Proportional relationship definition and example of analogy

Against proportional four-part analogy (and what to replace it with) | Roland Pooth -

escaped the necessity of considering the analogy or relation of the finite to the for since God is by definition self-existent being, to affirm that God is good is . 20 A further example of purely metaphorical proportionality is provided by. proportion definition: Proportion is defined as a numerical relationship that ( noun) An example of proportion is the number of girls in a class compared to the from Latin proportio (“comparative relation, proportion, symmetry, analogy”), from. a definition similar to what Speech Com- munication . itself, proportional analogy is a relation that underlies and . may serve better than invented examples to.

Since then to fight against neighbours is an evil, and to fight against the Thebans is to fight against neighbours, it is clear that to fight against the Thebans is an evil. The argument from example thus amounts to single-case induction followed by deductive inference. In Aristotle's example, S the source is war between Phocians and Thebans, T the target is war between Athenians and Thebans, P is war between neighbours, and Q is evil.

The first inference dashed arrow is inductive; the second and third solid arrows are deductively valid. The paradeigma has an interesting feature: Instead of regarding this intermediate step as something reached by induction from a single case, we might instead regard it as a hidden presupposition.

This transforms the paradeigma into a syllogistic argument with a missing or enthymematic premise, and our attention shifts to possible means for establishing that premise with single-case induction as one such means.

The argument from likeness homoiotes seems to be closer than the paradeigma to our contemporary understanding of analogical arguments. The most important passage is the following. Try to secure admissions by means of likeness; for such admissions are plausible, and the universal involved is less patent; e. This argument resembles induction, but is not the same thing; for in induction it is the universal whose admission is secured from the particulars, whereas in arguments from likeness, what is secured is not the universal under which all the like cases fall.

Topics b10—17 This passage occurs in a work that offers advice for framing dialectical arguments when confronting a somewhat skeptical interlocutor. In such situations, it is best not to make one's argument depend upon securing agreement about any universal proposition.

The argument from likeness is thus clearly distinct from the paradeigma, where the universal proposition plays an essential role as an intermediate step in the argument.

The argument from likeness, though logically less straightforward than the paradeigma, is exactly the sort of analogical reasoning we want when we are unsure about underlying generalizations. In Topics I 17, Aristotle states that any shared attribute contributes some degree of likeness: We should also look at things which belong to the same genus, to see if any identical attribute belongs to them all, e.

Topics a13 It is natural to ask when the degree of likeness between two things is sufficiently great to warrant inferring a further likeness. In other words, when does the argument from likeness succeed? Aristotle does not answer explicitly, but a clue is provided by the way he justifies particular arguments from likeness. As Lloyd has observed, Aristotle typically justifies such arguments by articulating a causal principle which governs the two phenomena being compared.

For example, Aristotle explains the saltiness of the sea, by analogy with the saltiness of sweat, as a kind of residual earthy stuff exuded in natural processes such as heating. The common principle is this: Mete a17 From this method of justification, we might conjecture that Aristotle believes that the important similarities are those that enter into such general causal principles.

Summarizing, Aristotle's theory provides us with four important and influential criteria for the evaluation of analogical arguments: The strength of an analogy depends upon the number of similarities. Similarity reduces to identical properties and relations.

Good analogies derive from underlying common causes or general laws. A good analogical argument need not pre-suppose acquaintance with the underlying universal generalization. These four principles form the core of a common-sense model for evaluating analogical arguments which is not to say that they are correct; indeed, the first three will shortly be called into question.

The first, as we have seen, appears regularly in textbook discussions of analogy. Versions of the third are found in most sophisticated theories. The final point, which distinguishes the argument from likeness and the argument from example, is endorsed in many discussions of analogy e.

A slight generalization of Aristotle's first principle helps to prepare the way for discussion of later developments. As that principle suggests, Aristotle, in common with just about everyone else who has written about analogical reasoning, organizes his analysis of the argument form around overall similarity.

In the terminology of section 2. Hume makes the same point, though stated negatively, in his Dialogues Concerning Natural Religion: Wherever you depart, in the least, from the similarity of the cases, you diminish proportionably the evidence; and may at last bring it to a very weak analogy, which is confessedly liable to error and uncertainty.

Disagreement relates to the appropriate way of measuring overall similarity. Some theories assign greatest weight to material analogy, which refers to shared, and typically observable, features.

Others give prominence to formal analogy, emphasizing high-level structural correspondence. The next two sub-sections discuss representative accounts that illustrate these two approaches.

Hesse offers a sharpened version of Aristotle's theory, specifically focused on analogical arguments in the sciences. She formulates three requirements that an analogical argument must satisfy in order to be acceptable: Requirement of material analogy.

The horizontal relations must include similarities between observable properties. The essential properties and causal relations of the source domain must not have been shown to be part of the negative analogy. Material analogy is contrasted with formal analogy. A second example is the analogy between the flow of electric current in a wire and fluid in a pipe. This has the same mathematical form as Poiseuille's law for ideal fluids: Both of these systems can be represented by a common equation.

These are horizontal relationships of similarity between properties of objects in the source and the target. Similarities between echoes sound and reflection lightfor instance, were recognized long before we had any detailed theories about these phenomena. There are good reasons not to accept Hesse's requirement of material analogy, construed in this narrow way. First, it is apparent that formal analogies are the starting point in many important inferences.

That is certainly the case in mathematics, a field in which material analogy, in Hesse's sense, plays no role at all. Analogical arguments based on formal analogy have also been extremely influential in physics Steiner With reference to this broader meaning, Hesse proposes two additional material criteria.

Causal condition Hesse requires that the hypothetical analogy, the feature transferred to the target domain, be causally related to the positive analogy. She states the requirement as follows: The vertical relations in the model [source] are causal relations in some acceptable scientific sense, where there are no compelling a priori reasons for denying that causal relations of the same kind may hold between terms of the explanandum [target].

proportional relationship definition and example of analogy

It derives support from the observation that many analogies do appear to involve a transfer of causal knowledge. The causal condition is on the right track, but is arguably too restrictive. For example, it rules out analogical arguments in mathematics. Even if we limit attention to the empirical sciences, persuasive analogical arguments may be founded upon strong statistical correlation in the absence of any known causal connection.

Electrical fluid agrees with lightning in these particulars: Colour of the light. Being conducted by metals. Crack or noise in exploding. Subsisting in water or ice.

proportional relationship definition and example of analogy

Rending bodies it passes through. Let the experiment be made. Benjamin Franklin's Experiments, Franklin's hypothesis was based on a long list of properties common to the target lightning and source electrical fluid in the laboratory. Analogical arguments may be plausible even where there are no known causal relations.

Once it was discovered that heat was not conserved, however, the analogy became unacceptable according to Hesse because conservation was so central to the theory of fluid flow. This requirement, though once again on the right track, seems too restrictive. It can lead to the rejection of a good analogical argument. Consider the analogy between a two-dimensional rectangle and a three-dimensional box Example 7.

This does not mean that we should reject every analogy between rectangles and boxes out of hand. The problem derives from the fact that Hesse's condition is applied to the analogy relation independently of the use to which that relation is put.

What counts as essential should vary with the analogical argument. The causal condition and the no-essential-difference condition incorporate local factors, as urged by Norton, into the assessment of analogical arguments. These conditions, singly or taken together, imply that an analogical argument can fail to generate any support for its conclusion, even when there is a non-empty positive analogy.

They propose formal criteria for evaluating analogies, based on overall structural or syntactical similarity. Let us refer to theories oriented around such criteria as structuralist. A number of leading computational models of analogy are structuralist.

They are implemented in computer programs that begin with or sometimes build representations of the source and target domains, and then construct possible analogy mappings. First, the goodness of an analogical argument is based on the goodness of the associated analogy mapping. Second, the goodness of the analogy mapping is given by a metric that indicates how closely it approximates isomorphism. The most influential structuralist theory has been Gentner's structure-mapping theory, implemented in a program called the structure-mapping engine SME.

In its original form Gentnerthe theory assesses analogies on purely structural grounds. Analogies are about relations, rather than simple features. No matter what kind of knowledge causal models, plans, stories, etc. She further distinguishes among different orders of relations and functions, defined inductively.

Consider the sentence, Gravitational attraction between the sun and a planet, and the fact that the mass of the sun is much greater than that of the planet, causes the planet to orbit the sun. Gentner represents this in the following form: An analogy mapping M is a one-to-one function from the items in the source domain to those in the target, such that if R holds of objects a1, …, an in the source domain, then M R holds of objects R a1…, R an in the target.

The order of M R must be the same as the order of R. M may be a partial mapping; not every item in the source domain needs to have a target image. The best mapping M is determined by systematicity: Properties and functions are unimportant, unless they are part of a relational network. Gentner's Systematicity Principle states: A predicate that belongs to a mappable system of mutually interconnecting relationships is more likely to be imported into the target than is an isolated predicate.

And a systematic analogy one that places high-order relations and their components in correspondence is better than a less systematic analogy. Hence, an analogical inference has a degree of plausibility that increases monotonically with the degree of systematicity of the associated analogy mapping. Gentner's fundamental criterion for evaluating candidate analogies and analogical inferences thus depends solely upon the syntax of the given representations and not at all upon their content.

The contrast with Hesse's approach is striking. Still, some of the fundamental difficulties with the structure-mapping approach are easiest to appreciate if we focus on the early version. There is an obvious worry about hand-coded representations of source and target domains.

Supposing suitable representations of the two domains, does the value of an analogy derive entirely, or even chiefly, from systematicity? There appear to be two main difficulties with this view.

The idea is to compare the artifacts in the archaeological record to similar items in existing cultures. The strength of these analogies is based, to a considerable degree, on surface resemblances between the two artifacts, regardless of whether these resemblances are known to participate in elaborate relational networks.

Second and more significantly: Greater systematicity is neither necessary nor sufficient for a more plausible analogical inference. It is obvious that increased systematicity is not sufficient for increased plausibility.

An implausible analogy can be represented in a form that exhibits a high degree of structural parallelism. More pointedly, increased systematicity is not necessary for greater plausibility. Indeed, in causal analogies, it may even weaken the inference. They think, however, that this is a mistake; analogy is and remains a matter of terms. Properly, therefore, analogy is to be treated by logic.

There has been as yet no general and sharply focused Thomistic critique of this theory, but it would seem to raise serious difficulties with the traditional notion of logic as a science of second intentions, as well as to question the possibility of sciences other than logic, such as theology, to attain valid knowledge of the real. In the last analysis, ambiguous terms may serve to confuse or to persuade, perhaps even to suggest; but they would seem to have no place in anything like a science whose object is to know.

Only if analogy reaches into our knowledge and even into the objects of our knowledge can analogical predication be admitted as scientific. Thomas on Analogy For St. Thomas analogy is a kind of predication midway between univocation and equivocation.

Since analogy is the use of a term to designate a perfection found in a similar way in two or more subjects in which it is found partly the same and partly different, the first step is to clarify what is meant by the words "partly the same, partly different.

In univocal predication, predicates have an absolute meaning and can be accurately and distinctly defined in themselves. But strictly analogous predicates cannot be so defined; their meaning is proportional to the subjects of which they are predicated.

The reason for this difference is that univocal terms arise by abstraction from the particular subjects in which the perfection is present, so that the difference in the subjects does not enter into their meaning.

proportional relationship definition and example of analogy

Analogous terms do not arise by abstraction but rather by "separation," or negation In Boeth. A consequence of this is that there is no single clear meaning for an analogous predicate ST 1a, Nevertheless, analogous terms can be validly used in argumentation on condition that they are not mistaken for univocal concepts of essences. Even if we cannot judge the conclusion of a univocal argumentation without considering the premises from which it was derived, we can use that conclusion without keeping the original premises in mind as a premise for further argumentation.

But in an argument using an analogous term, the conclusion cannot be abstracted from its premises, because the secondary analogate arrived at by the argument must include the primary in its definition ST 1a, That is why our statements concerning the existence and nature of God are limited in scope ST 1a, 2.

Thomas made no concessions to rationalism, whether heterodox or Christian in intent, Averroistic or Anselmian. The kind of abstraction called separation arrives at terms which in all their major uses are analogous and which can therefore be considered primarily analogous terms.

In addition, some terms are first univocal but become analogous in other applications; these are secondarily analogous terms. Thus, some qualities of sensible things can be simply abstracted and are therefore univocal; but they can also be used to signify something that is primarily analogous. But in a second use, it can signify the kind of being that lives, and thus we use the term analogously when we speak of intellectual life. The most fundamental division of analogy is into intrinsic, or proper, and extrinsic, or improper De prin.

An analogy is intrinsic when the perfection which is predicated is really found in both of the analogates; it is extrinsic when it is really found in only one but imposed by the mind on others.

Analogy and Analogical Reasoning

Thus the term "living" is analogously applied to angels and animals by an intrinsic analogy, for both live but in an irreducibly different way. But when it is applied to the language of Shakespeare, it is applied by an extrinsic analogy; for life is only attributed, it is not found there in reality cf. The second way of dividing analogy is based on the relation between the analogates themselves De pot.

This relation can be directly between the two analogates, a "one-to-one," two-term analogy, as substance and accident are related to each other. Again, it can be a relation, not between the analogates themselves but between the analogates and some third object, as two accidents of a thing may be unrelated to each other but both be related to the same substance; this is a "many-to-one," three-term analogy.

Finally, there may be no direct relationships of the analogates at all, but each of them may contain a relation that is similar to the relation in the other, a "many-to-many," four-term analogy, also called "proportionality.

What he means is that as vision is to the power of sight, so direct understanding is to the intellect, not that the intellect is similar to the eye or vision to understanding.

Analogy and Analogical Reasoning (Stanford Encyclopedia of Philosophy)

Two- Three-and Four-Term Analogies. In twoand three-term analogies there is necessarily an inequality between the terms, so that the terms can always be compared to each other as greater and lesser De prin. Consequently one of the analogates will be "prior" to the other or others in time, in understanding, in perfection, in causality In 3 meta. Moreover, when there is a direct order between the terms, one of them will be defined by the other; that is, all the posterior analogates will be defined through the first.

But since definition is relative to knowledge In 5 meta. Thus we define accidents through their dependence on substance, for we know substance as prior; but we "define" God through the creatures' dependence on Him, for creatures are first in our knowledge and we know God only through them ST 1a, In four-term analogies, on the other hand, there is no direct relationship between the analogates; hence, priority of one over the other is not necessary and one is not defined through the other De ver.

Mutual determination and eminence. Two-and three-term analogies can be further divided according to the kind of relation that is in question. Two-term analogies are sometimes definite proportions which are mutually determining In 1 sent. For example, knowledge can be possessed habitually or actually exercised. In the former case, "knowing" is predicated as in potency and potency is always proportioned to the act of which it is the potency ; in the latter, as in act in creatures, an act is always the act of some potency.

So, too, analogous causes are often strictly proportioned to their effects In 4 meta. A direct relationship, however, need not be understood as a definite, interdetermining proportion. Indeed, the term "proportion" itself is often used by St.

Thomas to indicate an indefinitely greater perfection in the prior analogate In 3 sent. In such cases, the prior analogate possesses a perfection eminently, in a higher degree, more perfectly; whereas the others possess it deficiently, in a lesser degree, less perfectly C.

This language must not be allowed to mislead us. The expression "degree of difference" may refer to a difference that is directly quantitative or at least based on a directly quantitative one; and then there is no analogy, but univocation, for the perfection in question is reducible to a single one ST 1a, But at other times we speak of degrees of difference when the differences are greater than merely specific ones and cannot be reduced to univocal genera and differences De pot.

Similarly, we sometimes use the terms "perfect" and "imperfect" to refer to stages of one and the same perfection, as when we say that a baby has only an imperfect control of his limbs, whereas the grown man has perfect control; this also is univocation.

On the other hand, we might say that animals have an imperfect spontaneity because they are not merely passive to outside influences; whereas the spontaneity of a man is perfect, in the sense that his spontaneity is truly a freedom, not only from external violence, but also from other predeterminations cf.

One kind of analogy of eminence has additional characteristics. For the prior analogate can be more eminent because it is the analogous perfection by its essence, whereas the imperfect analogates are such because they possess that perfection as distinct from themselves, as received, and so as limited by their own proper nature. The primary analogate, then, is identically its perfection and so is unlimited in its order; if we are talking about being and the properties of being, the being by essence is simply infinite.

The secondary analogates, which have the perfections as received and limited, are being good and so on, by participation De pot.

In other perfections, too, a similar relation can be found. Thus, the acts of reason itself are reasonable by their essence, whereas the desires of a virtuous man are reasonable— truly enough and intrinsically—only by participation, inasmuch as through obedience to reason they possess some order, structure, and so forth, that is derived from reason De virt.

Three-term analogies are sometimes a set of two-term analogies with a common primary analogate which is numerically one and the same, as medicine, health, food and complexion are each called "healthy" by their various relations to the health of the animal C.

Such a form of analogy is not really distinctive, since it can be simply reduced to the two-term analogies which make it up.

At other times, however, the common term is not itself one of the analogates that are immediately understood but is entirely outside the predication or is a whole made up of all the analogates In 1 sent. In the latter case, evidently, the parts cannot be equal or quantitative parts; otherwise there would simply be univocation. Thomas, analogy is not simply a formal structure of predication to be treated in logic. When analogy is "applied" to a particular case, the content or matter of what is said must be taken into account.

For this reason, analogy is properly treated in metaphysics. The analogon most often and most fully discussed by St. But there is no single analogy of being; rather, the various beings have different relations to each other. Following the lead of Aristotle, St. Thomas finds in each being a set of internal components. Of these, the most thoroughly discussed principles are substance and accident.

proportional relationship definition and example of analogy

Moral particularism accepts analogical moral reasoning, rejecting both deduction and induction, since only the former can do without moral principles. Law[ edit ] In lawanalogy is primarily used to resolve issues on which there is no previous authority.

A distinction can be made between analogical reasoning employed in statutory law and analogical reasoning present in precedential law case law. Analogies in statutory law[ edit ] In statutory law analogy is used in order to fill the so-called lacunas or gaps or loopholes. First, a gap arises when a specific case or legal issue is not explicitly dealt with in written law. Then, one may try to identify a statutory provision which covers the cases that are similar to the case at hand and apply to this case this provision by analogy.

Such a gap, in civil law countries, is referred to as a gap extra legem outside of the lawwhile analogy which liquidates it is termed analogy extra legem outside of the law. The very case at hand is named: Second, a gap comes into being when there is a statutory provision which applies to the case at hand but this provision leads in this case to an unwanted outcome.

Then, upon analogy to another statutory provision that covers cases similar to the case at hand, this case is resolved upon this provision instead of the provision that applies to it directly. This gap is called a gap contra legem against the lawwhile analogy which fills this gap is referred to as analogy contra legem against the law. Third, a gap occurs when there is a statutory provision which regulates the case at hand, but this provision is vague or equivocal. A gap of this type is named gap intra legem within the law and analogy which deals with it is referred to as analogy intra legem within the law.

The similarity upon which statutory analogy depends on may stem from the resemblance of raw facts of the cases being compared, the purpose the so-called ratio legis which is generally the will of the legislature of a statutory provision which is applied by analogy or some other sources. Statutory analogy may be also based upon more than one statutory provision or even a spirit of law.

In the latter case, it is called analogy iuris from the law in general as opposed to analogy legis from a specific legal provision or provisions. In statutory law analogy is also sometimes applied in order to liquidate the so-called conflicting or logical gap i.

The judge who decides the case at hand may find that the facts of this case are similar to the facts of one of precedential cases to an extent that the outcomes of these cases are justified to be the same or similar. Such use of analogy in precedential law pertains mainly to the so-called: Second, in precedential law, reasoning from dis analogy is amply employed, while a judge is distinguishing a precedent.

That is, upon the discerned differences between the case at hand and the precedential case, a judge reject to decide the case upon the precedent whose ratio decidendi precedential rule embraces the case at hand. Third, there is also much room for some other usages of analogy in the province of precedential law.

One of them is resort to analogical reasoning, while resolving the conflict between two or more precedents which all apply to the case at hand despite dictating different legal outcome for that case. Analogy can also take part in ascertaining the contents of ratio decidendi, deciding upon obsolete precedents or quoting precedents form other jurisdictions. It is too visible in legal eductaion, notably in the US the so-called 'case method'. An argument from analogy employed in precedential law is called case analogy as opposed to analogy employed in statutory law which is accordingly termed statutory analogy.

Then, there are compared instances to which a given rule applies with certainty with the facts of the case at hand. If the sufficient relevant similarity between them obtains, the rule is applied to the case at hand. Otherwise, the rule is deemed as inadequate for this case. Such analogy becomes a legal method.