Interpreting logic diagrams: a comparison of two formulations of diagrammatic representations

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1 Iterpretig logic diagrams: a compariso of two formulatios of diagrammatic represetatios Yuri Sato, Koji Mieshima, ad Ryo Takemura Departmet of Philosophy, Keio Uiversity {sato, miesima, bstract I the cotet of the cogitive study of diagrammatic represetatios for deductive reasoig, the availability of sytactic maipulatio of diagrams has played a key role i accoutig for their efficacy. urretly, however, little has bee kow about the iterface betwee such sytactic or proof-theoretical aspects ad the correspodig sematic or iformatioal aspects of diagram use. The preset paper ivestigates the cogitive processes of iterpretig diagrammatic represetatios uderlyig deductive reasoig, combiig the isights from both logical ad cogitive studies of diagrams. sematical aalysis of two differet ways of formalizig logic diagrams is provided. ased o it, a multiple stage model of cogitive processes of etractig iformatio from logic diagrams is proposed, ad evidece was foud to support this model. cosequece for the way the abstract syta ad sematic of diagrammatic represetatios are costraied is also eplored. Keywords: Diagrammatic reasoig; Logic diagram; Sematic iterpretatio of seteces ad diagrams; Deductive reasoig; Eteral represetatio. Itroductio Over the past few decades, may researchers have show a iterest i logical ad cogitive aspects of reasoig with diagrammatic represetatios (e.g. Glasgow, Narayaa, & hadrasekara, 1995). I particular, diagrammatic represetatios used for deductive reasoig, which were traditioally regarded as a auiliary device to help uderstadig of quatificatioal ad set-theoretical formalisms i logic tetbook, have bee itesively studied usig the method of mathematical logic (e.g. llwei & arwise, 1996; Shi, 1994). The formal ad mathematical study of diagrammatic represetatios have yielded fruitful isights ito the cogitive aspects of the use of diagrammatic represetatios. s a illustrative eample, let us eplai with Euler ad Ve diagrams that eterally support processes of solvig deductive reasoig tasks. What plays a crucial role i accoutig for efficacy of diagrams i problem solvig is the eistece of sytactic maipulatios of diagrams, whose structure has bee studied i the traditio of the logical study of diagrams. Whe logic diagrams are eterally give to reasoers, solvig processes of deductive reasoig tasks could be replaced by processes of combiig premise diagrams ad etractig the relevat iformatio. Thus a syllogism cosistig of two premises, ll are ad No are, ad a coclusio No are has a diagrammatic derivatio with Euler diagrams, as illustrated to the left i Figure Here the uified diagram D e 3 is obtaied by idetifyig the circle i the two premises ad readig off the relatio betwee the circle ad, i.e., the disjoitess relatio, which is automatically determied by the uifyig process (cf. Shimojima, 1996). ased o eperimets o syllogism solvig tasks aided by logic diagrams, Sato, Mieshima ad Takemura (2010a,b), provided evidece which suggested that the sytactic maipulatio of logic diagrams could be available eve to utraied users. ll are. No are. ll are. No are. D e 1 Uificatio D e 2 D v 1 Superpositio D v 2 D e D 3 v 3 No are. No are. Figure 1: diagrammatic derivatio of a syllosigm ll are, No are ; therefore No are with Euler diagrams (left) ad Ve diagrams (right). I the traditio of the logical ad cogitive study of diagrams, particular emphasis has bee o compariso betwee diagrammatic ad setetial (liguistic) represetatio systems. I additio to the availability of cocrete maipulatios, a umber of properties which distiguish diagrammatic from setetial represetatios have bee proposed, seekig to accout for what advatages diagrammatic represetatios i geeral have over setetial oes (e.g. Steig, 2000; Shimojima, 2001). y cotrast, relatively few attetio has bee paid to compariso betwee differet diagrammatic represetatio systems. However, such a compariso might be potetially importat to provide a more fie-graied aalysis of efficacy of various diagrams i huma problem solvig. s a crucial eample, cosider a solvig process of the syllogism we saw above with Ve diagrams. It is show to the right i Figure I Ve diagrams, every circle partially overlaps each other, ad the meaigful relatios amog circles are epressed by shadig uder the covetio that shaded regios deote the empty set. Give this sematics, the process of composig two premise diagrams automatically yields the iformatio correspodig to the correct coclusio, i a similar way to the process i Euler diagrams. Ituitively, however, reasoig with Ve diagrams appears to be relatively more difficult to hadle i reasoig. s emphasized i Gurr, Lee ad Steig (1998), what differetiates the two cases is the process of iterpretig (eterally give or iterally costructed) diagrammatic represetatios. I particular, a covetioal device such as shadig ivolved might cause complicatio i iterpretatio processes of Ve diagrams. This suggests that i order to obtai a more comprehesive accout of diagrammatic reasoig, we eed to 2182

2 take a closer look at cogitive processes uderlyig iformatio etractio from diagrams. Traditioally, the otio of similarity has played a role i accoutig for differeces i efficacy of sematic aspects of setetial ad diagrammatic represetatios. Geerally speakig, if a certai structural similarity holds betwee a represetatio ad what it represets, the represetatio could be effective i iterpretatio ad commuicatio eve for users who do ot lear covetios goverig its use eplicitly. I the literature, such a otio of similarity has bee specified i various ways: homomorphism (arwise & Etchemedy, 1991), directess (Gurr, Lee, & Steig, 1998; Steig, 2000); structural similarity (Gattis, 2004). However, a precise characterizatio of the otio of similarity that could be applied to a varieties of diagrammatic represetatios remais to be eplored. I particular, how a cogitive accout of iterpretatio processes could be coected to the formal sematics of diagrams still remais uclear. The questio of iformatio etractio has also bee ivestigated i the study of the cogitive role of relatively simple use of diagrams, such as charts, maps, ad graphs (e.g. Ratwai, Trafto, & oehm-davis, 2008; Shimojima & Katagiri, 2010). What plays a importat role here is the distictio betwee lower-level ad higher-level iformatio. For eample, a scatter plot cotais the lower-level iformatio about specific data ad the overall distributio of the dots delivers the higher-level iformatio about the structural properties of data (Kossly, 1994). urretly, however, applicatios of these fidigs to a aalysis of higher cogitive processes, such as deductio reasoig, were ot fully eplored. The preset paper aims to ivestigate the cogitive processes of iterpretig diagrammatic represetatios uderlyig deductive reasoig, combiig the isights from these differet research traditios. I particular, based o a sematical aalysis of diagrams, we argue that a certai structural correspodece betwee a diagrammatic represetatio ad its sematic cotets plays a crucial role i both iterpretatio ad iferece processes with the represetatios. Our approach ca also provide a further costrait o the choice of differet ways of formalizig the abstract syta ad sematics of diagrammatic represetatios, motivatig a more fruitful way of approachig to the logical study of diagrams. This could cotribute to establishig a closer coectio betwee the logical ad cogitive approach to huma problem solvig with diagrammatic represetatios. Geeral Hypothesis mai goal of the preset study is to eplore the hypothesis that the matchig relatio betwee the diagrammatic represetatio used i deductive reasoig ad the coveyed iformatio available to users plays a importat role i effective diagrammatic reasoig. s a case study, we focus o the use of logic diagram i syllogistic reasoig. I order to make clear what is the relevat structural relatioship betwee logic diagrams used ad their sematic cotets, it is helpful to first look at what sematic iformatio is carried by syllogistic seteces, usig the isight obtaied i the sematics of atural laguage. The relatioal aalysis of quatified setece. From a geeral viewpoit, syllogistic ifereces as ivestigated i the psychology literature ca be regarded as a special case of ifereces with quatificatioal seteces i atural laguage. ccordig to the stadard tetbook treatmet, such seteces are aalyzed usig represetatios i first-order predicate logic, which essetially ivolve quatificatio over idividuals as sematic primitives. I the field of atural laguage sematics, by cotrast, quatifiers i atural laguage, such as all, some ad o, are aalyzed as deotig relatios betwee sets, i.e., what is called geeralized quatifiers (arwise & ooper, 1981). Thus, a setece of the form ll is aalyzed as, rather tha as the first-order represetatio ( ). Similarly, No ca be aalyzed as epressig = /0. Here the sematic primitives of quatificatioal seteces are cosidered as the relatios betwee sets, such as subset relatio ad disjoitess relatio. Iterestigly, the moder recostructios of ristotelia categorical syllogisms (Łukasiewicz, 1958 ) ad recet developmet of the so-called atural logic (va ethem, 2008) take as a primitive logical form the relatioal structure of a quatified setece, which is schematically represeted as Q(,). These logical fidigs suggest that syllogistic ifereces ca be formulated as trasitive ifereces i a perspicuous way, without referece to idividuals terms. It should be emphasized that such a relatioal formulatio of the meaig of a quatified setece could capture ot oly the truth coditio or logical form of a setece, as is traditioally assumed, but also how speakers metally represet such a truth coditio or logical form (Hackl, 2008; Pietroski et al., 2009). The relatioal approach to quatificatioal seteces has also bee successfully applied to the psychological study of deductio, resultig i a processig model based o the assumptio that ifereces with quatifiers are doe i terms of sets rather tha idividuals (Geurts, 2003). For a discussio o the role of relatioal kowledge i geeral i higher cogitio, see Halford, Wilso, & Phillips (1998, 2010) I sum, we may plausibly assume that the sematic primitives of quatificatioal seteces i atural laguage are relatios betwee sets, ad that people s ifereces with quatified costructios are sesitive to such a relatioal structure. Relatioal aalysis of Euler diagrams. I the logical study of Euler ad Ve diagrams (e.g. Shi, 1994), diagrammatic represetatios have bee give their ow formal syta ad sematics, i a similar way as for formulas i mathematical logic. What is remarkable here is that a diagram may have several equivalet formalizatios. s a illustrative eample, cosider a simple Euler diagram E i Figure This diagram ca be aturally iterpreted as deotig the subset relatio betwee sets ad, i.e.,. ut it is also possible to iterpret it as epressig that = /0, where 2183

3 E V Figure 2: Eamples of Euler diagram E ad Ve diagram V that correspod to the setece ll are. The diagrams P is a so-called plai diagram, which epresses a tautology. deotes the complemet of the set. orrespodigly, there are two ways of formalizig the abstract syta of Euler diagrams (Mieshima, Okada & Takemura, 2010; 2011). ccordig to what is called a relatioal approach, Euler diagrams are abstractly specified as a set of topological relatios holdig betwee objects i the diagrams. For eample, the diagram E i Figure 2 is represeted as { }, where meas that circle is iside circle. other approach is a regio-based approach, which is fairly stadard i the logical study of diagram (e.g. Howse, Stapleto, & Taylor, 2005). Here diagrams are abstractly defied i terms of regios ad emptiess. Thus the diagram E i Figure 2 ca be represeted by specifyig the regio (,), the regio iside circle ad outside circle, as a missig regio. learly, these two ways of defiig Euler diagrams predict, for each diagram, the equivalet truth-coditio. The differece cosists i the way these truth-coditios are give. Now the questio is: which formulatio (or possibly others) reflects the way the user represets the sematic cotet of a give diagram? Here the regio-based formulatio appears to be more atural for the meaig of the Ve diagram such as the diagram V show i Figure Give the covetio that the shaded regio deotes the empty set, = /0 has the sytactic readig the regio iside the circle but outside the circle deotes the empty set, or more colloquially, there is othig which is but ot. It should be oted here that throughout our discussio, we are assumig that both Euler ad Ve diagrams adopt the covetio that each ushaded regio lacks eistetial imports, i.e., may deote the empty set. Thus the diagram P i Figure 2, where the circles ad partially overlap each other, coveys sematically tautologous iformatio. I other words, this diagram meas that the sematic relatioship betwee ad is idetermiate. I this respect, our sematics differs from the oe discussed i Johso-Laird, 1983 (for a more detailed discussio of the sematics of Euler ad Ve diagrams, see Hammer & Shi, 1998 ad Sato, Mieshima, & Takemura, 2010a). The two ways of formulatig logic diagrams were the summarized as follows. buildig blocks meaigful uits (sematic premitives) Relatio-based circles relatios aalysis betwee circles Regio-based regios o-emptiess aalysis of miimal regios Our basic hypothesis is that a Euler diagram like E i Figure 2 epresses relatioal iformatio that could be accouted P for by the relatio-based aalysis; it triggers a relatioal represetatio such as to the users. y cotrast, Ve diagrams are subject to a regio-based aalysis, triggerig the sematic iformatio such as = /0 to their users. We saw above that syllogistic seteces are quatificatioal seteces of the relatioal form, schematically represeted as Q(, ), ad that such seteces force a reasoer to form ad operate o relatioal represetatios i reasoig. If our basic hypothesis above is correct, Euler diagrams directly epress the topological relatioship betwee circles. Thus it is hypothesized that whe a reasoer is asked to match a syllogistic setece with a correspodig Euler diagram (or vise versa), he could appeal to the process of readig off the relatioal iformatio from a Euler diagram immediately, i.e., without ay itermediate steps, ad the verifyig that it is the same as the iformatio coveyed by the setece. I cotrast, Ve diagrams have a fied cofiguratio of circles ad represet set relatioships idirectly, by stipulatig that shaded regios deote the empty set. ccordigly, the process of etractig the relevat relatioal iformatio from a Ve diagram would be epected to proceed i several steps. s a cocrete eample, cosider how the reasoer could etract the correct relatioal represetatio from the Ve diagram V i Figure 2 above. Let us call a regio iside of some circles ad outside of the rest of the circles (possibly oe) i a diagram a miimal regio. Thus the diagrams V has four miimal regios. Firstly, the reasoer eeds to check each miimal regio whether it is shaded or ot, as lowerlevel iformatio. I this eample, oly the regio (, ) is shaded. Net, the reasoer iterally builds the segmets that lumps together the shaded miimal regios cotiuous with each other, as well as those which lump together the ushaded miimal regios. This step makes it possible for the reasoer to coclude that the diagram delivers the higher-level iformatio There is othig which is but ot. The the reasoer would be able to paraphrase it as ll are, which correspods to the required iformatio i syllogistic ifereces. It is thus hypothesized that such compleities would cause some difficulties i etractig the required iformatio from Ve diagrams. Eperimet 1 s a iitial test of our hypothesis, we coducted a setecediagram matchig test, i which participats were preseted with a syllogistic setece ad asked to choose the diagram epressig the same iformatio. I Eperimet 1, we used a simple form of Ve diagrams cosistig of two circles (see the diagram V i Figure 2), rather tha three. I order to eclude eteral factors such as familiarity with preseted diagrams, participats were provided with sufficiet iformal eplaatio of the sematics of diagrammatic represetatios. Method Participats Twety-seve udergraduates ad graduates (mea age 234 ± 27 SD) took part i the eperimet. They gave a coset to their cooperatio i the eperimet, 2184

4 ad were give small moey after the eperimet. ll the studets were ative speakers of Japaese ad task seteces ad istructios were give i Japaese. The participats were divided ito two groups: Euler group (13 studets) ad Ve groups (14 studets). Materials The syllogistic seteces used i the eperimet are divided ito eistetial ad o-eistetial seteces. They are of the followig patters: No-eistetial seteces Eistetial seteces (1) ll are. (5) Some are. (2) ll are. (6) Some are. (3) No are. (7) Some are ot. (4) No are. (8) Some are ot. The participats were preseted with oe setece i a P moitor ad required to choose the correspodig diagram (if ay). Figure 3 shows templates of tasks for the two groups. syllogistic setece is iserted i this area. syllogistic setece is iserted i this area. Figure 3: Templates of task seteces with Euler diagrams (left) ad Ve diagrams (right) used i the eperimet. Here, i both diagrams, setece ll are correspods to swer 1, No are ad No are to swer 2, Some are ad Some are to swer 3, ad Some are ot to swer Recall that a diagram i which circles partially overlap each other does ot epress ay specific sematic relatioship betwee them (see the diagram P i Figure 2). I order to epress the eistece of objects (i.e., the o-emptiess of a set), the, we use the poit. s a cosequece, i Euler ad Ve diagrasm, eistetial seteces are represeted i the same way as idicated i Figure Note also that the seteces ll are ad Some are ot have o correspodig diagram, hece the correct aswer is 5 ( Noe of them ). Stimuli were preseted radomly. fter a task setece ad four diagrams appeared, the participats were asked to press oe of five buttos. There is o time limit to solve the matchig tests. Procedure The eperimet was coducted idividually. (1) Istructio ad pretest. efore the test, the participats were provided with istructios o the meaig of seteces ad diagrams used. pretest to check whether they uderstad the istructios correctly was coducted; they were preseted with te diagrams of basic forms ad asked to choose all the seteces (if ay) that have the same meaig as give diagrams. fter the pretest, the eperimeter told the participats whether they aswered to the problems correctly. Whe a icorrect aswer was foud, they were asked to reread the istructio ad to select the correct aswer. (2) The matchig task. Oe task eample was displayed i a P moitor. total of eight differet types of seteces were prepared. The participats were asked to press, as quickly ad as accurately as possible, a butto with the umber represetig the aswer they reached. Predictio It is predicted that for o-eistetial seteces, the respose time to choose Euler diagrams would be shorter tha the respose time to choose Ve diagrams. For eistetial seteces, Euler ad Ve diagrams have the same form (see Figure 3 above), hece it is predicted that there would be o differece betwee the two cases. Results ad discussio mog the twety-seve studets, we ecluded four studets (oe i the Euler group ad three i the Ve group), who did ot aswer correctly at all or gave oly oe correct aswer. Table 1 shows the average respose times i the setecediagram matchig tasks for Euler ad Ve diagrams. Table 1: The respose times of the setece-diagram matchig task with Euler ad Ve diagrams. o-eistetial eistetial setece setece Euler diagrams s s Ve diagrams 1057s s These data were subjected to a two-way alysis of Variace (NOV) for a mied desig. There was o sigificat mai effect for the differece betwee Euler ad Ve diagrams (F (1, 21) = 20 p >.10). There was o mai effect for the differece betwee eistetial ad o-eistetial seteces (F (1, 21) = 0.48 p >.10). There was a sigificat iteractio effect for these two factors (F (1, 21) = 57 p <.10). Multiple compariso tests were coducted by Rya s procedure. The results idicated that (i) regardig o-eistetial seteces, the respose times i the setece-diagram matchig task for Euler diagrams were sigificatly shorter tha those for Ve diagrams (F (1, 42) = 730, p <.05), ad that (ii) regardig eistetial seteces, there was o sigificat differece i performace betwee the Euler group ad the Ve group (F (1, 42) = 0.270, p >.10). We ote that the average accuracy rates for both types of diagrams were very high (more tha 82 %). Furthermore, o sigificat differece was show by chagig the order of terms i preseted seteces, for eample, betwee No are ad No are. The overall results provide partial evidece for our hypothesis that the process of etractig relatioal iformatio from Euler diagrams to match it with setece meaig would be simple ad immediate, whereas i the case of Ve diagrams it could be more complicated. 2185

5 (1) (2) Eperimet 2 I order to provide a further support for our hypothesis, we coducted a diagram-setece matchig test, i which participats were preseted with a diagrams ad asked to select the setece coveyig the same iformatio. I Eperimet 2, we used Euler ad Ve diagrams cosistig of three circles as i Figure 1, which is epected to be more sesitive to the differece i compleity of iformatio-etractig processes for the two types of diagrams. h ll are. No are. Some are. Some are ot. orrect swer: 1 (3) (4) ll are. No are. Some are. Some are ot. orrect swer: 4 ll are. No are. Some are. Some are ot. orrect aswer: 1 ll are. No are. Some are. Some are ot. orrect aswer: 4 Figure 6: Eamples of the diagram-setece matchig task with Euler diagrams (1) (2) ad Ve diagrams (3) (4). Method Participats Twety-three udergraduates ad graduates (mea age 273 ± 41 SD) took part i the eperimets. They gave a coset to their cooperatio i the eperimets, ad were give small moey after the eperimets. The subjects were ative speakers of Japaese, ad the task seteces ad istructios were give i Japaese. The participats were divided ito two groups: Euler group (12 studets) ad Ve groups (11 studets). Materials Eleve Euler diagrams ad the correspodig eleve Ve diagrams were used i the diagram-setece matchig task. They are show i Figure 4 ad Figure (1) (2) (3) (4) j kk l l l l (5) (6) (7) (8) k k k k (9) (10) (11) Figure 4: The Euler diagrams used i Eperimet 2 (1) (2) (3) appeared, the participats were asked to press, as quickly ad as accurately as possible, oe of the five buttos with the umber correspodig to the aswer they chose. There is o time limit to solve the tasks. Procedure The eperimet was coducted i the same maer as Eperimet The istructios of seteces ad diagrams were provided, pretests were coducted, ad the the matchig task were imposed. Predictio It is predicted that whe diagrams do ot cotai a poit, the respose time for Euler diagrams would be shorter tha that for Ve diagrams. Whe Ve diagrams cotai a poit, what users eed to do is just to recogize the relatioship betwee the poit ad a relevat circle. I such cases, the relatioship betwee circles is irrelevat, ad thus the processes of idetifyig each miimal regio as shaded or ushaded ad costructig the relevat segmets could be simply omitted. Hece it is epected that the respose time for Ve diagrams that cotai a poit would be shorter tha that for those which do ot. y cotrast, whe a Euler diagram cotais a poit, there would be o differece with respect to whether it cotais a poit or ot, sice i both cases what the users eed to do is to check the relatioship betwee two objects, i.e., the oe betwee two circles or the oe betwee a circle ad a poit. (4) Results ad discussio (5) (6) (7) (9) (10) (11) Table 2 shows the average respose times i the diagramsetece matchig tasks with Euler ad Ve diagrams. I this table, o-poit stads for diagrams that do ot cotai poit ad poit stads for those which cotai poit. These data were subjected to a two-way alysis of Variace (NOV) for mied desig. (8) Table 2: The respose times of the diagram-setece matchig tasks with Euler ad Ve diagrams. Figure 5: The Ve diagrams used i Eperimet 2 Task eamples for Euler ad Ve groups are show i Figure 6. The participats were preseted with eleve Euler diagrams (Ve diagrams), ad were asked to choose all the seteces (if ay) that epress the same iformatio as a give diagram. There were five aswer optios: ll-, No-, Some-, Some-ot, ad Noe of them, as idicated i Figure 6. Stimuli were preseted radomly. Whe task diagrams ad seteces Euler diagrams Ve diagrams o-poit s s poit 1946s 1022s There was a sigificat mai effect for the differece betwee Euler ad Ve diagrams, (F (1, 21) = p <.05). There was o sigificat mai effect for the differece betwee di- 2186

6 agrams which cotai poits ad those which do ot, (F (1, 21) = 03 p >.10). There was a sigificat iteractio effect for these two factors, (F (1, 21) = p <.05). Multiple compariso tests were coducted by Rya s procedure. The results idicated that (i) regardig diagrams without poit, the respose times for Euler diagrams were sigificatly shorter tha those for Ve diagrams (F (1, 42) = 1919, p <.005), ad that (ii) regardig Ve diagrams, the respose times for diagrams that cotai a poit were sigificatly shorter tha those which do ot (F (1, 21) = 7.885, p <.05). (iii) regardig Euler diagrams, there was o sigificat differece i the respose betwee cases that cotai a poit ad those which do ot (F (1, 21) = 0.627, p >.10). These results clearly support our predictios. It is oted that the average accuracy rates for both diagrams were very high (more tha 83 %). Geeral discussio The overall results of the two eperimets provide evidece for our hypothesis about the multiple stage view o iformatio-etractio from Euler ad Ve diagrams. ccordig to this view, a process of etractig relatioal iformatio from Euler diagrams cosists of a sigle step, whereas that from Ve diagrams cosists of multiple steps, i.e., idetifyig shaded ad ushaded miimal regios ad costructig the segmets correspodig to the terms i questio. These results are compatible with the fidigs i Sato, Mieshima ad Takemura (2010b), where the performaces i syllogisms solvig tasks with Euler diagrams were compared with those with Ve diagrams ivolvig three circles, ad it was show that the former was better tha the latter. Sato, Mieshima ad Takemura (2010b) also reported that the performaces i syllogism solvig with Ve diagrams ivolvig two circles (see Figure 2) were worse tha the three-circle case. They argued that this differece could be ascribed to the availability of iteral sytactic maipulatios of diagrams: i the case of Ve diagrams with two circles, such a sytactic maipulatio (i.e. the process of combiig two premise diagrams) would be simply uavailable to utraied users. However, the differece i performace betwee Euler ad Ve diagrams with three circles remaied to be accouted for, sice both types of diagrams could easily trigger sytactic maipulatios because of their uiform forms. The preset paper provides a accout of it based o the differece i iformatio-etractio processes for these diagrams. Together with these fidigs, the preset study suggests a possibility of providig a model o the overall processes of reasoig with diagrams for deductio, where ot oly the availability of sytactic maipulatios of diagrams but also a subtle differece i processes of etractig iformatio from diagrams could make differece i effective diagrammatic ifereces. It should be emphasized here that the preset cogitive study could be fruitful i providig a further costrait or data o theorizig about proof theory (syta) ad model theory (sematics) for higher-level diagrammatic represetatios. ogitive or procedural differeces i sematically equivalet ways of specifyig truth-coditios or logical forms of diagrams (ad for that matter, seteces) ted to be eglected i theorizig about formal sematics ad proof theory for those represetatios. Eve i the study of diagrammatic logic, such theoretical ivestigatios could be coducted idepedetly of ay cogitive cosideratios. The preset study suggests a possibility of a more costraied ad itegrated framework to deal with both logical ad cogitive aspects of diagram use. Refereces llwei, G. & arwise, J. (Eds.). (1996). Logical Reasoig with Diagrams. New York: Oford Uiversity Press. arwise, J. & ooper, R. (1981). Geeralized quatifiers ad atural laguage. Liguistics ad Philosophy, 4, arwise, J., & Etchemedy, J.(1991). Visual iformatio ad valid reasoig. I G. llwei & J. arwise (Eds). Logical Reasoig with Diagrams (pp.3-26). Oford Uiversity Press. Gattis, M. (2004). Mappig relatioal structure i spatial reasoig. ogitive Sciece, 28, Geurts,.(2003). Reasoig with quatifiers. ogitio,86, Glasgow, J., Narayaa, N.H., & hadrasekara,. (Eds.). (1995). Diagrammatic Reasoig. ambridge, M: MIT Press. Gurr,.., Lee, J., & Steig, K. (1998). Theories of diagrammatic reasoig. Mids ad Machies, 8, Hackl, M. (2009). O the grammar ad processig of proportioal quatifiers. Natural Laguage Sematics,17, Halford, G.S., Wilso,W.H., & Phillips, S. (1998). Processig capacity defied by relatioal compleity. ehavioral ad rai Scieces, 21, Halford, G.S., Wilso,W.H., & Phillips, S. (2010). Relatioal kowledge. Treds i ogitive Sciece, 14(11), Hammer, E. & Shi, S. (1998). Euler s visual logic. History ad Philosophy of Logic, 19, Howse, J., Stapleto, G., & Taylor, J. (2005). Spider diagrams. LMS Joural of omputatio ad Mathematics, 8, Johso-Laird, P. N. (1983). Metal Models. Harvard Uiv. Press. Łukasiewicz, J. (1957). ristotle s Syllogistic from the Stadpoit of Moder Formal Logic. Secod editio, Oford: laredo Press. Kossly, S.M. (1994). Elemets of Graph Desig. Freema & o. Mieshima, K., Okada, M., & Takemura, R. (2010). Two types of diagrammatic iferece systems: Natural deductio style ad resolutio style. Diagrams 2010, LNI 6170, (pp ), Spriger. Mieshima, K., Okada, M., & Takemura, R. (2011). diagrammatic reasoig system with Euler circles. Joural of Logic, Laguage ad Iformatio, to appear. Pietroski, P., Lidz, J., Huter, T., & Halberda, J. (2009). The meaig of most. Mid & Laguage, 24(5), Ratwai, R.M., Trafto, J.G., & oehm-davis, D.. (2008). Thikig graphically. J. Ep Psychology: pplied, 14(1), Sato, Y., Mieshima, K., & Takemura, R. (2010a). The efficacy of Euler ad Ve diagrams i deductive reasoig: Empirical fidigs. Diagrams 2010, LNI 6170, (pp. 6-22), Spriger. Sato, Y., Mieshima, K., & Takemura, R. (2010b). ostructig iteral diagrammatic proofs from eteral logic diagrams. 32d oferece of the ogitive Sciece Society (pp ). Shimojima,. (1996). O the Efficacy of Represetatio. PhD thesis, Idiaa Uiversity. Shimojima,. (2001). The graphic liguistic distictio. rtificial Itelligece Review, 13, Shimojima,. & Katagiri, Y. (2010). eye-trackig study of itegrative spatial cogitio over diagrammatic represetatios. Spatial ogitio VII, LNI 6222 (pp ), Spriger. Shi, S.-J.(1994). The Logical Status of Diagrams. ambridge U.P. Steig, K. (2000). Distictios with differeces. Diagrams 2000, LNI 1889 (pp ), Spriger. va ethem, J. (2008). rief History of Natural Logic, Techical Report PP , Istitute for Logic, Laguage ad omputatio. 2187

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