On the logic of design - Science Direct

On the logic of design Y. Zeng and G.D. Cheng Research Institute of Engineering Mechanics, Dalian University of Technology, Dalian 116023, China

A new logic scheme called recursive logic is proposed, which is different from the widely accepted logics: deductive logic, inductive logic and abductive logic. It is suggested that recursive logic is the logic of design. The rationality of this proposal is then discussed based on the investigation of the facets of design activity. Finally, it is pointed out that many well-known design methodologies such as the pattern language given by Alexander can find their explanations in this logic scheme. Keywords: recursive logic, logic of design, design methodology

PROPOSAL OF THE PROBLEM

For a logic of design, the following features are essential:

In the 1960s, as the systems approach came of age, design methodology research came into fashion and the 'design method movement' began. The major goal of this movement was the rationalization of the design process. During the last three decades, many aspects of design and designing have been studied and investigated, including the management of the design process, the structure of design problems, the nature of design activity and the philosophy of the design method ]. As a result, many models about designing have been developed, each of which highlights some special aspects of the design and has found its own way in different domains and diverse applications. A universal design schema which implies all these methods is essential to give impetus to the deeper and further development of design studies, that is the task of the logic of design. Recently, as expert systems have been successfully applied to classification and diagnosis problems, the design type expert system has been paid more and more attention. Different from classification and diagnosis, design consists of a creative process which should be firstly subjected to formalization and mechanization. This brings difficulty to the problem. Indeed, the formalization of a problem solving process is composed of two parts with the first one being the logic of the process and the second the knowledge based on the logic. This paper is concerned with the former.

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• it is itself a general scheme to represent the form of the reasoning generating the design • it gives the demarcation between logic, science and design.

ARTIFICIAL

SYSTEMS

The purpose of design is to make an artifact that can carry out the expected function. Therefore the artifact should firstly adapt to the goals and requirements of humans. On the other hand, however, all artifacts are parts of the nature from which they can never be separated. So the artifact has to obey natural laws and rules at the same time. Hence, the artifact constitutes an artificial system as Simon 2 has pointed out.

Definition 1.

The form of an artifact, F , refers to its configuration or composition in the time-space coordinate system. The geometry of a building, the political and economic structure of a society, the composition of music and so forth are the representatives of form adhering to different artifacts. It is the true arbitrariness of the form that makes design difficult and unsatisfactory.

Definition 2.

The function of an artifact, R, refers to its

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response to the environmental actions according to some natural laws, rules and principles• To cite some examples, we have the deflection of a building under the external loadings, the living condition in a society, the audience's reaction to a concert, etc.

a

SA b

F~ml. Definition 3. The environment of an artifact, E , refers to the working conditions in which the artifact lies. It includes the natural laws and the natural actions, which are denoted by L and A, respectively E = < L, A >

(1)

The rectangle shown in Figure 1 can be seen as an artificial system. The form of this artificial system is the shape and the metric of the polygon, as given in Equation (6). F -- < rectangle, a, b >

(6)

The environment of a building is the whole natural world, that of a social structure is the citizens, and an art work the human minds.

which can be translated into a logical statement as

Definition 4. A form, the natural action and the function in the environment constitute the basic elements of the natural laws which dominate the behaviour of the form

The function of the artificial system is theoretically infinite, but in an application what it is concerned with is restricted. In the case of the rectangle, we just list, for example, the area as its function, as in (8),

L=

(2)

The metric of a rectangle is a and b

(7)

R = < rectangle, area >

(8)

which can be interpreted as a logical statement as From the logical point of view, the natural law can be seen as a statement. It may assume different types of statement, such as the universal positive one, the compound one, etc, in different situations 3. For example, IF the temperature is below 0 degree, T H E N the water will be frozen•

(3)

belongs to the category of compound statements which is a fundmental physical rule. But All people who live eat

(4)

falls into the universal positive statement, which is a basic principle in social science•

Definition 5. An artificial system, S A, is a triple which is made up of three parts that are form, function and environment. SA = < F,R,E

>

(5)

The form is the core of an artificial system• Each form has to be subjected to the action of the environment and results in some functions• Each function has to be connected to one form or another. When we are talking about a function, we are referring to the function of a form. When we give a form, we are defining the form of some functions. Hence, we have.

Definition 6. T h e ultimate goal of designing is to create a form which displays the prescribed functions in its environment.

LOGIC OF D E S I G N To investigate the intrinsic logic of design, we give an intuitive example as follows.

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The function of the rectangle is area

(9)

Finally, in regard to the environment of this artificial system, we assume the measurement system, that is, the area formulae as the natural laws and the empty set as the natural action. L = < rectangle, S = a × b >

(10)

A=~

(11)

which is equivalent to the following statement• T h e area of the rectangle is the result of the multiplication of its metric (12) The Equations (12), (7) and (9) just comprise a perfect syllogism 3, based on which we can distinguish the three kinds of logic 4 suggested by Peirce as follows

• Deductive reasoning Major p r e m i s e L = < rectangle, S = a × b > (13) Minor premise (F = < rectangle, 1, 3 > , A = ~ ) (14) Conclusion R -- < rectangle, S -- 3 > (15) • Inductive reasoning { Minor premise, conclusion ) { (F = , A = qb) R = } (16) { (F = rectangle, a = 2, b -- 3 > , A -- ~ ) R = )

(17)

{ (F -- rectangle, a = 2, b = 7 > , A -- O) R = }

(18)

Major premise ? L = < rectangle, S = a × b >

(19)

• Abductive reasoning Major premise L = < rectangle, S = a × b > Minor premise R -- < rectangle, S = 2 0 > , Conclusion ? (F = < rectangle, a, b > , A = • )

(20) (21) (22)

DESIGN STUDIES

March 5 has taken abductive reasoning as the logic of design. As abductive reasoning requires, a particular theory should hold in the inference. Considering Equation (2), March's proposition implictly assumes that the form of the design is a priori determined, which is not the case in design. Indeed, design is a process to simultaneously produce both the artifact and its behaviour system which is here the major premise of the logic. As a result, the theory dominating the inference isn't known. Therefore, as in the general design problem, the logic form becomes

• Design reasoning Major premise ? L = < F , S = S(X)> Minor premise (A -- ~ , R = < F , S = 20> ) Conclusion ? F --

(23) (24) (25)

The above example illustrates the fact that design is largely different from deduction, induction and abduction in that the conclusion of the reasoning is recursively dependent on the major premise of the reasoning. Based on this observation, we can give a new viewpoint on the logics, and hence on the logic of design.

Definition 7. An artificial system can be defined as a logical system of deduction, where the major premise is the behaviour principle of the system subjected to the natural law, the minor premise is the form of the system and the natural actions alike exerted on the system, and the conclusion is the function of the system in the environment. It is represented symbolically as Major premise < F , L > , or < F , A > -> < F , R > Minor premise < F , A > Conclusion < F , R >

(26) (27) (28)

Definition 8. In the case given the form, the natural action and the function of a system, the reasoning process to construct the analytic reasoning of the system, i.e. to search for the natural laws which ascribe the behaviour of the system, is defined as inductive logic, which can be symbolically represented as { (Minor premise, conclusion) } { (F, A, R) } Major premise < F , ?L>

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THE INTRINSIC NATURE OF RECURSIVE LOGIC The existing three modes of reasoning are easy to understand. Since we define design logic as recursive reasoning, we should firstly give the justification of how designing is recursively implemented. We take it for granted that any compound proposition can be resolved into the singular ones whose truth value determines that of the former8, thereupon it is natural to redefine the term in Equations (34)-(36) as Fn = (F0, Fn_~) Ln = (Lo, Ln_l) An = (Ao, An-I) Rn = (Ro, Rn-~)

(37) (38) (39) (40)

any design problem, there must exist an atomic design set Fo of a certain type

For

F0 = {a0, a l ,

a2,.

• -,

an}

(41)

which satisfy the logic in Equations (42)-(44)

(31) (32) (33)

Definition 10. In the case given the environment and the function of a system, the reasoning process to construct the analytic reasoning of the system, i.e. to find the form which dominates the whole system, is defined as recursive logic, which can be symbolically represented as

(34) (35) (36)

Of the four logics illustrated above, only deductive reasoning is logically determinate. The other three are synthetic, and belong to the category of plausible reasoning. 'Induction is the inference of the rule from the cause and the results. Abduction is the inference of a case from a rule and result. '5 Recursion, however, is the inference of a case and partial rule from a result, which reflects the designer's presumption that a certain form might exist to satisfy the functions required. The recursive explanation of the design activity corresponds to the observation of many designers in different fields. Hein 6, the noted Danish poet and scientist, put it this way: 'Art is solving problems that cannot be formulated before they have been solved. The shaping of the question is part of the answer.' When Ritte 7 talked about the traits of designing, he said that 'you cannot understand the problem without having a concept of the solution in mind; and you cannot gather information meaningfully unless you have understood the problem but you cannot understand the problem without information about it. It is interesting to note that when these authors describe design verbally, the vocabulary and the sentence structure they use express themselves as something of recursive or 'dead-loop' characteristic.

(29) (30)

Definition 9. In the case given the form, the natural laws and the function of a system, the reasoning process to construct the analytic reasoning of the system, i.e. to find the cause which brings out the conclusion, is defined as abductive logic, which can be symbolically represented as Major premise Minor premise < F , R > Conclusion < F , ?A>

Major premise < ?F, L > Minor premise (?F, A, R) Conclusion ?F

Major premise Minor premise Conclusion

(42) (43) (44)

Thereupon, the recursive logic shown in the expressions (34)-(36) can be further resolved into the following process. (1)

Substitute (37)-(41) into (34)-(36) Major premise < (a0, ?Fn-1), (Lo, Ln_l) >

(45)

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(2)

Realization of atomic design

Minor premise ( (a0, ?Fn_l), (A0, An_l), (R0, R,_I) ) Conclusion

(46) (47)

Substitute (42)-(44) into (45)-(47) Major premise < ?Fn_l, L n_l > Minor premise ( ?Fn_l, An_l, R~_l) Conclusion ?Fn_x

(48) (49) (50)

(3)

Repeat steps (2) and (3) for Equations (48)-(50) until n = 1. Just as the prediction problem can be deductively solved, the empirical problem can be inductively solved and the diagnosis problem can be abductively solved, the design problem can be recursively solved as shown in the above process. It is the very intrinsic nature of the recursive logic. March 5 has put forward the apparent differences between these three regions: Logic has interests in abstract forms. Science investigates extant forms. Design initiates novel forms. A scientific hypothesis is not the same thing as a design hypothesis. A logical proposition is not to be mistaken for a design proposal. To give the further demarcation, he referred to abductive reasoning as the characteristic of design. 'Science must employ inductive reasoning in order to generalize, and design must use productive inference to particularize.' As shown in the third part of this paper, abductive (productive in March) reasoning is not the logic of the design. It can only reflect the researcher's presumption that a certain phenomenon might exist to account for his observations, which is in fact a diagnosis problem. So it can not be seen as the demarcation between design and science, and logic alike. The recursive process in Equations (45)-(50) qualifies recursive reasoning as the demarcation of design from science and logic, through which the design proposal is produced and tested. From the logical point of view, only when each atomic design and the recursive resolution of the problem remains true, the final design proposal becomes true. Each of the atomic designs and resolutions satisfies the design requirements from which one can produce and test the design proposal. But the conjecture and the refutation of the scientific hypothesis are two separated processes, and it can only be falsified9.

Looking at Equations (42)-(44), we can see clearly that to depart from the recursive loop of the design process the atomic form must be determinately produced. Generally speaking, it is a process involved in creative leaping. Our interest, however, is to put forward its mechanical realization. By now, the most commonly accepted design strategy is trial-and-error, by which a group of candidate design proposals may be firstly conjectured, then deductive logic is used to justify or reject them based on the design requirements. The third generation of design method whose basic model is conjecture-analysis assumes this form ~°'1~, and so are most numerical optimization techniques 12. The other feasible strategy is the pattern language proposed by Alexander 13'~4, where a deductive relation, between the design requirements, or as he called it the tendency, and the form, is constructed. In designing, the designer can use the disclosed patterns to compose the whole design solution. The acquisition of the pattern, according to Alexander, is a question of fact instead of a question of value. In this case the design problem becomes a science problem. The prototype model of design has drawn many researchers' attention recently to the development of design type expert systems ~5. It is based on the assumption that the expert designs with preconception and typology. Here the prototype in the expert's mind bears some resemblence to the pattern in 'pattern language' except that the prototype is subjectively determined. As the design cases and experiences accumulate, the complexity of either pattern or prototype increases, thereby, the recursion depth to solve design problem will decrease, considering the recursive process in Equations (37)-(50). It is the very phenomenon of design activity. It should be emphasized that the form cannot be generated by abduction even in this case. As Equations (42)-(44) tell, the major premise is still problematic, which doesn't satisfy abductive reasoning. Hence it can be concluded that not only in external appearance but also in internal nature abductive reasoning is not the logic of design.

Resolution of problem DESIGN METHODOLOGY As portrayed by definition 10, the difficulty of designing is the recursive property of the relation between the major premise and the conclusion of the logic. So the goal of design methodology is to find a way to jump out of such a loop. Most researchers in design studies during the last three decades have engaged in solving this problem. To effect the logic process given in (37)-(50), there are two features to be taken into account which are both the realization of atomic design and the recursive resolution of a design problem.

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However, the 'trial-and-error' method or 'pattern language' strategy can be successfully applied to many design problems, but there must be some situations in which the initial design candidate is almost impossible to be generated totally, so the technique of problem resolution has to be used instead. The first generation design method, that is the systematic design method, mainly dealt with this problem, where the first task is to gather all the design requirements and then classify them in the light of a certain specification. The solution of a design problem comes from the synthesis of all the subproblems resolved which are weakly dependent on each other 16'17.

DESIGN STUDIES

The traditional systems approach just provided a little strict resolution strategy. Most design requirements actually interconnect so strongly that the resolution is not such an easy task. In the case of recursive resolution, the need for independency is not necessary, only if in problem resolving the relation between two parts is taken as the new constraints exerted on the problem, which has been successfully i m p l e m e n t e d in a real design situation is. On the other hand, the depth of the resolution is varying even for a problem which relies on the form of the existing atomic design. If the atomic design represents a relatively complex relationship between the requirements and design forms, the design work is more directly done.

ACKNOWLEDGEMENTS The authors are grateful to the Council of the Chinese National Science Foundation for funding this project. Special thanks should be given to Prof. X.L. Liu, Prof. G.Q. He, Prof. X.H. Liu, Prof. G.Y. Wang, Prof. K.C. Shen, Prof. D.L. Li, Dr. G.X. Cao, Dr. M. Yan and Dr. Q. Chen.

REFERENCES 1 Cross, N (ed.), Developments in design methodology, Wiley, Chichester (1984) 2 Simon, H A, The sciences of the artificial, MIT Press, Cambridge, Mass (1969) 3 Mekeon, R (ed.), The basic works of Aristotle, Random House (1941) 4 Quoted by March, L J, from Hartsborne, C and Weiss P (eds), Collected papers of C S Peirce, Harvard University Press, Cambridge, Mass. (1931-5)

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March, L J, 'The logic of design', in Cross, N (ed.), Developments in design methodology, Wiley, Chichester (1984) 6 Quoted in Ching, F D K, Architecture:form, space and order, Van Nostrand Reinhold, New York (1979)

Rittel, H W J, 'Second-generation design methods', in Cross, N (ed.), Developments in design methodology, Wiley, Chichester (1984) Wittgenstein, L, Tractatus logico-philosophicus, Routledge and Kegan Paul, London (1922) (Chinese version) Popper, K, The logic of scientific discovery, Science Editions, New York (1961) 10 Hillier, B, Musgrove, J, O'Sullivan, P, 'Knowledge and design', in Cross, N (ed.), Developments in design methodo/ogy, Wiley, Chichester (1984) 11 Broadbent, G, 'The development of design methods', in Cross, N (ed.), Developments in design methodology, Wiley, Chichester (1984) 12 Cheng, G D, The foundation of engineering optimization, Press of Water Conservancy & Power (1984) (in Chinese)

13 Alexander, C, The timeless way of building, Oxford University Press, New York (1979) 14 Alexander, C, A pattern language, Oxford University Press, New York (1977) 15 Gero, J S, Maher, M L and Zhang, W, 'Design knowledge and representation', in Gero, J S (ed.), Artificial intelligence in engineering: design, Computational Mechanics Publications, Southampton (1988)

16 Alexander, C, Notes on the synthesis of form, Harvard University Press, Cambridge, Mass. (1964) 17 Jones, J C, 'A method of systematic design', in Cross, N (ed.), Developments in design methodology, Wiley, Chichester (1984) 18 Zeng, Y, Cheng, G D, Liu, H B, 'The logic, representation and implementation of design', Working Paper No. 90-3018, Research Institute of Engineering Mechanics, Dalian University of Technology, China

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