entering the exponentially decaying part of the mass function. The MS population dominates the SFR density at all masses, whereas offsequence starburst galaxies contribute almost constantly ∼ 10% of the total, or even less since some outliers may actually be MS objects with exceptionally large errors (in either SFR or M * ). If a top-heavy IMF were really to apply to starbursts, then this fraction would even be largely reduced. The number density of offsequence sources is also very small, varying between 2 and 3% as a function of the stellar mass. Only in SFR-limited samples (Fig. 4), off-sequence galaxies become important representing 46 ± 20% of the galaxies with SFR> 1000 M ⊙ yr -1 and 20±4% of those with SFR> 100 M ⊙ yr -1 . This suggests that even among SMGs (or luminous Herschel selected populations) only a fraction of the galaxies are strong MS outliers.

COMBINING UV AND IR SFR TRACERS

An important aspect of this work concerns the combination of UV-based and IR-based SFR indicators for the sources for which both data are available. Indeed, about 70%(80%) of the COSMOS(GOODS) PACS sources have a BzK counterpart. For ∼ 30% of these PACS-detected galaxies the IR-based SFRs exceed the UV-based SFRs by factors of 4-10, that we qualify as optically thick starbursts. For these sources the SFR(UV) is totally unreliable. However, these objects are almost all outliers from the MS, and they represent only ∼ 2% of the total BzK population. For the other objects, a direct comparison of UV-based and far IR-based SFRs for GOODS BzKselected galaxies shows reasonably good agreement, consistently with the results of Nordon et al. (2010) and Reddy et al. (2011 in prep.).

To further test the reliability of the SFR(UV), we analyzed the BzK population with SFR(UV) in the same SFR range accessible by PACS in the deep GOODS field. We restricted our analysis to 60 <SFR(UV)< 300M ⊙ /yr objects in two mass bins: a) 10.0 <log(M)< 10.75M ⊙ and b) 10.75 <log(M)< 11.0M ⊙ , and then stacked on the 160µm map the BzK PACS-undetected sources combining them with PACS detections to get a mean far-IR SFR for the BzK samples (as in Rodighiero et al. 2010a). For the two mass bins the result is: a) <SFR(UV)>= 79 ± 1M ⊙ /yr and <SFR(PACS)>= 69 ± 10M ⊙ /yr; b) <SFR(UV)>= 126 ± 1M ⊙ /yr and <SFR(PACS)>= 113 ± 6M ⊙ /yr. Reassuringly, in the common mass and SFR ranges, the PACS and UV SFR are in excellent agreement (with only ∼10% of UV overestimation in the highest mass bin that would not affect the conclusion of this work).

Summarizing, we found that PACS and UV based SFRs are in good agreement along the MS, while the off-sequence discrepancies (i.e., objects for which UV can dangerously underestimate the actual SFR) are due to dust obscuration at high SFR levels. These objects represent ∼ 2% of star-forming galaxies at z ∼ 2 and M * > 10 10 M ⊙ .

THE MOST EXTREME STAR FORMING

SOURCES AND THE OBSCURED AGN ACTIVITY We have shown that the distribution of SSFRs in z ∼ 2 massive galaxies are largely Gaussian with σ of 0.24 dex, and relatively minor deviations only detected above 0.6 dex of the main trend. Although some of these high SSFR galaxies will just be normal MS galaxies being scattered there by measurement errors, we conservatively identify all of them as (merging-driven) starbursts. Of course, also galaxies that are still on the MS can be witnessing merging events, given that not all mergers are expected to produce strong SFR enhancements (di Matteo et al. 2008), hence our off-sequence galaxies can be seen as objects where a major merger event has substantially boosted the SSFR, increasing it by a factor of ≥ 4.

In order to investigate the nature of off-MS galaxies we have focused on the most extreme 28 objects, lying a factor 10 above the MS. We performed a simultaneous three components SED fitting analysis including the emission of stars and star-heated dust, and allowing also for the presence of a dusty torus (AGN), using the method described in Fritz et al. (2006). In Fig. 5 we report an example of the resulting SED fitting obtained for 9 representative objects, selected among the 28 sources. The presence of an AGN is required in most cases (25 out of 28) to reproduce the 24µm emission, while its contribution to the L[8 -1000µm] is always lower than 5-10%, thus negligible. None of the 16 galaxies observed by Chandra is detected (Elvis et al. 2009), suggesting that the AGNs are extremely obscured. These objects are probably similar to the Fe K α emitting Compton thick QSO at z = 2.5 from Feruglio et al. (2011), that qualifies as a starburst galaxy according to our criterion. Only 2 of our 28 sources fall in the AzTEC area (Scott et al. 2008), and one of them is detected in the millimeter catalog. Fig. 5 also reports ACS-F775W cutouts for the example objects. A few objects are consistent with advanced stage mergers (double components on the I-band ACS COSMOS image, Koekemoer, A.M. et al. 2007), others are quite compact (consistent with the analysis of Elbaz et al. 2011), whereas some remain undetected. A more extended morphological analysis is postponed to a future paper (but see also Wuyts et al., 2011b).

DISCUSSION

The redshift range 1.5 < z < 2.5 corresponds to a ∼ 2 Gyr cosmic time interval, hence our star-forming galaxies have on average spent in it ∼ 1 Gyr. Therefore, the fact that only ∼ 2% of the massive galaxies are off the main sequence implies that on average each galaxy spends ∼ 20 Myr in such a phase. This is actually much shorter than both the gas depletion timescale and the (outer rotation) dynamical time in starburst galaxies (Daddi et al. 2010;Genzel et al. 2010). This is also much shorter than the expected duration of the SFRexcess phase in mergers based on numerical simulations (e.g. di Matteo et al. 2008, Martig et al. 2010), where the duration of the phase with SFR in excess of > 4 over the pre-merger SFR is of order of 200-300 Myr (Bournaud et al. 2011). Most likely this is because only a fraction of massive star-forming galaxies undergoes major mergers during this time interval, and/or because most mergers do not produce a substantial increase of the SFR (consistent with the simulations of di Matteo et al. 2008).

All in all, our results quite clearly show that the merger-enhanced SFR phases are relatively unimportant for the formation of stars in z ∼ 2 galaxies, and probably at all redshifts given that z ∼ 2 is known to be the 'prime time' for SMGs (Chapman et al. 2005), and that this is similar to what is observed in the local Universe (e.g. the percentual contribution of starburst is very small, Sanders & Mirabel, 1996). Still, going through this merging-driven starburst phase may be a critical phase for the transformation of star-forming galaxies into passive ellipticals. Hence, we maintain that off-sequence galaxies are likely to be crucial objects for our understanding galaxy formation and evolution.

GR acknowledges support from ASI (Herschel Science Contract I/005/07/0). ED acknowledges funding support from ERC-StG grant UPGAL 240039 and ANR-08-JCJC-0008. PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF-IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain). We thank the anonymous referee for a constructive report.

Figure 5. Spectrophotometric properties of 9 out of 28 representative sources lying a factor 10 above the MS. We performed a simultaneous three components SED fitting analysis including stellar (blue), dusty torus (green) and starburst (red) emissions. The sum of the three components is also reported (black). A cutout of the ACS i band (5" × 5") is reported.

Figure 2 .

 2 Figure2. Distribution of the SSFR for the four samples, corrected by the corresponding comoving volumes and after accounting for volume and selection incompleteness, splitted in four mass bins. The dotted-black, solid-black, cyan and red histograms correspond to the BzK-GOODS, BzK-COSMOS, PACS-GOODS South and PACS-COSMOS data-sets. Error bars are Poisson. The green curves show the Gaussian fits to the black-solid histograms. The green dashed vertical lines mark the vertex of each best-fit Gaussian, and the dotted black vertical lines +0.6 dex above show our choice to separate ON-and OFF-sequence galaxies along the SSFR distribution.

Figure 3 .

 3 Figure 3. SFR density (top panel) and number density (bottom panel) of ON sequence sources (blue open squares) and OFF sequence sources (red filled squares), in the four mass bins considered in Fig. 2. Error bars are Poisson.

Figure 4 .

 4 Figure 4. Contribution of OFF sequence galaxies to the total SFR density in different SFR bins (top panel) and stellar mass bins (middle panel). In the bottom panel we also report the number density percentage of OFF sequence sources. Error bars are Poisson.

Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.


Adrien Barton
email: adrien.barton@gmail.com

Olivier Grenier

Ludger Jansen

Jean-François Ethier
email: ethierj@gmail.com.

The Identity of Dispositions

Keywords: Disposition, Identity, Causality, Trigger, Realization, Multiplicativism

Clear criteria for the identity of dispositions are still lacking, and this has been presented as one of the main challenge raised by such entities. It is of prime importance to identify or distinguish dispositions such as diseases or risks. This article first introduces conventional ways to refer to a disposition (such as "fragility") and canonical ways (such as "disposition to break in case of a strong shock"). This raises the issue of how should exactly be defined a "disposition d to R when TR", where R is a realization specification and TR a trigger specification. Two ontological frameworks are distinguished. The first framework, which has been largely used so far in the literature on dispositions, interprets d as a disposition which can only be triggered by instances of TR, and can only be realized by instances of R. The second, new framework introduces the notion of "minimal trigger" and "maximal realization", and interprets TR as a parent class of a class of processes that have as part a minimal trigger, and R as a parent class of a class of processes that are parts of a maximal realization. We then discuss several criteria of identity, including the criterion according to which two dispositions are identical iff they have the same categorical basis, the same class of minimal triggers and the same class of maximal realizations. We show on several examples that the second framework avoids the disposition multiplicativism that is introduced by the first framework.

Introduction

Dispositions are entities such as fragility, inflammability, solubility, or vulnerability to poison, which can be triggered by some process, leading to a realization process. They may exist even if they are not realized or even triggered: I am vulnerable to arsenic even if I never ingest any arsenic in my life. Dispositions are causal properties, and for this reason, they are of central importance for scientific ontologies. They have been used to formalize diseases [START_REF] Scheuermann | Toward an ontological treatment of disease and diagnosis[END_REF], risks [START_REF] Barton | What is a risk? A formal representation of risk of stroke for people with atrial fibrillation[END_REF], or probabilities [START_REF] Barton | Probability assignments to dispositions in ontologies[END_REF]. A formalization of dispositions has been proposed in the framework of the Basic Formal Ontology (BFO) [START_REF] Röhl | Representing dispositions[END_REF], and a theory of mereology of dispositions has been sketched [START_REF] Barton | A taxonomy of disposition-parthood[END_REF].

However, dispositions have some disturbing characteristics. A common worry is that there are many of them: dispositions are often identified with causal powers (the capacity of an entity to cause some effect), and the world is replete with causal powers. Their identity is one of the main argument that has been raised against the relevance of dispositions in applied ontologies: as stated by Guarino [START_REF] Guarino | BFO and DOLCE: So Far, So Close[END_REF], "the difficulty of distinguishing one disposition from another […] is a good evidence of their problematic status". Moreover, in some cases, what we would identify as one causal power seems to give rise to several dispositions. For example, OGMS, the Ontology for General Medical Science [START_REF] Scheuermann | Toward an ontological treatment of disease and diagnosis[END_REF], formalizes a disease as a disposition realized by the whole disease course, the latter being composed by several pathological processes. But does it mean that there is another disposition realized by each of those pathological processes? If yes, we have a case of "disposition multiplicativism". Another example concerns risks, which have been identified with dispositions [START_REF] Barton | What is a risk? A formal representation of risk of stroke for people with atrial fibrillation[END_REF]. Consider a person with atrial fibrillation who has a risk to have a stroke. Obviously, the probability of his risk of stroke over 6 months is different from the probability of his risk of stroke over 12 months; but does it mean that he has two dispositions, one risk at 6 months, and another risk at 12 months? This leads to a question about the identity of dispositions: what are the conditions for two dispositions to be identical (that is, for them to be the same entity)?

This article will analyze this question. The next part will distinguish a disposition from its categorical basis. A third part will differentiate two ways to refer to dispositions, and analyze how to specify the triggers and realizations of a disposition, contrasting the framework accepted so far, labelled here "ONLY", with a new framework labelled "PARTHOOD". A fourth part will argue that PARTHOOD is more ontologically economical than ONLY. A conclusion will follow.

Next to standard first-order logic, we will make use of the Manchester Syntax [START_REF] Horridge | OWL 2 web ontology language manchester syntax[END_REF] for the Web Ontology Language (OWL), which is a description language for ontologies based on Description Logics. Terms for universals will be italicized and terms for particulars will be written in bold. If a and b are particulars, a+b is defined as the mereological sum of a and b. Similarly, if A and B are classes or universals, A+B is defined as the class of mereological sums of one instance of A and one instance of B.

Dispositions and their categorical bases

Formalization of dispositions in BFO

Our ontological framework builds on a model of dispositions (and processes) proposed in the context of the Basic Formal Ontology (BFO) [START_REF] Arp | Building Ontologies with Basic Formal Ontology[END_REF], but should be adaptable to other upper ontologies. BFO defines a disposition as a realizable entity "that exists because of certain features of the physical makeup of the independent continuant that is its bearer" ( [START_REF] Barton | What is a risk? A formal representation of risk of stroke for people with atrial fibrillation[END_REF], p. 178). To be realized in a process, a disposition needs to be triggered by some other process: for example, strong_shock 0 can trigger the disposition fragility 0 of glass 0 (fragility 0 has_trigger strong_shock 0 , or strong_shock 0 trigger_of fragility 0 ) which is then realized by glass 0 _breaking (fragility 0 has_realization glass 0 _breaking or glass 0 _breaking realization_of fragility 0 ) -where has_trigger and has_realization are primitive relational predicates for causal relations.

What are categorical bases?

The fragility of a glass exists because of a certain molecular structure of this glass. The inflammability of a match exists because of its chemical constitution. Whatever constitutes a disposition is sometimes called the "causal basis" of the disposition [START_REF] Prior | Three theses about dispositions[END_REF]. There are several proposals concerning the nature of causal bases. For one, BFO [START_REF] Arp | Building Ontologies with Basic Formal Ontology[END_REF] introduces the "material basis" of a disposition as a material entity. E.g., the morphine contained in opium is the material basis of opium's dormitive virtue, and my weak eye muscle is the material basis of my short-sightedness. The material basis of a disposition is a material entity. This, however, cannot always capture finely enough the causal structure relevant for a disposition. Imagine a glass whose molecular structure makes it both fragile and electrically resistive: the material basis of both its fragility and its electrical resistivity is the whole glass -but because of different properties. Therefore, following Röhl & Jansen [START_REF] Röhl | Representing dispositions[END_REF], we introduce the notion of "categorical basis", which is a quality (or a sum of qualities) of the disposition bearer. The categorical basis of the glass fragility is the sum of qualities of the glass that make it fragile, and the categorical basis of its electrical resistivity is the sum of qualities of the glass that make it electrically resistive. Thus, the notion of categorical basis enables a finer identification of the specific causal factors underlying a disposition.

Are categorical bases the same entities as dispositions?

Aiming at parsimony, it is tempting to identify dispositions with their categorical bases. Guarino [START_REF] Guarino | BFO and DOLCE: So Far, So Close[END_REF] argues in favor of the identification, as the "truth-maker of the property being fragile seems to be the same as the truth-maker of the property having a certain crystalline structure." Prior, Pargetter & Jackson [START_REF] Prior | Three theses about dispositions[END_REF] argue against this identification, but their arguments rely on the ontological hypothesis that properties exist only as types, or universals, and other debatable hypotheses about the nature of identity between types. Mumford [START_REF] Mumford | Dispositions[END_REF] replies to them that dispositions and categorical bases are different at type level, but the same at token level, i.e., that categorical terms and dispositional terms are just two ways to refer to the same entity at the token level.

There is, however, a good reason not to subscribe to this token-identity thesis, as dispositional and categorical tokens seem to satisfy different relations. For example, one can define mereological relations on dispositions [START_REF] Barton | A taxonomy of disposition-parthood[END_REF], such that a disposition of glass 0 to break-or-crack has as proper parts a disposition to break and a disposition to crack; however, the categorical basis of the disposition of glass 0 to break and the categorical basis of the disposition of glass 0 to crack are not proper parts of the categorical basis of the disposition of glass 0 to break-or-crack. Those three bases are rather identical.

In contrast, imagine that the very same qualities underlie the fragility and electrical resistivity of glass 0 -and, thus, that we cannot distinguish the categorical basis of its fragility from the categorical basis of its electrical resistivity. Still, we can distinguish fragility (as a disposition to break) from electrical resistivity (as a disposition to block electric current). Therefore, we have good reasons not to identify dispositions with their categorical basis.

Defining trigger specification, realization specification and bearer

Conventional and canonical dispositions: definitions

We need to distinguish two ways to refer to dispositions [START_REF] Choi | Dispositions[END_REF]. Conventional descriptions are terms or phrases regimented in common language or scientific discourse, such as "fragility" of "solubility" not explicitly referring to their triggers or realizations. In contrast, canonical descriptions explicitly describe a disposition in terms of their triggers and realizations. For example, "a disposition to R when TR" is characterized by the trigger specification TR and the realization specification R (we will call the association of a trigger specification and a realization specification a "track"). Consider a magnet magnet 0 , and let Attraction be the class of attraction processes between two magnets, Repulsion the class of repulsion processes between two magnets, Unlike poles approach the class of processes during which two magnets are approached with opposite poles facing, and Like poles approach the class of processes during which two magnets are approached with similar poles facing. Then, examples of canonical dispositions include the single-track attraction_power 0 of magnet 0 to Attraction when Unlike poles approach, its single-track repulsion_power 0 to Repulsion when Like poles approach, and its multi-track attraction_and_repulsion_power 0 to Attraction when Unlike poles approach and to Repulsion when Like poles approach. Following the above-mentioned theory of mereology among dispositions [START_REF] Barton | A taxonomy of disposition-parthood[END_REF], the following parthood relations (named "mod-part_of") obtain between these three canonical dispositions:

• attraction_power 0 mod-part_of attraction_and_repulsion_power 0 • repulsion_power 0 mod-part_of attraction_and_repulsion_power 0 Put differently, attraction_and_repulsion_power 0 is a mod-complex (that is, a disposition with a proper mod-part) that has two possible pathways, or modes, of being realized: via attraction_power 0 or via repulsion_power 0 .

Canonical descriptions are typically useful to analyze the structure of dispositions referred to by conventional names. For example, we might analyze the conventional disposition magnetism 0 of magnet 0 by stating that it is identical with a canonical multitrack disposition to Attraction when Unlike poles approach and to Repulsion when Like poles approach. We will here deal with canonical dispositions that are sure-fire, i.e., that are invariably realized when triggered [START_REF] Röhl | Representing dispositions[END_REF], and for which there are no masks, i.e., no entities blocking the process that would otherwise lead from a trigger to a realization, like a cushion would mask the breaking of glass 0 [START_REF] Choi | Dispositions[END_REF]. As we will see, there are two nonequivalent ways to formalize canonical dispositions: ONLY and PARTHOOD.

Defining canonical dispositions

ONLY: A first framework for canonical dispositions

Following Röhl & Jansen [START_REF] Röhl | Representing dispositions[END_REF], we consider first a single-track disposition d that has x as a bearer, R as its realization specification and TR as its trigger specification, and which can thus be described as "a disposition d of x to R when TR"; and we will ignore the distinction between "trigger" and "background conditions". A trigger specification and a realization specification are formalized as follows: However, such a reading would be insufficient in an ontology in which we would quantify over actual entities only. As a matter of fact, any disposition d that is never triggered and never realized would trivially satisfy these two conditions. So, for example, if glass 0 is never hit by a strong shock during its history, we could state that its disposition_to_break 0 has the trigger specification Heart_beating and has the realization specification Headache, or any other unrelated processes. Therefore, the quantification above should hold as a matter of necessity: the right-hand side in (ONLY TR Instance ), for example, should be read as holding that necessarily any trigger of d is an instance of TR (and any instance of TR is an instance of Process). This would mean that disposition_to_break 0 could only be realized by a breaking process, and could only be triggered by a strong shock. Such a reading is implicit in (ONLY TR Class ), if we accept the distinction drawn between classes and sets by Johansson [START_REF] Johansson | Four Kinds of 'Is_A' Relation[END_REF]: a set is identified by its members, whereas the identity of a class goes beyond its actual instances. We consider that this holds for all classes, including fully defined ones. Thus, a class such as 'trigger_of d' share some similarities with the "terminological units" (as defined in [START_REF] Schulz | Higgs Bosons, Mars Missions, and Unicorn Delusions: How to Deal with Terms of Dubious Reference in Scientific Ontologies[END_REF]), as they might have no extension in the actual world. In the following, we will call "ONLY" the framework composed by the conjunction of (ONLY R Class ) and (ONLY TR Class ). In general, not any instance of a trigger specification TR would trigger d. For example, not every instance of Strong_shock would trigger disposition_to_break 0 : clearly, an instance of Strong_shock in which glass 0 does not participate would not trigger this disposition. For the same reason, not any instance of a realization specification R would be a realization of d. This means that in general, we do not have '(trigger_of d) EquivalentTo TR', nor '(realization_of d) EquivalentTo R'. It is important, that is, to distinguish a trigger specification (resp. realization specification) for a disposition from its class of actual triggers (resp. class of actual realizations).

One could suggest that any instance of TR in which b participates is a trigger of d (and that any instance of R in which b participate is a realization of d). But this must also be rejected. As a matter of fact, glass 0 could participate in a Strong_shock in a way different from the one intended -for example, by being used as a tool to hit a more fragile glass 1 that would break, while glass 0 would remain intact. Therefore, if we want to find a trigger specification such that every instance of this class would trigger d, and a realization specification such that every instance of this class would realize d, we should define them by referring to the particular(s) involved. For example, we could define disposition_to_break 0 as a disposition to glass 0 _Breaking when Strong_shock_on_glass 0 (which are both defined classes, rather than universals). We will call this a full specification of a disposition:

(FULL SPECIFICATION) The phrase "a disposition to Rfull when TRfull" is a full specification of d iff:

TRfull EquivalentTo (trigger_of d) and Rfull EquivalentTo (realization_of d) Consider three dominos, domino 0 , domino 1 and domino 2 , which are placed in a line from left to right (cf. the left part of configuration 1 in Figure 1), such that if we push domino 0 to the right (the class of such processes is named "Push →0 "), it will fall (Fall →0 ) and push domino 1 , that will fall (Fall →1 ) and push domino 2 that will fall (Fall →2 ). More precisely, we define for i {0, 1, 2}:

Disposition multiplicativism in ONLY

Push →i = def a process during which domino i is pushed to the right in configuration 1.

• Fall →i = def a process during which domino i falls to the right onto the floor in configuration 1.

According to those definitions, Fall →0 SubClassOf Push →1 , because the process during which domino 0 falls to the right in the configuration 1 is a process during which domino 1 is pushed to the right. Similarly, Fall →1 SubClassOf Push →2 . (Of course, it would not be the case if the domino configuration was different from configuration 1; this configuration should therefore be seen as a background condition for the dispositions mentioned below -and, as suggested earlier, is here integrated into the trigger.) Consider, e.g., the following four dispositions: 1) d →0;→2 to Fall →2 when Push →0 2) d →0;→1,2 to Fall →1,2 when Push →0 (where Fall →1,2 is the class of falls of domino 1 and fall of domino 2 where the former causes the latter) 3) d →0;→0,1,2 to Fall →0,1,2 when Push →0 4) d →0;→0,2 to Fall →0,2 when Push →0 . According to ONLY, those four dispositions are different from each other, as they do not have the same classes of realizations. Suppose for example that domino 0 is pushed, causing a chain of falls of domino 0,1,2 (= domino 0 +domino 1 +domino 2 ) named fall 0,1,2 ; then fall 0,1,2 is a realization of d →0;→0,1,2 , but it is not a realization of d →0;→1,2 nor a realization of d →0;→2 .

Suppose now that domino 3 is also next to domino 0 as in figure 1. Consider the following dispositions: d →0;→0 to Fall →0 when Push →0 ; and d →0,3;→0 to Fall →0 when Push →0 +Push →3 . According to ONLY, those two dispositions are different from each other, as they do not have the same class of triggers. For example, if domino 0 is pushed (during the process push 0 ) at the same time as domino 3 (during the process push 3 ), then push 0 is a trigger of d →0;→0 , but push 0 +push 3 is not. For the same reasons, d →3;→3 and d →0,3;→3 are different. Thus, ONLY commits to a relatively strong form of disposition multiplicativism. We will now discuss an alternative, more economical ontological framework.

PARTHOOD: A second framework for canonical dispositions

One might want to endorse a more encompassing conception of triggers and realizations of a disposition than the one implied by ONLY. We might accept that whenever a disposition d is triggered by a process tr, it is also triggered by processes having tr as a part; for example, the mereological sum of a strong shock on glass 0 and John's heart beating (or any other unrelated process) is also a trigger of disposition_to_break 0 ; or in Figure 1, push 0 +push 3 is also a trigger of d →0;→0 . Similarly, we might say that whenever d is realized by a process r, it is also realized by processes that are parts of r; e.g., any part of the breaking of glass 0 caused by a strong shock would also be a realization of disposition_to_break 0 ; or, in Figure 1, fall 0 and fall 1,2 are also realizations d →0;→0,1,2 .

Formally, this could be expressed by stating that a process that would have as part a trigger of d is a trigger of d, and a process that would be a part of a realization of d is a realization of d: Intuitively, an instance trmin of TRmin(d) is a smallest part of a trigger of d, such that the interaction of trmin with the categorical basis of d is enough to cause a realization of d. Similarly, an instance rmax of Rmax(d) is a largest whole that has a realization of d as part, such that there is an instance trmin of TRmin(d) whose interaction with the categorical basis of d causes rmax (or, put differently, it is the mereological sum of all processes that are caused by the interaction of trmin and b). Note that when we use the word "caused", we do not only mean "proximally caused", but also "distally caused": that is, if there is a causal chain due to the categorical basis of d which is triggered by trmin, then the whole causal chain of events belongs to the realization rmax of d. However, if some process is caused by the interaction not only of the categorical basis of d and trmin, but also by other factors, then it is not a part of rmax.

(
Reciprocally, by application of (HAS-PART TR ) and (PART-OF R ), the triggers of d are exactly the processes that have as part a minimal trigger of d, and the realizations of d are exactly the processes that are a part of a maximal realization of d: Note that we chose here to speak of a class of minimal triggers, and of classes of triggers which have as parts those minimal triggers; but alternatively, we might have reserved the term "triggers" for the minimal triggers, and called processes that include those triggers "over-triggers" (in which case, if ot is an over-trigger of d, then there is a part t of ot such that d has_trigger t). The choice between both vocabularies has no real ontological import, so we will chose the first option to be in the continuity of Röhl & Jansen [START_REF] Röhl | Representing dispositions[END_REF] model and other later works that built on it.

In this framework, a disposition is characterized by its classes of minimal triggers and maximal realizations. However, it is still possible to characterize a disposition by its trigger specification and realization specification in order to describe, for example, that disposition_to_break 0 is a disposition to Breaking when Strong_shock. A trigger specification TR of a disposition d is any class of processes that has a subclass TR' whose instances all have as part some minimal trigger of d:

(HAS-PART TRMIN ) A disposition d has a trigger specification TR iff:

TR SubClassOf Process, and there is a class TR' such that: TR' SubClassOf TR and TR' SubClassOf (has_part some TRmin(d)).

For example, according to (HAS-PART TRMIN ), disposition_to_break 0 has the trigger specification Strong_shock because Strong_shock is a parent class of Strong_shock_on_glass 0 , whose instances all have as part an instance of Pressure>P 1 _on_glass 0 (that is, a minimal trigger of disposition_to_break 0 ). Similarly, we can define a realization specification of a disposition d as any class of process such that any maximal realization of d has as part some element of this class:

(PART-OF RMAX ) A disposition d has a realization specification R iff: R SubClassOf Process, and there is a class R' such that: R' SubClassOf R and Rmax(d) SubClassOf (has_part some R'). E.g., according to (PART-OF RMAX ), disposition_to_break 0 has the realization specification Breaking because Breaking has as subclass glass 0 _Breaking, and every maximal realization of disposition_to_break 0 has as part an instance of glass 0 _Breaking. We will call "PARTHOOD" the framework composed by the assumptions (HAS-PART TR ), (PART-OF R ), (HAS-PART TRMIN ) and (PART-OF RMAX ). The two different ontological frameworks ONLY and PARTHOOD thus both define what is a trigger specification and a realization specification -though, as we will see, a disposition is more finely characterized by its full specification in ONLY, and by its classes of minimal triggers and realizations in PARTHOOD. We will now specify the maximal realizations on a few domino examples.

Specifying the maximal realization: Domino examples

On the domino example in Figure 1, TRmin(d →0;→1 ) might be the application of a momentum above M 0 on domino 0 ; let's call it Minimal_push →0 . We have seen above that an instance Rmax is the mereological sum of all processes that are caused by the interaction of a minimal trigger trmin and the categorical basis of d →0;→1 . So what would be Rmax(d →0;→1 )? To answer this, we need to specify further d →0;→1 , as it has been so far ambiguously defined. We can first define d →0;→1 do0,1,2 as the disposition to Fall →1 when Push →0 inhering in domino 0,1,2 . But we can also define d →0;→1 do0,1 as the disposition to Fall →1 when Push →0 inhering in domino 0,1 only. The interaction of Push →0 with the categorical basis of d →0;→1 do0,1,2 borne by domino 0,1,2 will cause an instance of Fall →0,1,2 . However, the interaction of an instance of Push →0 with the categorical basis of d →0;→1 do0,1 borne by domino 0,1 will only cause an instance of Fall →0,1 . Thus, Rmax(d →0;→1 do0,1,2 ) = Fall →0,1,2 _caused_by_Push →0 (where the latter is a subclass of Fall →0,1,2 , when those falls are caused by an instance of Push →0 ) and 1,2 and d →0;→0 do0,1,2 all have the same class of maximal realization, namely Fall →0,1,2 _caused_by_Push →0 : the class of realizations systematically depends on the categorical basis and the class of minimal triggers. To determine whether these dispositions are identical, though, we will need a criterion of identity between dispositions, that we will devise in the next section.

Rmax(d →0;→1 do0,1 ) = Fall →0,1 _caused_by_Push →0 . Therefore, d →0;→0,1,2 do0,1,2 , d →0;→0,2 do0,1,2 , d →0;→1,2 do0,1,2 , d →0;→0,1 do0,1,2 , but also d →0;→2 do0,1,2 , d →0;→1 do0,

Criteria of identity for dispositions

We will now formulate several identity criteria for disposition instances, and evaluate whether they are acceptable in ONLY or PARTHOOD. We will first discuss singletracks dispositions, and then turn to multi-tracks dispositions. (ID 1 ) A disposition with basis cat to R when TR and a disposition with basis cat' to R' when TR' are identical iff: cat and cat' are identical, R and R' are identical, and TR and TR' are identical. This is, however, not a sound criterion of identity. As a matter of fact, let R# be any parent class of R such that R# SubClassOf Process, and TR# any parent class of TR such that TR# SubClassOf Process. Then, whether we accept ONLY or PARTHOOD, it is easy to see that a disposition to R when TR is also a disposition to R# when TR#.

Identity of single-track dispositions

Do we need to specify the bearer or the categorical basis in identity criteria?

A second proposal would be that two dispositions are identical iff they have the same class of triggers and the same class of realizations (that is, if they have the same full specification):

(ID 2 ) d is identical to d' iff [(trigger_of d) EquivalentTo (trigger_of d') ∧ (realization_of d) EquivalentTo (realization_of d')] Figure 2.
However, we can construct in ONLY a counter-example to (ID 2 ) illustrated by the domino arrangement in Figure 2. If domino 4 is pushed to the right, then it falls on both domino 5 and domino 6 , which both falls on domino 7 , making it fall. However, even if only domino 5 or domino 6 would fall, domino 7 would fall. Then, there are two causal chains from Push →4 to Fall →7 -one involving Fall →5 , and the other involving Fall →6 . In ONLY, one could distinguish two dispositions with the same class of triggers and realizations: the disposition d →4;→7 do4, 5,7 to Fall →7 when Push →4 due to the first causal chain; and the disposition d →4;→7 do4, 6,7 to Fall →7 when Push →4 due to the second causal chain. Clearly, these two dispositions are not identical, as they do not have the same bearer: the first one can disappear (if domino 5 would disappear) while the second one still exists, and vice versa. However, they have the same class of triggers Push →4 , and the same class of realizations Fall →7_caused_by_Push→4 ; that is, they have the same full specification. Thus, (ID 2 ) is false in ONLY.

However, Figure 2 is not a counter-example of (ID 2 ) if we accept PARTHOOD: in such a case, d →4;→7 do4,5,7 and d →4;→7 do4,6,7 do not have the same class of realizations. For example, an instance of Fall →5 caused by an instance of Push →4 would belong to realization_of d →4;→7 do4,5,7 , but not to realization_of d →4;→7 do4,6,7 . We did not find any real-world counter-example to (ID 2 ) in PARTHOOD, but they are theoretically possible.

Identification by class of triggers, class of realizations and categorical basis

This leads us to (ID 3 ), according to which two dispositions are identical if they have the same class of triggers, the same class of realizations and the same categorical basis: ). Such a structure would provide a counter-example to (ID 3 ). However, it is not clear that such a model would exist in the actual world.

(ID 3 ) d is identical to d' iff ∃cat (d has_basis cat ∧ d' has_basis cat) ∧ [(trigger_of d) EquivalentTo (trigger_of d')] ∧ [(realization_of d) EquivalentTo (realization_of d')]
Figures 3a and3b.

Consider for example the scenario pictured in Figure 3b, in which two switches are each connected to two light bulbs by different wires, such that switching either switch 1 or switch 2 (or both) would be enough to light up both bulb 1 and bulb 2 . Consider now the disposition for both bulbs to light up when both switches are switched up whose categorical basis include blue_wires a and blue_wires b , and the other disposition for both bulbs to light up when both switches are switched up whose categorical basis include red_wires a and red_wires b . Then, both dispositions do not have the same categorical basis, and, thus, Figure 3b does not present a counter-example to (ID 3 ). However, scenarios like Figure 3a are at least theoretically possible. Therefore, we can suggest the following criterion of identity, according to which two dispositions are identical iff they have the same basis, and any process that triggers the first disposition into a realization does trigger the second disposition into the very same realization, and vice versa: Again, the reading of this quantification should be modal (otherwise, two dispositions with the same basis that would never be triggered would trivially be identical). We suggest that (ID 4 ) is the correct criterion of identity for disposition instances. For all practical purposes, however, (ID 3 ) seems to be usable until a convincing real-world counter-example is found. (ID 3 ) and (ID 4 ) show, as we stated in section 2.3, that a disposition should not be identified with its categorical basis. As a matter of fact, according to both criteria, two dispositions can have the same categorical basis but still be different, in case they have different classes of triggers or different classes of realizations.

Identification by classes of minimal triggers and maximal realizations

Within PARTHOOD, we can show that the following criterion (ID 3 ') is equivalent to (ID 3 ) by applying (TRIGGER), (REALIZATION), (TRMIN) and (RMAX):

(ID 3 ') d is identical to d' iff: ∃cat (d has_basis cat ∧ d' has_basis cat) ∧ (TRmin(d) EquivalentTo TRmin(d')) ∧ (Rmax(d) EquivalentTo Rmax(d'))
Finally, as suggested during the analysis of the domino example in subsection 3.3, the class of maximal realizations Rmax(d) of a disposition systematically depends on its class of minimal triggers TRmin(d) and its categorical basis. Therefore, the following criterion of identity (ID 3 '') is also equivalent to (ID 3 ) and (ID 3 '):

(ID 3 '') d is identical to d' iff: ∃cat (d has_basis cat ∧ d' has_basis cat) ∧ TRmin(d) EquivalentTo TRmin(d')

Identity of multi-track dispositions

The former criteria apply only to single-track dispositions. Some dispositions, however, are multi-tracks, having several single-track dispositions as mod-parts. Classically, two entities are identical iff they have the same proper parts (PP):

x is identical to y iff ∀w (PPwx PPwy)

This criterion can be adapted to multi-track dispositions by stating that two multi-track dispositions are identical iff they have the same mod-parts:

Avoiding disposition multiplicativism

Now that we have criteria of identity for the two frameworks ONLY and PARTHOOD, we can evaluate them by applying them to the examples mentioned in section 3. We had seen that according to ONLY, and independently of any criterion of identity we endorse, dispositions such as d →0;→2 do0,1,2 , d →0;→1,2 do0,1,2 , d →0;→0,1,2 do0,1,2 , d →0;→0,2 do0,1,2 are all distinct disposition instances, as they all have different classes of realizations. However, if we accept PARTHOOD and (ID 4 ) -or any other acceptable criterion of identity mentioned above -they are all identical to each other. We had also seen that according to ONLY, d →0;→0 and d →0,3;→0 are different disposition instances, as they have different classes of triggers; however, in PARTHOOD with (ID 4 ), they are identical. PARTHOOD would thus be able to limit disposition multiplicativism.

Discussion and conclusion

We have distinguished two ways to refer to dispositions: the conventional terms like "fragility", and canonical terms like "disposition to R when TR", where TR is a trigger specification and R is a realization specification. We have proposed two ontological frameworks for the interpretation of canonical specifications: the framework ONLY, that has been endorsed until now in most of the applied ontology literature on dispositions; and the new framework PARTHOOD. The latter introduces minimal triggers, which are the smallest processes that would trigger a disposition, and maximal realizations, which are the largest processes that would realize a disposition. We have seen that PARTHOOD largely avoids the disposition multiplicativism that is implied by ONLY; for this reason, the former may be preferred when ontological parsimony is desired.

Regardless of whether we accept ONLY or PARTHOOD, any single-track disposition is a disposition to Process when Process -that is, Process is always both a trigger specification and a realization specification (but not, of course, a class of triggers or a class of realizations). If a speaker wants to refer to a disposition d, it would be a very unspecific characterization of d to state that it is a disposition to Process when Process, and an interlocutor could not identify which disposition the speaker has in mind. Although a trigger specification and a realization specification do not usually point to a unique disposition instance, the interlocutor can sometimes identify the disposition relying on the pragmatics of communication. So, for example, when the speaker mentions the disposition of glass 0 to Breaking when Strong_shock, the interlocutor can understand that what is referred to is the disposition with full specification "disposition to glass 0 _Breaking when Strong_shock_on_glass 0 ", although this was not explicitly specified [START_REF] Jansen | On Ascribing Dispositions[END_REF]. But to avoid misunderstanding when mentioning a disposition, the speaker can refer to a disposition instance by providing a full specification of it (in the framework ONLY) or a specification of its classes of minimal triggers and maximal realizations (in the framework PARTHOOD). In some rare cases, it might even be necessary to specify the categorical basis of the disposition; and if the categorical basis and a list of trigger-realization associations (in both ONLY and PARTHOOD), or a categorical basis and a specification of the class of minimal triggers (in PARTHOOD) are provided, then it is possible to identify dispositions unambiguously, according to the criteria of identity (ID 4 ) or (ID 3 ''). In general, the more specific the trigger and realization specifications are, the more likely the disposition is going to be correctly identified by the interlocutor.

Note also that in our framework, the bearer of a disposition does not participate in all realizations. Indeed, the bearer of d →0;→2 do,1,2 , which is domino 0,1,2 , does not participate in Fall →2_due_to_a_push→0 (a subclass of Fall →2 ) which is the class of realizations of d →0;→2 do0,1,2 according to ONLY, and a subclass of the class of realizations of d →0;→2 do0,1,2 according to PARTHOOD (unless one would consider that whenever x part_of y and x participates_in r, then y participates_in r, but this would imply that the whole universe participates in any process, which would probably lead to a too broad theory of participation). Instead, we might suggest the axiom according to which the bearer of a disposition always participates in any of its maximal realizations.

In the PARTHOOD framework, it can be very useful to identify a disposition by its minimal triggers and maximal realizations, when we are interested in determining the causal factors with a high precision. However, the description of a disposition by a trigger specification and a realization specification can also be very useful, when we do not know what are the exact causal factors of some effect, and how far their effects extend. The usefulness of minimal triggers and maximal realizations depends on the domain of applications, and whether we want to describe triggers and realizations in a more finegrained or coarse-grained way. Consider dispositions such as Fragility of a Glass, which can have as trigger instances of Fall_glass or of Heavy_object_put_on_glass. Such triggers will be more useful when we deal with, e.g., data concerning handling or shipment of the glasses. But when we deal with, e.g., engineering specifications, we would be more interested in the minimal triggers of Fragility, such as instances of Pressure>P 1 _on_glass.

Some aspects need to be examined in future work. First, our analysis of disposition identity could be combined with the mereology of dispositions [START_REF] Barton | A taxonomy of disposition-parthood[END_REF] and applied, e.g., to biomedical dispositional entities such as predispositions to diseases, to diseases themselves, or medical risks. A second question concerns the diachronic identity of dispositions -namely: what makes that a disposition d at time t and a disposition d at time t' are the same entity? Third, the connection between dispositions and causality needs more elaboration, so that the informal characterization of TRmin and Rmax that has been proposed here could be more systematically spelled out.

(

  ONLY TR Class ) d is a disposition with the trigger specification TR iff (TR SubClassOf Process) and (d has_trigger only TR). (ONLY R Class ) d is a disposition with the realization specification R iff (R SubClassOf Process) and (d has_realization only R). The last clause of (ONLY TR Class ) can be rephrased as '(trigger_of value d) SubClassOf TR', where 'trigger_of value d' (abbreviated in the following 'trigger_of d') is the class whose instances tr are such that 'tr trigger_of d', that is, the class of triggers of d. Analogously, the last clause of (ONLY R Class ) can be rephrased as '(realization_of value d) SubClassOf R', where 'realization_of value d' (abbreviated in the following 'realization_of d') is the class whose instances r are such that 'r realization_of d', that is, the class of realizations of d. The two conditions (ONLY TR Class ) and (ONLY R Class ) can be translated into first-order logic by quantifying over instances: (ONLY TR Instance ) d has the trigger specification TR iff for all tr: (tr instance_of TR → tr instance_of Process) and (d has_trigger tr → tr instance_of TR) (ONLY R Instance ) d has the realization specification R iff for all r: (r instance_of R → tr instance_of Process) and (d has_realization r → r instance_of R))

Figure 1 .

 1 Figure 1. Domino configuration 1

  HAS-PART TR ) (Process and (has_part o trigger_of d)) SubClassOf (trigger_of d) (PART-OF R ) (Process and (part_of o realization_of d)) SubClassOf (realization_of d) (HAS-PART TR ) clearly implies that some parts of the trigger might not play any causal role in leading the disposition to be realized. What causally matters in the mereological sum of a strong shock on glass 0 and my heart beating in triggering disposition_to_break 0 is only the strong shock. Actually, what causally matters might even be only a proper part of the strong shock, such as the final part of the shock, during which is applied a pressure above a threshold value P 1 -let's call the class of such processes Pressure>P 1 _glass 0 . This leads to define the class of minimal triggers of d: the class of triggers of d for which no proper part is a trigger of d. Therefore, we can define the class of minimal triggers of d named "TRmin(d)" as follows: (TRMIN) TRmin(d) EquivalentTo [(trigger_of d) and not (has_proper_part o trigger_of d)] Similarly, we can define the class Rmax(d) of maximal realizations of d as follows as realizations of d who are not proper parts of another realization of d: (RMAX) Rmax(d) EquivalentTo [(realization_of d) and not (proper_part_of o realization_of d)]

(

  TRIGGER) (trigger_of d) EquivalentTo [Process and (has_part some TRmin(d))] (REALIZATION) (realization_of d) EquivalentTo [Process and (part_of some Rmax(d))]

4. 1 . 1 .

 11 Identification by trigger and realization specifications Consider the disposition d 0 inhering in b to R when TR, and the disposition d 1 inhering in b' to R' when TR'. What are the necessary conditions for d 0 and d 1 being identical? A first necessary condition is that b and b' are identical: since dispositions are specifically dependent continuants, two dispositions cannot be identical if they inhere in two different bearers. More specifically, we could impose that the categorical basis cat of d and the categorical basis cat' of d' are identical. This in turn implies that b and b' are identical, since cat inheres_in b and cat' inheres_in b', and a quality instance inheres in only one bearer. A first, natural candidate criterion of identity would be:

  However, we can construct a theoretical model of two different dispositions d and d' that would have the same categorical basis, the same class of triggers (with instances tr1 and tr2) and the same class of realizations (with instances r1 and r2), as pictured on figure 3a. Let us introduce the ternary relation real such that real(tr 1 ,d,r 1 ) iff r 1 is the realization of d when it is triggered by tr 1 . This figure 3a represents the following relationships: real(tr 1 ,d,r 1 ); real(tr 2 ,d,r 2 ); real(tr 1 ,d',r 2 ); and real(tr 2 ,d',r 1

(ID 4 )

 4 d is identical to d' iff: ∃cat (d has_basis cat ∧ d' has_basis cat) ∧ ∀r ∀t (real(tr,d,r) real(tr,d',r))

(ID 5 )

 5 If d and d' are mod-complexes, then d is identical to d' iff: ∀ d'' (d'' proper_mod-part_of d d'' proper_mod-part_of d')

We thank Barry Smith and three reviewers for useful comments. AB and OG acknowledge financial support by the CIHR-funded Quebec SPOR Support Unit.