O Quénard

O Dorival

Ph Guy

MEASUREMENT OF FRACTURE TOUGHNESS OF METALLIC MATERIALS PRODUCED BY ADDITIVE MANUFACTURING

Keywords: Additive Layer Manufacturing, Laser Beam Melting, Ti-6Al-4V, fracture toughness, mechanical properties ALM -Additive Layer Manufacturing AM -Additive Manufacturing

This study focuses on the microstructure and mechanical properties of metallic materials produced by additive layer manufacturing (ALM), especially the Laser Beam Melting process. The influence of the specimen orientation during the ALM process and that of two post-build thermal treatments were investigated. The identified metal powder is Ti-6Al-4V (titanium base). Metallographic analysis show their effects on the microstructure of the metals. Mechanical experiments involving tensile tests as well as toughness tests were performed according to ASTM (American Society for Testing and Materials) norms. The results show that the main influence is that of the thermal treatments; however the manufacturing stacking direction may lead to some anisotropy in the mechanical properties.

INTRODUCTION

For a few decades, additive manufacturing (AM) appeared as on the way of revolutionizing both the design possibilities and the manufacturing standards. It is now used as a regular technique in space, aircraft and dental industry. Design limits were incredibly stretched thank to unprecedented freedom and part complexity. Moreover AM totally fits the use of CAD tools, thus matching topology (and weight) optimization approach well. Global manufacturing time and cost were deeply changed due to the decrease in the number of processing operations and number of parts. Additionally, AM leads to an optimal use of raw material. This paper focuses on the Laser Beam Melting AM technique dedicated to Ti-6Al-4V. AM also means a different manufacturing process resulting in a different microstructure due to rapid solidification rate. Consequently the mechanical properties have to be precisely determined with respect to lots of parameters available (powder properties, machine properties, batch design, post-processing, loading orientation, etc.). Much work has been done to investigate the process-microstructure-tensile property relations for LBM and shed light on the influence of the thermal treatment [START_REF] Thijs | A study of the microstructural evolution during selective laser melting of Ti-6Al-4V[END_REF][START_REF] Facchini | Ductility of a Ti-6Al-4V alloy produced by selective laser melting of prealloyed powders[END_REF][START_REF] Vrancken | Heat treatment of Ti6Al4V produced by Selective Laser Melting: microstructure and mechanical properties[END_REF] or the building orientation [START_REF] Qiu | Microstructure and tensile properties of selectively laser-melted and of HIPed laser-melted Ti-6Al-4V[END_REF][START_REF] Simonelli | Effect of the build orientation on the mechanical properties and fracture modes of SLM Ti-6Al-4V[END_REF][START_REF] Palanivel | Spatially dependent properties in a laser additive manufactured Ti-6Al-4V component[END_REF]. Similar studies were done for EBM [START_REF] Hrabe | Effects of processing on microstructure and mechanical properties of a titanium alloy (Ti-6Al-4V) fabricated using electron beam melting (EBM), Part 2: Energy input, orientation, and location[END_REF][START_REF] Suo | Microstructure and Mechanical Properties of Ti-6Al-4V by Electron Beam Rapid Manufacturing[END_REF][START_REF] Tan | Graded microstructure and mechanical properties of additive manufactured Ti-6Al-4V via electron beam melting[END_REF][START_REF] Formanoir | Electron beam melted Ti-6Al-4V: Microstructure, texture and mechanical behavior of the as-built and heat-treated material[END_REF]. Carroll et al. [START_REF] Carroll | Anisotropic tensile behavior of Ti-6Al-4V components fabricated with directed energy deposition additive manufacturing[END_REF] gave a summary of the tensile properties available in literature for Powder Bead Fusion, Direct Energy Deposition, and Electron Beam Melting techniques. Several papers dealt with the fatigue properties [START_REF] Van Hooreweder | Analysis of Fracture Toughness and Crack Propagation of Ti6Al4V Produced by Selective Laser Melting[END_REF][START_REF] Rafi | A comparison of the tensile, fatigue, and fracture behavior of Ti-6Al-4V and 15-5 PH stainless steel parts made by selective laser melting[END_REF][START_REF] Leuders | On the mechanical behaviour of titanium alloy TiAl6V4 manufactured by selective laser melting: Fatigue resistance and crack growth performance[END_REF][START_REF] Leuders | Fatigue strength prediction for titanium alloy TiAl6V4 manufactured by Selective Laser Melting[END_REF][START_REF] Gong | Influence of defects on mechanical properties of Ti-6Al-4V components produced by selective laser melting and electron beam melting[END_REF][START_REF] Cain | Crack propagation and fracture toughness of Ti6Al4V alloy produced by selective laser melting[END_REF][START_REF] Edwards | Effect of build direction on the fracture toughness and fatigue crack growth in selective laser melted Ti-6Al-4 V[END_REF][START_REF] Seifi | Evaluation of Orientation Dependence of Fracture Toughness and Fatigue Crack Propagation Behavior of As-Deposited ARCAM EBM Ti-6Al-4V[END_REF][START_REF] Zhao | Comparison of the microstructures and mechanical properties of Ti-6Al-4V fabricated by selective laser melting and electron beam melting[END_REF][START_REF] Mower | Mechanical behavior of additive manufactured, powder-bed laser-fused materials[END_REF][START_REF] Kasperovich | Correlation between porosity and processing parameters in TiAl6V4 produced by selective laser melting[END_REF][START_REF] Greitemeier | Fatigue performance of additive manufactured TiAl6V4 using electron and laser beam melting[END_REF]. Li et al. [START_REF] Li | Critical assessment of the fatigue performance of additively manufactured Ti-6Al-4V and perspective for future research[END_REF] compared fatigue properties published in open literature for LBM and EBM.

However to the author knowledge only few papers focused on the fracture toughness of AM Ti-6Al-4V [12, 17-19, 23, 25], although this is a key point in the damage tolerance approach. Table 1 gives an overview of the data available in open literature for Ti-6Al-4V toughness obtained with Laser Beam Melting (LBM) and Electron Beam Melting (EBM) techniques. There is huge scatter between the various results for a given technique; however the dependency on the heat treatment and on the layering direction was reported by most of the authors.

The present contribution focuses on the microstructure-tensile-toughness relations of Ti-6Al-4V with respect to sample orientation and heat treatment used. LBM technique was used to manufacture several specimens, including metallography cube, tensile and fracture toughness specimen. The various specimens were produced in the same batch to ensure unique manufacturing conditions. Each batch received, or not, a particular heat treatment. In the end, samples had to experience several stages: ALM process, thermal treatments, machining, fatigue pre-cracking, final mechanical tests. The statistical analysis required a fine tracking of each specimen to be sure the conclusions could be drawn concerning the influence of the parameters of interest.

The paper is organized as follows: after introducing the samples preparation and the tests set-up, results of metallographic analysis, tensile tests, and toughness tests are presented, followed by a discussion.

SAMPLES PREPARATION AND TESTING METHODS

Materials, manufacturing parameters and thermal treatment

Two batches of Ti-6Al-4V specimens were manufactured. Each batch was manufactured during a single job to ensure that the same manufacturing parameters were used for all the samples. For the same reasons the manufacturing parameters were also kept the same for the two batches, and the jobs were done in a row to avoid too much recycling of the powder between the two batches.

Batch #1 and Batch #2 were manufactured from Ti-6Al-4V powder from Advanced Powders and Coatings, with 15-45 µm particle size. The powder had been recycled 14 times for batch #1 and 15 times for batch #2. The batches were manufactured in a EOS M280 machine using a 1060-1100 µm wavelength laser. Layers were 60 µm thick. Both batches followed a stress relieving treatment on the base platform (670°C / 5h) to release residual stresses. The as-obtained Batch #2 was submitted to an additional Hot Isostatic Pressing (HIP) process (920°C / 2h / 1020 bars).

Both batches were then ready to be machined to obtain the expected shape and surface quality (Ra = 0.8), allowing mechanical testing. (**) Edwards et al. [START_REF] Edwards | Effect of build direction on the fracture toughness and fatigue crack growth in selective laser melted Ti-6Al-4 V[END_REF] used plain stress specimens instead of plain strain ones; this is likely to have given higher toughness values.

(***) Seifi et al. [START_REF] Seifi | Evaluation of Orientation Dependence of Fracture Toughness and Fatigue Crack Propagation Behavior of As-Deposited ARCAM EBM Ti-6Al-4V[END_REF] used bending toughness specimen whereas most authors used CT specimens.

Tensile and toughness specimens and specimen orientations

The aim of the project required to get all the samples of a given batch in one job. For the sake of a statistical analysis and due to the dimensions of the manufacturing zone, dimensions of the samples had to be sufficiently small. For tensile tests, standard round specimens were designed using the following dimensions specified by ASTM E8 norm [START_REF]Standard Test Methods for Tension Testing of Metallic Materials[END_REF]: diameter for the section: 6 mm, length of the reduced section: 27 mm, total length: 60 mm. Seven samples were manufactured for each orientation/material/thermal treatment. After the ALM process and the thermal treatments, specimens had to be machined (mainly by turning process) to achieve good dimensional quality and to avoid any dependency on the quality of the surface shape.

For toughness tests, compact (CT) specimens were designed according to ASTM E399 norm with 32 mm specimen width (W32-B16). This allowed to manufacture five samples for each combination of material/orientation/thermal treatment. After the ALM process and the thermal treatments, specimens were machined to get the corresponding dimensions and to create the notch. A fatigue crack was created and propagated according to ASTM E399 norm recommendations. Unfortunately, manufacturing difficulties appeared during Ti-6Al-4V process and lead to several cracks in the samples during or after the manufacturing stage. Consequently such samples were missing for the toughness tests, as will be detailed later. For each group of samples, the number of tested specimens will be given in the corresponding tables.

The ALM process consists in melting the part layer by layer, favouring the anisotropy of direction z of the "stacking". Consequently, three particular orientations of the samples were identified for the tensile specimens, as shown in Figure 1. Note that for tensile specimens, this direction corresponds to the loading axis: vertical (denoted by "V"), horizontal ("H"), slanted at 45° ("O"). Note that inplane directions x and y were not distinguished since the laser advance direction is turning of 30° when changing for the next layer. For the toughness specimens however, one needs to report both the building plane and the loading axis due to multiple crack orientations. Consequently four different orientations were identified: horizontal (resp. oblique) specimens gave only horizontal (resp. oblique) loading direction denoted by "H" (resp. "O"), whereas vertical specimens could lead to both a vertical notch and horizontal loading direction (denoted by "V/H"), and a horizontal notch and vertical direction ("V/V").

Figure 1: Orientations of the samples for tensile and toughness tests 2.3 Metallographic analysis

Preparation of the samples consisted in polishing, under water as lubricant, using a PRESI device (reference: MECAPOL P 220 U). Sequential grinding with silicon carbide (SiC) abrasive paper was performed for the preliminary steps, using water as lubricant. Four grades were used (P320, P800, P1200 and P4000), the corresponding abrasive grains median diameter of which are 46.2 µm, 21.8 µm, 15.3 µm and 2.5 µm respectively.

A specific Silica suspension (particle size: 0.04 µm, PRESI reference: SPM) was used for the finishing.

No additional lubricant was necessary in this case. The best results were obtained by controlling the parameters (rotation speed v, load F and duration t) as follows:

 for P320: v=300 rpm, F=25 N, t=1 min;

 for P800: v=300 rpm, F=25 N, t=0.5 min;

 for P1200: v=300 rpm, F=25 N, t=0.5 min;

 for P4000: v=300 rpm, F=25 N, t=0.5 min;

 for silica suspension, v=150 rpm, F=15 N, t=2 to 4 min.

To reveal the microstructure of the specimen, Kroll''s reagent (6 mL of HNO3, 3 mL of HF and 91 mL of H2O) was used: 2 min 30 sec at room temperature.

Optical observations were performed with a ZEISS Stemi 2000 binocular (for magnifications ranging between x0.65 to x5) and a ZEISS Axiotech microscope (for magnifications ranging between x5 to

V H O V/V V/H O H direction z
of layer stacking x100). In both cases, a PRESIVIEW software allowed acquiring images. For larger magnification (up to x10 000), a HITACHI TM1000 scanning electron microscope (SEM) was available.

One must note that observations were performed along two orientations:

 perpendicularly to the plane of manufacturing. The corresponding observation (horizontal cross section) can then reveal the microstructure of a layer;

 perpendicularly to the layers of the material. The corresponding observation (vertical cross section) reveals the homogeneity along the height of the sample, from the first to the last fused layer.

For practical reasons, we will call these two orientations as horizontal and vertical respectively in the following.

Tensile and toughness tests set-up

Tensile tests were performed on a Instron electromechanical testing machine of type UTM-EM.

Machine driving was performed by Bluehill 2.21 software. A 2525-801 type 100 kN Instron load cell was used to record the tensile force. A 12.5mm 2620-601 Instron clip-on extensometer was used to measure the axial strain of the specimen. Tests were performed according to recommendation of norms ASTM E8 [START_REF]Standard Test Methods for Tension Testing of Metallic Materials[END_REF] and NF EN 10002-1 [START_REF]Matériaux métalliques -Essai de traction -Partie 1 : méthode d'essai (à la température ambiante)[END_REF]. Load rate of 2 MPa/s was used till reaching 70 MPa.

Then strain rate of 0.00025 s -1 was prescribed. At about 2% strain, the clip-on extensometer was removed manually, and strains were measured directly by the displacement of the load frame.

Toughness tests were performed on the same Instron testing machine using Bluehill software 2.21.

Apparatus involved the same 100 kN load cell and a 2670-130 Crack Opening Displacement Gauge from Instron. Tests were performed following the ASTM E399 recommendation [START_REF]Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness KIc of Metallic Materials[END_REF]. Load rate of 0.33 kN/s was used up to global failure of the specimen.

RESULTS FOR TI-6AL-4V

Results of metallographic analysis for Ti-6Al-4V alloy

Batch #1: Stress-relieved Ti-6Al-4V

Observations at low magnification revealed a good microstructural homogeneity of the sample, whatever the orientation. This result could prove that the cooling rate was similar during the manufacturing, whatever the number of layers. The mass of obtained sample has no (or not optically detectable) influence on the resulting microstructure.

Horizontal: At low magnification, each layer is composed of lines parallel one to each other, each of them showing a succession of more or less regular squares (see Figure 2). This typical microstructure reflects the successive scanning orientations of the laser beam during the manufacturing. Indeed, for each layer, the laser beam produced a melted bead and, after recrystallization, a line of solid material (the width of which is ranging between 100 and 120 µm) aligned to the direction of the laser beam displacement. The direction of the beam changing from layer to layer, and the induced fusion affecting not only the concerned layer but also the 2-3 lower layers, the recrystallization produced square grains detectable along each line. Observing several orientations of the lines indicates that the polishing affected several successive layers. is composed of layers, the obtained grains after recrystallization have a columnar structure (see Figure 4), the orientation of which is perpendicular to the layers. This microstructure is only possible by fusion of the lower layers, leading to a melted bead with a "keyhole" section when scanning the laser beam [START_REF] Paradis | Monographies techniques du DFTN : soudage par laser[END_REF]30]. Figure 4 confirmed that the grains are made of ' martensitic phase. As a result, it appears that the microstructure of the Ti-6Al-4V samples obtained by LBM is quite homogeneous, even if it does not seem to be isotropic, nor in the layer plane, nor in the height plane.

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Batch #2: Ti-6Al-4V after stress-relief and HIP

The HIP treatment was expected to eliminate, or at least reduce, the porosity rate in the obtained samples. Among all the potential techniques, we estimated the porosity rate by observing the polished surface of the sample and detect the porosity by difference of colour. The surface of the sample appears dark while porosity appears as white dots (see Figure 5). As a first result, we can say that the HIP finally did not reduce the porosity rate, maybe due to the very low proportion of porosity (lower than 0.2%) in the initial material. sample (see Figure 6), even if some residual anisotropy can be detected. The lines and squares largely vanished. One can note at low magnification (Figure 8, left) that the anisotropy is not fully blurred, as layers are still slightly visible, as well as some residual columnar grains. We could then estimate that the anisotropy is more strongly present in the vertical plane than in the horizontal one. Higher magnification (Figure 8, right) allows to conclude that the  phase appears along the grain boundaries between the columnar grains, as expected.

 phase  phase

Results of tensile tests

Two typical stress-strain curves obtained for samples of batches #1 and #2 are shown in Figure 9.

Detailed results including Young's modulus (E), tensile yield stress (YS), ultimate tensile stress (UTS) and elongation (A%) for batches #1 and #2 are reported in Table 2 and Table 3 respectively. Unlike toughness specimen, tensile specimens did not experience cracking during the ALM process thank to more slender shape, and all specimens manufactured could be tested. Except for elongation, results for batch #1 were rather repeatable. For Young's modulus, yield stress and ultimate stress, the slight differences between the various orientations are hardly significant and the material can be considered as quasi-isotropic for those properties. However a moderate 16% increase in the elongation tends to appear for the vertical direction (building direction), although low repeatability maintains doubt about this.

Samples of batch #2 were found to provide very repeatable results including the elongation. Again the material looks almost isotropic including the elongation. This tends to confirm that the HIP treatment removed the slight z-anisotropy noted in the elongation in batch #1.

Comparisons between batch #1 and batch #2 lead to the most interesting conclusions. Young's modulus seemed to have slightly decreased (by 4.6%) after the HIP treatment, but the difference is not high enough to be sure that it is relevant. Important drops were noticed concerning the yield stress (-19.2%) and the ultimate stress (-14.7%). On the opposite elongation increased by 59%. As a conclusion, the HIP treatment did not lead to any improvement in the tensile characteristics, except the large 59% increase in the inelastic deformation of the material. (3.6%) 17.1% (6.4%)

Fracture surface analysis of tensile tests

Batch #1

The overall view of the tensile fracture surface of the vertical as built samples is presented in Figure 10a. It appears that failure mechanism was mostly ductile as a transgranular fracture surface with a network of dimples can be largely observed (see Figure 10b). Some defects can also be observed, such as porosity (resulting from microvoid coalescence during the manufacturing process), mostly located in the center part of the section of the sample (see Figure 10b), even if this porosity does not seem to have a strong influence on the mechanical properties, at least for tensile tests. One could assume that the porosity could be the main cause of the fracture [START_REF] Rafi | A comparison of the tensile, fatigue, and fracture behavior of Ti-6Al-4V and 15-5 PH stainless steel parts made by selective laser melting[END_REF]. Nevertheless, another defect seems to have much more influence on the mechanical behaviour of the sample: the micro-cracks. They were mostly detected in the external part of the section of the sample (see Figure 10c), even if few of them can be observed in the centre of the section. These micro-cracks were attributed in our case to the residual stress in the sample after the laser manufacturing, which was not erased after the stress relieving treatment. The large number of such micro-cracks at the fracture surface should underline their strong influence on the mechanical performance of the stress-relieved Ti-6Al-4V sample. This conclusion was confirmed by the brittle surfaces detected around most of these micro-cracks, leading to cleavage facets and a layered profile (see Figure 10c-d). These brittle surfaces were attributed to cleavage within layers of the material, joining two ductile surfaces, which were the interface between two successive layers. Nevertheless, in such a mixed ductile-brittle behaviour, the ductility was largely dominating. Similar results were obtained by observing samples in other directions (horizontal and sidelong), confirming the quasi-isotropy of the obtained LBM samples.

Batch #2

As a first result, one must note that at the lowest magnification (x40), the whole fracture surface of the vertical sample after HIP could be observed (see Figure 11a), what was not possible for the stress-relieved sample. This remark means that the HIP treatment lead to a higher ductility, and so to a larger necking, confirming the larger elongation measured for such samples during the tensile tests.

Then, it appeared that there was no porosity detected on the fracture surface, proving that if there was residual porosity in the whole sample (as noted in Figure 5), it had no influence on the mechanical performances of the HIPed sample. The thermal treatment seems to be very efficient to increase the plastic deformation of LBM manufactured Ti-6Al-4V.

Increasing magnification during the observation allowed revealing a pure ductile behaviour of the sample, characterized by the dimples network (see Figure 11b). Such a behaviour could be confirmed by both a larger necking of the sample and a "cup and cone" aspect (typical of ductile material). Some few micro-cracks were still remaining (see Figure 11c), but they did not give rise to cleavage facets attributed to brittle behaviour. The final overloaded surface (see Figure 11d) also presented a ductile aspect, but the size of the dimples is much smaller. This result can be explained as the duration of final was too short to allow large plastic deformation of the corresponding area, and then to lead to larger dimples.

Similar results were obtained for other directions (horizontal and sidelong).

Results of toughness tests

Results of toughness tests for Ti-6Al-4V batches #1 and #2 are reported in Table 4 and Table 5 respectively. This includes the number of tested samples, the number of pre-cracking fatigue cycles and the results of plane strain fracture toughness KIC.

Unlike tensile samples, let us first remark that several CT samples experienced cracking during or immediately after the ALM process. This was due to residual stresses in such compact parts, although samples were "only" 19 mm thick (aiming at 16 mm after machining).

Batch #1 showed very repeatable values. The number of fatigue cycles was similar, except for the "V/V" orientation that was 20% lower. Fracture toughness was almost equal for "H", "O" orientation.

On the contrary, the "V/H" and "V/V" samples were found to show a 17% increase and a -4% decrease, respectively.

Results of batch #2 tests were also found to be very repeatable. Fatigue cycling was similar for all orientations, and was 19% lower than that of batch #1. Fracture toughness was equal for "H" and "O" orientations. Although "V/H" and "V/V" very low differences (+5% and -7.5% respectively) could be attributed to scatter, it is interesting to note that they have the same tendency than in batch #1, that is: slightly higher toughness for "V/H" samples, and lower but similar values for "V/V", "H" and "O" samples.

The most important point is the 57% average increase (+48% to +63% according to the orientations) in the average fracture toughness of batches #1 and #2. Unlike the tensile characteristics, the HIP treatment greatly improved the fracture strength of the samples. (**) The results for KIC for batch #2 were not all considered as valid according to norm ASTM E399, due to the Pmax/PQ condition. Specimens that failed to meet the condition were very close to reach it though.

(***) According to the Pmax/PQ condition.

DISCUSSION

One must first note that if the LBM process lead to metallographic anisotropy of the material, such anisotropy was not significantly observed in the measured tensile properties. Nevertheless, the layered microstructure has a notable influence on the toughness values, as the "V/H" samples exhibited higher characteristics than the other three sample orientations ("O", "H", "V/V"). These later presented similar toughness values. This could be easily explained by the fact that the interface between layers acted as grain boundaries, limiting the crack propagation and so leading to a higher toughness. This explanation could also explain why the "H" and "V/V" samples showed similar toughness (no additional layer interface to overcome). The "O" values were surprisingly similar to the "H" and "V/V" ones, although it could be expected to exhibit an intermediate behaviour between

the "H" and "V/H" ones.

The HIP treatment applied almost restored the complete isotropy to the samples from both a metallographic and tensile properties points of view. The anisotropic behaviour was still remaining in the toughness test for the "V/H" orientation. As presented in other works, the HIP treatment (the role of which is to reduce or erase any porosity) mostly allowed larger plastic deformation (+59%) by strongly reducing the residual stresses induced by the LBM manufacturing. This higher plasticity was twined to a notable decrease in the yield and ultimate stresses. Unlike in [START_REF] Qiu | Microstructure and tensile properties of selectively laser-melted and of HIPed laser-melted Ti-6Al-4V[END_REF]31], observations did not reveal any notable decrease in the porosity rate, but this may be attributed to a very low initial rate: 0.2%. Nevertheless, the potential residual porosity was not detected on the fracture surface. This indicates that the plasticity was sufficient to strongly reduce the influence of the porosity, which was not anymore a crack initiation location. This could explain the resulting 57% increase in the toughness of the HIPed samples. The HIP heat treatment is then recommended for LBM manufactured Ti-6Al-4V samples submitted to fatigue solicitations.

CONCLUSIONS

The obtained results seem to be promising for the Ti-6Al-4V material even if some other characterization are still missing. Further studied are under investigation concerning the toughness fracture surface and the influence of potential defects within the raw material.

Similar investigations are also in progress for AlSi7Mg0.6 material. The influence of building orientation and heat treatments after LBM manufacturing, as well as internal defects, will be soon presented in another paper.

Figure 2 :

 2 Figure 2: Microstructure observed for the Ti-6Al-4V sample at low magnifications

Figure 3 :

 3 Figure 3: Martensitic microstructure of the Ti-6Al-4V sample

Figure 4 :

 4 Figure 4: Microstructure along the height of the Ti-6Al-4V sample at low (left) and high (right) magnification

Figure 5 .

 5 Figure 5. Porosity (white dots) detected in the Ti-6Al-4V sample before (left) and after (right) the HIP treatment

Figure 6 .

 6 Figure 6. Microstructure observed for the Ti-6Al-4V sample after HIP at low magnifications

Figure 7 .

 7 Figure 7. Microstructure observed for the Ti-6Al-4V sample after HIP at high magnification

Figure 8 .

 8 Figure 8. Microstructure along the height of the Ti-6Al-4V sample after HIP at low (left) and high (right) magnification

Figure 9 -

 9 Figure 9 -Typical stress-strain results for titanium specimen

Figure 10 .

 10 Figure 10. Fracture tensile surface of the vertical stress-relieved sample. (a): overall view of the fracture surface. (b): defects detected in location 1 in (a). (c): microcracks detected in location 2 in (a). (d): brittle fracture surface in location 3 in (a).

Figure 11 .

 11 Figure 11. Fracture tensile surface of the vertical sample after HIP treatment. (a): overall view of the fracture surface. (b): observed ductile behaviour. (c): microcrack detected in location 1 in (a). (d): magnification in location 2 in (a).

Table 1 : Results of LBM and EBM Ti-6Al-4V toughness reported in literature AM technique Reference Scanning strategy Post processing

 1

	Sample
	orientation	Surface	KIC/KQ
	/ load	finishing	(*)
	orientation

Table 2 : Results of tensile tests for batch #1

 2

	Specimen orientation	H	O		V
	Number of tested samples	7	7		7
	E [GPa]	118.2 (7.4%) 118.0	(3.5%)	119.2	(2.5%)
	YS [MPa]	1085 (1.7%) 1098	(2.0%)	1116	(1.9%)
	Aver. (std dev %)
	UTS [MPa]	1166 (1.5%) 1166	(1.9%)	1155	(6.2%)
	A%	9.9% (5.3%) 9.9% (18.4%) 11.5% (10.7%)

Table 3 -Results of tensile tests for batch #2

 3

	Specimen orientation	H		O	V
	Number of tested samples	7		7		7
	E [GPa]	114.6	(1.6%)	111.6	(3.2%) 112.8 (0.9%)
	YS [MPa]	896	(0.6%)	909	(0.8%)	861	(4.3%)
	Aver. (std dev %)
	UTS [MPa]	995	(0.5%)	1013	(0.6%)	967	(2.9%)
	A%	16.2% (2.1%) 16.6%

Table 4 : Results of toughness tests for batch #1

 4

	Specimen orientation	H	O			V/H		V/V
	Nb of samples (*)	(valid) (***)	4	(4)	4	(4)	4	(4)	3	(3)
	Ncycles	Aver.		46275		48765	48639	40732
	KIC/KQ(**) [MPa.√m] Aver. (Std dev %)	40.3 (3.4%) 38.9 (6.3%) 45.6 (4.3%) 38.1 (1.6%)
	(*) Due to manufacturing hazards, several samples of each orientation in batch #1 were cracked
	and could not be tested.
	(**) The results for KIC for batch #1 were all valid according to ASTM E399.

(***) According to the Pmax/PQ condition in ASTM E399.

Table 5 -Results of toughness tests for batch #2

 5

	Specimen orientation	H	O			V/H		V/V
	Nb of samples (*)	(valid) (***)	5	(4)	2	(2)	4	(3)	2	(2)
	Ncycles	Aver.		38471		39998	38549	38050
	KIC/KQ(**) [MPa.√m] Aver. (Std dev %)	64.9 (1.6%) 63.6 (0.5%) 67.3 (5.3%) 59.4 (0.2%)
	(*) Due to manufacturing hazards, several samples of each orientation in batch #2 were cracked
	and could not be tested.

ACKNOWLEDGEMENTS

The authors wish to acknowledge Centre National des Etudes Spatiales (CNES) for support through grant N°160025/00 and implication during the recurrent informal meetings.

We also thank FusiA company for providing the heat treated manufactured samples, and Thales Alenia Space for the technical discussions.

Our acknowledgement also to Exova company for preparing the toughness samples.


Francesca Poggiolesi
email: poggiolesi@gmail.com

Bolzano, (the appropriate) relevant logic and grounding rules for implication

Keywords: Bolzano, Tennant, grounding, implication, relevant logic

In this paper our goal is double. First of all we aim to show that there is a deep link between Bolzano's notion of exact derivability and Tennant's relevant logic CR. Secondly, we aim to argue that Tennant's relevant logic CR is an adequate framework for developing interesting grounding rules for the implication connective.

Introduction

Although neglected for many decades, the great Bohemian thinker Bernard Bolzano is today the center of a renewed interest and enjoys the respect that he deeply deserves: he is indeed considered an outstanding mathematician, an exceptional philosopher and one of the greatest logicians who ever lived (see [START_REF] Morscher | Bernard Bolzano[END_REF]). This renewed enthusiasm towards Bolzano seems to go hand in hand with the contemporary attention to the notion of grounding, or metaphysical explanation (e.g. see [START_REF] Correia | Grounding: an opinionated introduction[END_REF]; [START_REF] Fine | Some puzzles of ground[END_REF]; [START_REF] Poggiolesi | On defining the notion of complete and immediate formal grounding[END_REF]; [START_REF] Schnieder | A logic for 'Because[END_REF]); Bernard Bolzano can indeed be seen as one of the major figures in the history of the philosophy of this notion. His conception of grounding, together with his attempt at logically characterizing it, are milestones in the analysis of this concept.

As pointed out in several studies, in Bolzano's conception of grounding, the notion of exact derivability (Ableitbarkeit), 1 that Bolzano himself introduced (Wissenschaftslehre II, §155), plays a special role. Roughly speaking the general idea is that the notion of complete grounds2 of some truth A always presupposes a relation of exact derivability (under certain restrictions) of A from the set of its complete grounds.

In the first part of this paper the main aim is to have a closer look at exact derivability and its relations to contemporary logic. More precisely, we will show that there exists a particular connection between Bolzano's notion of exact derivability and the relevant logic CR introduced by [START_REF] Tennant | Perfect validity, entailment and paraconsistency[END_REF]. These reflections will not only clarify the notion of exact derivability per se, but also, in virtue of the connection between exact derivability and grounding, shed further light on the general Bolzanian conception of metaphysical grounding.

In the second part of the paper we will focus on the recent studies on the logic of grounding and in particular on the grounding rules for implication, a topic that, as far as we know, has received relatively little treatment. We will try to argue that the logic CR can again play a role: it is indeed an useful and interesting framework for formulating the grounding rules for implication.

The paper is organized as follows. In Section 2 we will recall Bolzano's notions of derivability and exact derivability emphasizing their logical properties. In Section 3 we will introduce the logic CR, while in Section 4 we will explain why this is the adequate contemporary counterpart of Bolzano's notion of exact derivability. We will use Section 5 to discuss the issue of the grounding rules for implication. We will show that even in this context the logic CR has a role to play.

Deducibility and exact deducibility

In the Wissenschaftslehre Bolzano famously introduced two concepts that are now part of the history of logic, namely the notion of derivability and the notion of exact derivability. Many valuable studies (e.g. [START_REF] Morscher | Bernard Bolzano[END_REF]; [START_REF] Sebestik | Forme, variation et déducibilité dans la logique de Bolzano[END_REF]) have been dedicated to the analysis and reconstruction of these concepts: amongst other things, they have highlighted some important properties of derivability and exact derivability.

In this section our aim is double: on the one hand, we will introduce the concepts of derivability and exact derivability, on the other hand we will try to summarize the properties and the logical principles that have been associated with these concepts. In accomplishing these two tasks, we will mainly summarize some recent studies on Bolzano's work (e.g. [START_REF] Berg | Bolzano's Logic[END_REF]; [START_REF] Morscher | Bernard Bolzano[END_REF]; [START_REF] Rusnock | Bernard Bolzano, Theory of Science[END_REF]) without however entering into the details of their different interpretations.

Before starting dealing with our double task, let us underline two points. The first point concerns Bolzano's conceptual framework that is behind the notions of derivability and exact derivability, that is thus useful for understanding them but which, nonetheless, for reasons of space, we omit to present here. The reader who is not acquainted with such a framework is referred to the Introduction of this volume. The second point concerns the choice of not being entirely faithful to Bolzano's original work; in the following aspects we will indeed opt for a more modern and elegant formulation than the one used by Bolzano. (i) We follow the standard conception of propositions and not that of Bolzano according to which all propositions have the form [A has B], see [START_REF] Betti | Bolzano's universe: Truth, logic andmetaphysics[END_REF]; [START_REF] Casari | An interpretation of some ontological and semantical notions in Bolzano's logic[END_REF]; (ii) we work with sets of propositions, while Bolzano uses collections of propositions; (iii) we allow empty set of premises, while Bolzano doesn't.

In the Wissenschaftslehre §147-168 Bolzano introduces his theory of variation which is one of its central contributions to the field of logic. The heart of the theory analyses the semantic links that arise amongst propositions as soon as certains of their elements are substituted in a homogeneous way. In this framework Bolzano singles out several relationships, among which the relation of derivability stands out. He defines this notion in the following way.

Definition 2.1. Propositions A 1 , ..., A n are derivable from propositions B 1 , ..., B m with respect to the sequence of components P 1 , ..., P r ¡, which occur in the propositions A 1 , ..., A n as well as B 1 , ..., B m3 , if, and only if:

(1) the premisses B 1 , ..., B m are compatible with respect to P 1 , ..., P r ¡, which is to say that there exists a sequence of components Q 1 , ..., Q r ¡ that, if substituted with P 1 , ..., P r ¡, make simultaneously true all the premisses.

(2) each sequence that, substituted with P 1 , ..., P r ¡, make true all the premisses, make true all the conclusions too.

Let us make a few comments about this definition. First of all, the definition is composed of two items: the first expresses the idea of the premisses being compatible, the second expresses the idea of truth-preservation from premisses to conclusion. Truth-preservation as well as compatibility are relativized to a sequence of components of propositions; it is important to underline what kind of components might be part of that sequence. Translated in contemporary terms variation can concern the sequence of all individual, functional and predicative symbols, all molecular terms as well as all subsentences contained in propositions (in this case we have a notion of logical derivability). Variation can also concern a proper subsequence of that sequence (and in that case the descriptive symbols are constants as the logical symbols). 4Let us first give an exemple of a set of propositions which are compatible with respect to a sequence of components. We use Bolzano's exemple and we thus consider the propositions "this flower is red", "this flower has a pleasant fragrance", and "this flower belongs to the twelfth class of Linne's system": these propositions are compatible with respect to the variable idea "this flower", because all three are true if we substitute "this flower" with "this rose".

Let us now move to propositions which stand in a truth-preserving relation from premisses to conclusion with respect to a certain sequence of components. Let us consider the propositions "the triangle ABC is isosceles", "the triangle ABC is equilateral". Each variable idea that, substituted with ABC, makes true "the triangle ABC is equilateral", makes true "the triangle ABC is isosceles" as well. Thus "the triangle ABC is equilateral" is the premiss and "the triangle ABC is isosceles" the conclusion of a relation which is truth-preserving with respect to the variable-idea ABC. (The two propositions nevertheless do not stand in a logical derivability relation.)

As many (e.g. [START_REF] Etchemendy | The concept of logical consequence[END_REF]; [START_REF] Sebestik | Forme, variation et déducibilité dans la logique de Bolzano[END_REF]) have suggested, Bolzano's notion of derivability anticipates in many respects the Tarskian notion of logical consequence: a common idea that underlies both definitions is truth-preservation from premises to a conclusion under certain variations. While from the Tarskian perspective, variation concerns all non-logical elements, from the Bolzanian's perspective variation might vary and non-logical elements may remain constant. In this respect Bolzano's notion is much more general than the Tarski's idea of logical consequence. In another respect, however, the notion is narrower. For contrary to Tarski, Bolzano requires the premises to be compatible with respect to a certain sequence of components.

From now on we adopt the following notations. Let M, N, ... denote sets of propositions. We write:

-M | ù P1,...,Pr¡ N to denote that the set N is derivable from the set M with respect to the sequence of components P 1 , ..., P r ¡. -M * P1,...,Pr¡ N to denote that the set N is not derivable from the set M with respect to the sequence of components P 1 , ..., P r ¡.

-M * N to denote that the there exist no sequence of components P 1 , ..., P r ¡ with respect to which the set M is derivable from the set N .

Let us now move to the main properties that this relation of derivability satisfies. These properties have been emphasized by (Roski, 2014, Ch. 3.3.4), [START_REF] Siebel | Bolzano's concept of consequence[END_REF], [START_REF] Stelzner | Compatibility and relevance: Bolzano and Orlov[END_REF].

Restricted Monotonicity For all M, M 1 and N , if M | ù P1,...,Pr¡ N , M M 1 and the propositions contained in M 1 are compatible with respect to P 1 , ..., P r ¡, then M 1 | ù P1,...,Pr¡ N .

The relation of derivability is monotonic up to compatibility; indeed, if M | ù P1,...,Pr¡ N and the set M is enriched with some propositions A such that A is not compatible with M with respect to P 1 , ..., P r ¡, then N is not derivable from M Y A with respect to P 1 , ..., P r ¡.

Restricted Cut For all M, N, S, T and A, if M | ù P1,...,Pr¡ N, A and A, S | ù P1,...,Pr¡ T , and M and S are compatible with respect to P 1 , ..., P r ¡, then M, S | ù P1,...,Pr¡ N, T .

As it was the case for non-monotonicity, even what is known as cut only holds in a restricted form: one can cut a formula and thus unite two different derivability relations as long as the new set of premisses still satisfies compatibility with respect to a certain sequence of components.

Transitivity For all M, N, S, if M | ù P1,...,Pr¡ N and N | ù P1,...,Pr¡ S, then M | ù P1,...,Pr¡ S.

Bolzano's derivability relation is transitive without limitation. Let us remind the reader, since it will be useful in what follows, that transitivity is nothing but a special case of cut: it indeed amounts to cut with no added premisses in the middle step.

Let us now move to two logical principles that are associated with the derivability relation. These logical principles have been investigated by [START_REF] Berg | Bolzano's Logic[END_REF]; [START_REF] George | Bolzano's consequence, relevance, and ethymemes[END_REF]; [START_REF] Siebel | La notion bolzanienne de déductibilité[END_REF]; [START_REF] Stelzner | Compatibility and relevance: Bolzano and Orlov[END_REF]. They will be of crucial importance in the next section.

1. A _ B, A | ù A,B¡ B 2. A, A * B
The first principle is quite well-known and is standardly called disjunctive syllogism; it is also a valid principle in classical and intuitionistic logic. It says that if a disjunction is true and one of its disjunct is false, then the other disjunct must be true. It is quite straightforward to see its validity according to Definition 2.1.

Let us analyse the second principle which amounts to the block of famous Lewis's first paradox (see [START_REF] Lewis | Symbolic Logic[END_REF]); it indeed says that there exists no sequence of components with respect to which B is derivable from propositions A and A. This principle has been the object of debate. Indeed, although everybody agrees that Bolzano would have never accepted the passage from A and A to B as valid, there are two different interpretations concerning what Bolzano's reasons might have been. An accurate presentation of these different interpretations would go beyond the scope of the paper and thus we leave it aside; let us however underline that it is a very interesting debate (in particular, see [START_REF] George | Bolzano's consequence, relevance, and ethymemes[END_REF] and [START_REF] Stelzner | Compatibility and relevance: Bolzano and Orlov[END_REF] for one interpretation of the Bolzanian block of Lewis's first paradox; and [START_REF] Berg | Bolzano's Logic[END_REF] and [START_REF] Siebel | La notion bolzanienne de déductibilité[END_REF] for the other interpretation).

It is now time to move to the notion of exact derivability. Bolzano introduces two different notions of exact derivability: one in the Wissenschaftslehre §155.26, the other in the Von der mathematischen Lehrart, §8. According to (Rusnock and George, 2015, Vol. II, p. xxxvi), the latter notion of exact derivability is an improved version of the first: Bolzano spotted a flaw in the definition of exact derivability given in the Wissenschaftslehre. We thus embrace the definition of exact derivability given in Von der mathematischen Lehrart. Definition 2.2. Propositions A 1 , ..., A n are exactly derivable from propositions B 1 , ..., B m with respect to the sequence of components P 1 , ..., P r ¡, , which occur in the propositions A 1 , ..., A n as well as B 1 , ..., B m , if, and only if:

-A 1 , ..., A n | ù P1,...,Pr¡ B 1 , ..., B m -no A i , 1 ¤ i ¤ n,
may be removed from the set A 1 , ..., A n without the loss of derivability of B 1 , ..., B m from the remainder with respect to P 1 , ..., P r ¡.

Analogously to our notation for derivability, we write:

-M ( P1,...,Pr¡ N to denote that the set N is exactly derivable from the set M with respect to the sequence of components P 1 , ..., P r ¡; -M * P1,...,Pr¡ N to denote that the set N is not exactly derivable from the set M with respect to the sequence of components P 1 , ..., P r ¡; -M * N to denote that there is no sequence of components with respect to which the set N is exactly derivable from the set M .

The idea behind the notion of exact derivability is rather simple: exact derivability is derivability plus the condition that no premiss can be omitted without losing derivability itself. On the one hand, exact derivability keeps the most salient ingredients of derivability, namely compatibility of premisses and preservation of truth both relative to a sequence of components. On the other hand, all the properties of derivability discussed above are lost. As is easily seen, exact derivability does not enjoy monotonicity nor restricted monotonicity; it does not enjoy cut5 and thus, since transitivity is a special case of cut, as we have said above, it is not even a transitive relation.

As for the logical principles associated with exact derivability, they include the following three:

1. A _ B, A ( A,B¡ B 2. A, A * B 3. B * A _ A
The first two principles are those that were associated with derivability.

Principle 1, namely disjunctive syllogism, still holds: both A and A _ B are indispensable premisses to derive B. As for principle 2, namely the block of Lewis's first paradox, since exact derivability is a special case of derivability, it obviously still holds. Principle 3 is new and amounts to the block of Lewis's second paradox (see [START_REF] Lewis | Symbolic Logic[END_REF]); it can be thought of as the direct consequence of the second item of Definition 2.2.

We have thus finished our summary of the two Bolzanian notions of derivability and exact derivability, that are milestones in the history of logic. In what follows we will only focus on the notion of exact derivability. In particular we will try to answer the following question: which modern notion of derivability best approximates Bolzano's notion of exact derivability?

Of course we are not the first to be intrigued by this issue, e.g. see [START_REF] George | Bolzano's consequence, relevance, and ethymemes[END_REF]; [START_REF] Siebel | La notion bolzanienne de déductibilité[END_REF]; [START_REF] Stelzner | Compatibility and relevance: Bolzano and Orlov[END_REF]; several parallels have been drawn between the notion of exact derivability and the relavant logic R introduced by [START_REF] Anderson | Entailement[END_REF]. In the following section we will show that, although the parallel between relevant logic and Bolzano is appropriate, there is another relavant logic that matches the properties of the notion of exact derivability far better than Anderson and Belnap's: it is the logic CR introduced by [START_REF] Tennant | Perfect validity, entailment and paraconsistency[END_REF]. As we will show, the resemblance between Tennant's approach and Bolzano's approach is so striking that it is actually astonishing that nobody has noticed it before.

Tennant's relevant logic

Let us start with classical logic, probably the best known of all logics, characterized by a classical logical consequence relation and the corresponding material implication connective. Classical logic has been the object of several criticisms and revisions occasioned by the so-called fallacies of relevance. These fallacies can be expressed either in an implicational form or in a deductive form. Lewis's first paradox can thus either be expressed as the theorem pA ^ Aq Ñ B, or as the fact that B is deducible from the premisses A and A. Analogously, Lewis's second paradox can either be expressed as the theorem B Ñ pA _ Aq or as the fact that pA _ Aq is deducible from the premise B.

By the term relevant logics one generally indicates those substructural logics that attempt to block the fallacies of relevance. This has been done in two different ways. Relevant logic in the tradition of Anderson and Belnap attacks the problem of relevance from an implication point of view: their main concern is to describe a relevant conditional. Relevant logic in the tradition of Tennant attacks the problem of relevance from a deductive point of view: his main concern is indeed to describe a relevant deduction. Let us note that:

Solving the problem of relevance in the deductive form is arguably a precondition for solving it for the conditional, if we suppose (as is reasonable) that the relevant conditional is to be governed by anything like the rule of conditional proof. To assert the conditional A Ñ B, one will have to be able relevantly to prove B from A; and characterizing the notion of relevant deduction appealed to here is no more than what I have called the problem of relevance in its deductive form. (Tennant, 1987, p. 665) As emphasized above, a parallel has already been drawn between Bolzano's notion of exact derivability and relevant logic; in particular, it has been said

K A E . . . A . . . B A ^B ^I . . . A ^B A ^E . . . A ^B A ^E . . . A A _ B _I . . . B A _ B _I . . . A _ B & & rAs . . . C & & rBs . . . C C _E & & rAs . . . B A Ñ B ÑI . . . A Ñ B . . . A B
ÑE that Bolzano's notion of exact derivability captures an idea remarkably similar to the one that motivates the relevant logic R of Anderson and Belnap. But in view of what has been said in the paragraph above, we can already see that this is quite imprecise. Exact derivability is indeed a relation, Anderson and Belnap are mainly concerned with conditionals and it is Tennant who focuses on relevant derivation. Thus, if an analogy is to be drawn, it is far more appropriate to compare Tennant and Bolzano. As far as we know, this has never been done, but seems to be the correct direction to look at. Let us continue to explore this parallel and discover whether it is really appropriate.

In order to accomplish our task, let us consider the proof of Lewis's first paradox in its deductive form. We follow Tennant in first presenting it in an informal way and then turn to its natural deduction tree presentation. We want to show that from A and A, B can be deduced. Thus we accomplish the following four steps:

1. Assume A 2. By introduction of disjunction, obtain A _ B 3. Assume A

By disjunctive syllogism, obtain B

Let us now present the inference from A and A to B by using the rules of the classical natural deduction calculus as well as the properties of the classical derivability relation. For doing this, let us first of all introduce the language of classical logic and then the classical calculus of natural deduction which is formulated in that language.

Definition 3.1. The classical language L c is composed of a denumerable stock of propositional atoms (p, q, r, . . . ), the logical operators , ^, _ and Ñ, the parentheses (, ) and the square brackets [, ]. The connective Ø is defined as usual; the symbol K stands for the absurd and is defined as A ^ A. Propositional formulas are standardly constructed.

Definition 3.2. The classical calculus of natural deduction C, formulated in the language L c , is composed of the rules of Figure 1. We write M $ C A to denote that the formula A is derivable in the classical calculus of natural deduction C from the set of premisses M .

Let us underline a couple of things relative to the calculus C and the corresponding derivability relation. First of all, since we have assumed the symbol K to be defined as a conjunction of two contradictory formulas, in order to introduce it, we will use the rule ^I, e.g. in this application of the rule E no assumption is discharged since the formula B does not occur previously in the derivation.

We close this paragraph by making a third and final remark about the derivability relation of the classical calculus: this relation is notoriously transitive and enjoys cut.

We now have all the elements to clearly present the inference from A and A to B by using the rules of the classical natural deduction calculus as well as one property of the classical derivability relation. First of all, we consider an application of the rule introducing disjunction, namely the rule _I:

A A _ B _I
Secondly, using the rules ^I, E (with vacuous discharge, as illustrated in the previous page) and _E, from A_B and A, we infer B by disjunctive syllogism:

A _ B A, A K ^E B E B B

_E

Finally, by cut applied on the previous two derivations, we obtain the desired result. We call this derivation d 1 :

A A _ B _I A, A K ^I B E B B

_E

Let us assume that we accept the rule that introduces disjunction since it is clearly not problematic. Then, in order to block the paradox, only two strategies are available: either disjunctive syllogism is rejected or cut is rejected. While Anderson and Belnap's relevant logic is based on the rejection of disjunctive syllogism, Tennant's logic leads to the rejection of cut. Since Tennant's rejection of cut is elegant and subtle, let us better explain it.

In proof theory particular attention is and has always been devoted to redundancies in derivations. A formula occurring in a derivation (in natural deduction calculi) is said to be redundant, or more technically maximal, when it is both a conclusion of an introduction rule and a major premise of the corresponding elimination rule (e.g. see [START_REF] Prawitz | Natural deduction. A proof theoretic study[END_REF]

^E

the formula A ^B is redundant because it is at the same time the conclusion of the rule ^I and the major premise of the rule ^E. A derivation containing at least one maximal formula is said to be not-normal ; a derivation containing no maximal formula is said to be normal. Normal derivations are particularly attractive because they are derivations where each formula is relevant to derive the conclusion (e.g. see [START_REF] Tennant | Perfect validity, entailment and paraconsistency[END_REF]). In the example d above, the formula A^B is clearly not relevant to derive the conclusion A (from the assumptions A and B). In order to conclude A, we could have just stopped to the assumptions A and B. Thus d is not a normal derivation.

Another excellent example of a not-normal derivation is represented by the derivation d 1 from A and A to B that we have seen above. Let us consider this derivation again:

A A _ B _I A, A K ^I B E B B

_E

The derivation contains a namely the (maximal) formula A _ B which is at the same time the conclusion of the rule _I and the major premise of the rule _E. The formula A _ B has no relevant role to play in deriving the conclusion B. B could have been derived from the premisses A and A by means of the derivation d, that we have seen before, where there is no occurrence of the formula A _ B. 6Now that we have clarified what normal derivations are, we can explain Tennant's strategy for eliminating cut. Tennant's idea is indeed very simple: in his logic CR only derivations that are in normal form, i.e. that do not contain redundancies (or equivalently maximal formulas), are allowed. As a direct consequence, only a cut amongst derivations that do not create nonnormal derivations is allowed. To see this, consider again the derivation from A and A to B that contains the maximal formula A _ B. Such a maximal formula has been created by applying cut to the derivation from A to A _B and the derivation from A _ B, A to B. This application of cut is not permitted in CR since it creates a maximal formula, i.e. a non-normal derivation, and, as already explained, only normal derivations are allowed.

The insistence on normality of proofs together with the consequent failure of unrestricted cut is thus the first characteristic of Tennant's logic CR; there are two more properties that need to be mentioned in order to fully understand this logic. First of all, in the logic CR, in all applications of rules in which discharge is indicated, discharge is obligatory. In other terms, no vacuous discharge is allowed.

[...] there must be an undischarged occurrence of the assumption of the indicated form on which the subordinate conclusion depends. Upon application of the rule all such occurrences must be discharged. (Tennant, 1987, p. 672) For illustrating the point, let us consider again the Examples 1 and 2 above where applications of rules with vacuous discharge occurred. Let us start with Example 2. In this case the inference from A and A to B was obtained by applying first the rule ^I and secondly the rule E. In Tennant's logic this passage is no longer valid because it involves an application of the rule E with vacuous discharge. 7 Let us now move to Exemple 1 where we examined the inference from B to A Ñ B. Once again in Tennant's logic this inference is no longer valid because it involves an application of the rule Ñ I with vacuous discharge of assumptions.

Note that by preventing the possibility of vacuous discharge in the rule that introduces the implication connective, i.e. rule Ñ I, Tennant's logic straightforwardly contains a relevant implication rather than the material one of classical logic. Indeed as Dunn and Restall observe:

In the rule for introducing implication, a proviso has been attached which has the effect of requiring that the hypothesis A was actually used in obtaining B. This is precisely what makes the implication relevant. [START_REF] Dunn | Relevance logic[END_REF] Let us finally move to the third characteristic of Tennant's logic; the rule that eliminates disjunction has the following form: . . .

A _ B & & rAs . . . K{C & & rBs . . .

K{C K{C _E 1

where the slash notation K{C is to be understood as follows: we allow a subordinate conclusion of either one of the cases to be brought down as the main conclusion, if the other subordinate conclusion is K. If both subordinate conclusions are of the same form, the main conclusion has the same form. Also, in line with what has been said before, no vacuous discharge is allowed.

It is interesting to see why Tennant replaces the old rule _E of elimination of disjunction with the new rule _E 1 . The rule _E in a calculus where no vacuous discharge is allowed does not permit derivation of disjunctive syllogism. Indeed, as we have seen above, disjunctive syllogism can be proved to be derivable in classical logic by means of the following derivation:

A _ B A, A K ^I B E B B

_E

But because of the conclusion B in the application of the rule E, this derivation is no longer valid in the logic CR: vacuous discharge is indeed no longer allowed and B is a vacuous discharge. The new rule _E 1 easily solves this problem in the following way:

7 Once more the derivation from A and A to B is blocked in the logic CR.

ÑE

The overall restrictions are two: (i) every natural deduction is in normal form, (ii) no deduction contains vacuous discharge.

A _ B A, A K ^I B B _E 1
Let us sum up what we have said up to now in the following definition.

Definition 3.3. The calculus of natural deduction CR, formulated in the language L c , is composed of the rules of Figure 2 (the notation used for discharged assumptions is to be read in the same way as in the classical calculus, except that, as remarked in point (ii), no vacuous discharge is allowed). We write M $ CR A to denote that the formula A is derivable in the calculus CR from the set of premisses M .

As it has been proved by [START_REF] Tennant | Perfect validity, entailment and paraconsistency[END_REF] and as it has been (at least partially) explained in this section, in CR the following logical principles hold:

1. A _ B, A $ CR B 2. A, A & CR B

B & CR A _ A

There also exists a sequent calculus for the logic CR, that we do not introduce for a question of space. Tennant has proved the equivalence between the sequent calculus and the natural deduction calculus for the logic CR.

R, CR and exact derivability

We will use this section to list all the common points between the logic CR introduced by Tennant and Bolzano's notion of exact derivability. Moreover we will show the differences with respect to Anderson and Belnap's relevant logic.

As already emphasized at the beginning of the previous section, the first analogy between Bolzano's and Tennant's results consists in the fact that both philosophers focus their attention on the relevance of a derivability relation.8 Thus, while Anderson and Belnap are mainly concerned in identifying a relevant conditional, Tennant, just like Bolzano, wants to characterize a particular type of derivability relation. This difference is actually quite significant and, as far as we know, has never been highlighted before.

Secondly, Bolzano's notion of exact derivability is non-monotonic (and thus is associated with principle 3. above), avoids Lewis's first paradox (see principle 2.) and does not enjoy cut (nor, as a special case, transitivity). Both Tennant's logic CR and Anderson and Belnap's logic R are non-monotonic and avoid Lewis's first paradox; but while the latter allows unrestricted cut, the former characterizes itself precisely for the lack of a relation enjoying a full cut. Given that derivability relations which do not enjoy full cut9 are quite rare in the history of logic, this is a second important common point between Bolzano and Tennant.

Let us move to the third point. It concerns disjunctive syllogism (see principle 1.); this inference is indeed considered as valid by Tennant, as well as by Bolzano, while it is not valid in Anderson and Belnap's logic. Therefore this is another aspect that unites the two approaches and distinguishes them from Anderson and Belnap's. For the sake of clarity, let us remark that there is a notion of exact derivability introduced by Bolzano in the Wissenschaftslehre §155 (we mentioned it above), for which disjunctive syllogism is not a valid inference. This notion, that from now on we call ED2, is basically a special case of the notion of exact derivability introduced in Definition 2.2. Indeed it is obtained by adding to Definition 2.2 the following clause: no part of any premiss of an exact derivability relation can be deleted without losing derivability itself. Now, if one considers the premisses of disjunctive syllogism, namely A and A _ B, one quickly realizes that A_ can be deleted by these premisses whilst preserving derivability. From B and A one can indeed still derive B. Thus disjunctive syllogism is not an exact inference according to ED2.10 As remarked by [START_REF] Rusnock | Bernard Bolzano, Theory of Science[END_REF] and as already emphasized before, Bolzano preferred the notion of exact derivability as defined in Definition 2.2 over the ED2 notion. Let us illustrate whether, beyond Bolzano's preferences, there are good reasons for choosing one notion over the other. First of all, let us note that ED2 is, just like the relation of exact derivability introduced in Definition 2.2, a relation that does not enjoy cut. But then, if ED2 does not enjoy cut, it makes little sense for it not to enjoy disjunctive syllogism as well: Lewis's first paradox is already blocked by the lack of transitivity and thus there is no need to add a further restriction. Moreover, a derivability relation that enjoys neither cut nor disjunctive syllogism is a quite narrow relation and several doubts can be cast against its conceptual utility. Finally, and perhaps most importantly, disjunctive syllogism is to be counted as a relevant inference: indeed both its premisses A and A _ B are relevant to derive B.11 Thus, it is the notion of exact derivability introduced in Definition 2.2 the one that should be taken into account.

Let us pass to the fourth common point between Bolzano and Tennant. As we have seen in the previous section, Tennant insists on and defends the normality of derivations. In an analogous way, Bolzano (see (Rusnock and George, 2015, §378 and §609)) states principles that are very close to the subformula property.12 But the subformula property is precisely the property enjoyed by normal derivations (see [START_REF] Poggiolesi | Gentzen Calculi for Modal Propositional Logic[END_REF]). Hence, once more, the two philosophers seem to share the same point of view.

We have summed up in Figure 3 analogies and differences between the notion of exact derivability, the logic CR and the logic R. The relevant logic R and the notion of exact derivability share some important features: they both are non-monotonic and they both block Lewis's paradoxes. This is the reason why -we believe -several scholars have drawn a parallel between the two. However, as soon as other properties are taken into account such as disjunctive syllogism, cut or normality (subformula property), it is CR the logic that best matches the notion of exact derivability.

Some reflections about grounding rules for implication

As already emphasized, in the last ten years the notion of grounding has become a vibrant area of research. Amongst the several different approaches to this concept, one of the most interesting concerns the logics of grounding: several different logics of grounding have indeed been proposed, e.g. see [START_REF] Correia | Logical grounds[END_REF]; [START_REF] Fine | Guide to ground[END_REF]; [START_REF] Poggiolesi | On constructing a logic for the notion of complete and immediate formal grounding[END_REF]; [START_REF] Schnieder | A logic for 'Because[END_REF]. Most of them are characterized by grounding rules for the classical connectives: these rules give us the grounds of classical conjunction, disjunction and negation. Perhaps unsurprisingly, little has been said about the connective of implication. More precisely, [START_REF] Schnieder | A logic for 'Because[END_REF] is the only author who have explicitly formulated the grounding rules for implication, which are the following ones:

B A Ñ B because B Ñ1 A A Ñ B because A Ñ2
These rules should be read in the following way. Suppose that B is true, then

B is the ground of A Ñ B; suppose that A is true, then A is the ground of A Ñ B. Thus the grounds of A Ñ B are either A or B.
Our aim in this section is to further discuss the case of the implication connective in the framework of grounding, which in our opinion deserves further reflection. It will turn out that the reflections about grounds for implication are linked to the previous investigations on exact derivability.

Let us start by considering ordinary-language conditionals and the grounding intuitions concerning them. Let us for example consider the following three sentences:

1. "If it rains, then the road will be wet" 2. "If the ball is thrown, then it rolls" 3. "If the snow is white, then it is not black"

The question that we aim at answering concerns the grounds of 1-3. What are the reasons for the truth of these sentences that all have the form "if A, then B"? For each of the conditionals 1-3, the answer seems to be the same: the ground is a sentence C such that from A and C, B can be derived. Thus, more precisely -Ground For 1: "Water makes things wet" -Ground For 2: "The ball is a sphere" -Ground For 3: "Snow only has one color"

In each case from the ground and the antecedent, one can derive the consequent. Consider for example the case 2. Why is it true that "if the ball is thrown, then it rolls"? Because "the ball is a sphere." Indeed if the ball is a sphere and spheres if thrown, roll, then if the ball is thrown, then it rolls.

Let us make three remarks about what has just been said. First of all, these intuitions formulated in a completely non-formal way seem to be both natural and reasonable. Secondly, they only apply to a certain type of conditional, namely indicative conditionals characterized by a connection between antecedent and consequent. This connection is very important since it precisely represents that which is grounded. Thirdly, these intuitions seem to be quite distant from those that are behind Schnieder's rules: according to them, the grounds of an implication are basically its truth-conditions, while, according to the intuitions presented above, the grounds of an implication is a sentence C such that from the antecedent (of the implication) and C one can derive the consequent.

In front of these two divergent approaches, it seems worth deepening our analysis to understand whether there is a way to conciliate them; we will do this by moving to the formal level and thus situating the intuitions behind Schnieder's rules as well as the intuitions described above in the wide and long debate concerning the connective of implication. In this broader and more technical context they both will be clarified and better evaluated.

Let us thus remind the reader the main points of this debate. First of all there is the classical or material implication: a material implication is true if, and only if, the antecedent is false or the consequent is true. On the one hand the material implication has the great advantage of being easy and truth-functional; on the other hand, it does not take into account the connection between antecedent and consequent that we assume conditionals to have when we use them in natural language. Thus the conditional "if 2+2 =3, then the moon is yellow" intuitively seems to be false, while, when formalized by the material implication, it is taken as true. An important defense of the material implication is that of Grice (e.g. see [START_REF] Grice | Studies in the Way of Words[END_REF]; [START_REF] Thomson | In defense of Ñ[END_REF]). According to Grice, when conditionals are evaluated, a distinction should be drawn between semantics and pragmatics: on the one hand, lies the question of when we can say that a conditional is true and, on the other, lies the question of when a conditional can be reasonably asserted. These two levels often differ: for example the conditional "if 2+2 =3, then the moon is yellow" is true but it is not reasonable to assert it since it violates some basic rules on conversational cooperation. This defense of material implication is a milestone in the history of the subject and has attracted the interest and approval of many philosophers. Nevertheless it is not free from defects (e.g. see [START_REF] Gibbard | Two recent theories of conditionals[END_REF]) and several scholars have preferred different formalizations of conditionals. Let us mention those accounts that are, in our opinion, the most famous ones.

The first attempt to fix the inadequacies of the material implication was made by the aforementioned C. I. [START_REF] Lewis | A survey of symbolic logic[END_REF] with the definition of strict implication, intended to block the paradoxes of material implication, and go beyond the truth-functional analysis. Strict implication is defined as a material implication under the scope of the necessity operator, i.e. the famous A B := lpA Ñ Bq. An improved version of the strict implication is the variably strict implication introduced by Stalnaker (1968) and D. [START_REF] Lewis | Counterfactuals[END_REF]. While a strict implication says: "if A, C" is true provided C is true in all the worlds where A is true, the variably strict implication says "if A, C" is true provided C is true in all the closest worlds to the actual world where A is true (where closeness depends on the world of evaluation).

Alongside the work on (variably) strict implication, there is also the trivalent approach whose starting point is the proviso that a material implication is true whenever its antecedent is false. A common consideration is that an indicative conditional whose antecedent is not true cannot be evaluated as true or false, and so remains indeterminate in truth value. Trivalent logics implications are basically built to satisfy this criterion, e.g. see De [START_REF] De Finetti | La logique de la probabilité[END_REF]; [START_REF] Belnap | Conditional assertion and restricted quantification[END_REF][START_REF] Mcdermott | On the truth conditions of certain if-sentences[END_REF].

Relevant implication together with relevant logic, which have already been introduced in the previous sections, are the last account that we present. Relevant logic is an umbrella term for denoting several different logics whose main (but not only) characteristic is that of having a relevant conditional, i.e. a conditional where the antecedent is relevant for the consequent, or, in other terms, a conditional where antecedent and consequent are on the same topic. Note that there is formal principle that relevant logicians apply to force theorems and inferences to be on the same topic. This is the variable sharing principle.

The variable sharing principle says that no formula of the form A Ñ B can be proven in a relevant logic if A and B do not have at least one propositional variable (sometimes called a proposition letter) in common (and that no inference can be shown valid if the premises and conclusion do not share at least one propositional variable).

We have thus finished our brief panorama on the discussion concerning material implication, which involved an excursus through other types of implication. Let us now go back to our main concern namely the grounding rules for implication. Recall that we had identified two different ways of treating grounding of conditional sentences. On the one hand, according to Schnieder's approach, the grounds of a conditional are its truth-values, and on the other hand, our intuitions suggest that the grounds of a conditional are strictly related to the connection between antecedent and consequent of that conditional. Let us now link these insights with what has just been said about the implication connective. Two possible scenarios emerge.

Scenario number 1. We focus on the material implication adopting Grice's defense. In this context a conditional sentence like "if 2+2 =3, then the moon is yellow" is true. But why is it true? The only plausible answer seems to be the following: because the antecedent is false or the consequent is true. This answer completely matches with Schnieder's rules which identify the grounds of an implication with its truth-conditions. Thus in the framework depicted by material implication Schnieder's rules are the appropriate grounding rules for implication.

Scenario number 2. We do not adopt the material implication and we are thus faced to the other types of implication connective. In this second scenario Schnieder's intuitions are of course no longer appropriate and thus the challenge is to combine our intuitions on grounding rules for conditionals with the several connectives of the market. More specifically, the question which naturally arises seems to be the following: what kind of implication best allows us to formalise the intuitions on grounds for conditionals presented above? The trivalent analysis of conditionals fairs poorly since it mainly focuses on the situation where an implication has a false antecedent. Then we remain with the (variably) strict implication accounts and the relevant one. Both approaches appear to create a tighter link between antecedents and consequents of implication, so at the first glance they both seem a plausible choice. However relevant logic is definitely closer to our insights since it explicitly requires a connection between antecedent and consequent and that is exactly what is needed to find the grounds of a conditional. To see this in more detail, let us consider the case of a conditional where antecedent and consequent are tautologies with no atomic sentence in common, such as "if it is not the case that it rains and it does not rain, then the moon is yellow or the moon is not yellow". This conditional formalized in both Lewis and Stalnaker-Lewis accounts is true because both antecedent and consequent are true in any world; on the contrary if formalized as a relevant implication is not provable since it does not enjoy the variable sharing principle. The analysis offered by relevant logics is thus compatible with our perspective: according to our insights the lack of connection between antecedent and consequent in the sentence "if it is not the case that it rains and it does not rain, then the moon is yellow or the moon is not yellow" does not make it possible to formulate the grounds for such a sentence. So relevant logic seems to be the best option for formulating grounding rules for implication that reflect our intuitions. But since relevant logic is an umbrella term that gathers together several different formal systems, it would be interesting to identify the most appropriate. There are two good reasons for preferring the system CR, which has been introduced in Section 4, over the others. Let us explain these two reasons in detail. The first is simplicity: compared to the other relevant logics, CR has both an elegant and simple semantics and and elegant and simple proof theory. The proof theory has been introduced in Section 4 and it is striking for its clarity. The semantics has not been presented for brevity but we refer to [START_REF] Tennant | Perfect validity, entailment and paraconsistency[END_REF]. Grounding is a complicated and problematic object to treat formally; choosing a simple framework in which to formulate grounding rules is thus a primary requirement if we do not want the complexity of the topic to explode. Thus, CR appears to be the best relevant logic for this purpose.

A second reason for choosing CR is its peculiar link with normality of derivations: as already said (see Section 4) in CR all derivations are normal. There exists a viable tradition that starts with Aristotle, passes trough Bolzano, but is also present in the contemporary literature (e.g. see Casari (1987); [START_REF] Rumberg | Bolzano's theory of grounding against the background of normal proofs[END_REF]) that requires ground-revealing proofs to be normal in some sense. This aspect has been deeply discussed in the recent debate. Hence, from this perspective, CR seems to be an ideal framework where to develop grounding rules for implication since it ensures normality right from the start. This certainly is a second significant reason for preferring it over the other existing relevant logics.

We have thus terminated our reflections on grounding rules for implication. We have shown that depending on which type of implication is adopted the grounding rules for implication vary. While Schnieder's intuitions match very well with the material implication, our intuitions are captured very well by relevant logic and in particular by the system CR. Schnieder's intuitions have already been formalized by means of the rules Ñ 1 and Ñ 2. As for our intuitions (or analogous ones) no result has so far been presented. This is certainly an interesting and compelling line of work for future research.

Figure

  Figure 1: Classical Natural Deduction Calculus Introduction Rules Elimination Rules

E

  the rules I, _E and Ñ I we have used the notation & &rAs. This notation means that all (for this stand the square brackets) occurrences of the formula A are discharged (for this stands the bar), once the rule is used. If there is no occurrence of the formula A to discharge, each of the rules above can be used anyway: it is an application of the rule with vacuous discharge of assumptions. For the sake of clarity, let us make some examples of application of the rules I, _E and Ñ I with vacuous discharges.Exemple 1B A Ñ B ÑIIn this application of the rule Ñ I there is no discharge of assumptions: in the implication that occurs as conclusion of the rule the antecedent A is not previously used as a premise.In this derivation, let us call it d, first we introduce the absurd by means of the rule ^I and afterwards we introduce the formula B by means of the rule E:

Figure

  Figure 2: CR Natural Deduction Calculus Introduction Rules Elimination Rules

Figure

  Figure 3: Summary

In some texts (e.g.[START_REF] Roski | Bolzano's notion of grounding and the classical model of science[END_REF]) the term Ableitbarkeit is translated with deducibility, rather than derivability. Since the notion of deducibility involves the requirement of discharge of assumptions, which is absent in the notion of derivability as well as in the original notion of Ableitbarkeit, we prefer this second translation.

For a definition of the notion of complete grounds, as opposed to full grounds, seePoggiolesi (2016a).

As the editors made us notice, although not explicit about this point, it seems that Bolzano would require the sequence P 1 , ..., Pr ¡ to actually occur in the premisses as well as in the conclusion of the derivability relation.

We owe to[START_REF] Paoli | Bolzano e le dimostrazioni matematiche[END_REF] the translation in contemporary terms of the Bolzanian domain of variation. Let us also underline that logical constants do not belong to this domain of variation. The question is however complex since Bolzano could not decide how to draw the distinction between logical and not-logical ideas, see[START_REF] Rusnock | Bernard Bolzano, Theory of Science[END_REF], II, §148.

Let us underline that Bolzano himself realized that cut is not valid for exact derivability, see(Rusnock and George, 2015, II, §155.32).

Thus, d and d 1 are two different derivations from the formulas A and A to the formula B; d is normal, while d 1 is not-normal.

[START_REF] Tennant | The relevance of premises to conclusions of core proofs[END_REF] the reader can find a very accurate analysis of the relevance of the derivability relation. This analysis is carried out in the so-called core logic, a refined version of the logic CR mentioned in this article.

On this point see also[START_REF] Tennant | Rule-irredundancy and the sequent calculus for core logic[END_REF].

For a precise analysis of this point see(Rusnock and George, 2015, p. xxxvi, II volume).

This, we think, is Tennant's central insight.

Note that many scholars have investigated the importance of the subformula property for Bolzano's grounding proofs, see in particular[START_REF] Rumberg | Bolzano's theory of grounding against the background of normal proofs[END_REF] (Roski, 2014, Ch. 5.2.3) and the chapter of A. Tatzel contained in this volume.