Locke claims that ideas of primary qualities resemble the qualities themselves, while ideas of secondary qualities do not.
What does he mean by this?
Does he succeed in establishing it?

The claim mentioned in the Essay title is from Section 15 of Chapter 8 in Book II of Locke's "Essay Concerning Human Understanding"-

Ideas of primary qualities are resemblances; of secondary, not. From whence I think it easy to draw this observation,�" that the ideas of primary qualities of bodies are resemblances of them, and their patterns do really exist in the bodies themselves, but the ideas produced in us by these secondary qualities have no resemblance of them at all. There is nothing like our ideas, existing in the bodies themselves. They are, in the bodies we denominate from them, only a power to produce those sensations in us: and what is sweet, blue, or warm in idea, is but the certain bulk, figure, and motion of the insensible parts, in the bodies themselves, which we call so. [Essay II,viii,15](1)

To understand what Locke means by this claim, we must go back a bit, and start with what Locke understands by "qualities".   And to do that, we must understand the background of his Essay.   One of the primary purposes of his Essay was to provide a sound philosophical foundation for the "Corpuscular Hypothesis"of Boyle.   In Locke's judgement, the then new mechanistic / corpuscularian (atomic) physics was the best science available.   This was in specific contrast with the then traditional Scholastic notions of the teleology of Aristotelean "formal"and "final"causes.   The new mechanistic view of physics meant that the cause of all change was to be found in the basic properties of the fundamental properties of the "atoms"that constituted all things material.

Locke further maintained that we have privileged access to our internal mental environment.   We have access to the external environment only through the ideas that are caused in our minds by that external environment.   The things we think normally of as "qualities"of things, Locke maintained are actually ideas in our minds.   Those ideas are caused by things in the external environment.   So for Locke, ideas are the primary existents of our mental worlds.   "Qualities"are not what we think they are.

Whatsoever the mind perceives in itself, or is the immediate object of perception, thought, or understanding, that I call idea; and the power to produce any idea in our mind, I call quality of the subject wherein that power is. Thus a snowball having the power to produce in us the ideas of white, cold, and round,�" the power to produce those ideas in us, as they are in the snowball, I call qualities; and as they are sensations or perceptions in our understandings, I call them ideas; which ideas, if I speak of sometimes as in the things themselves, I would be understood to mean those qualities in the objects which produce them in us. [Essay II,viii,8 -- my emphasis]

Locke here draws a clear distinction between the ideas that we have of the nature of things, and the powers that are in the things that are the causes of the ideas that we have.   The former -- the ideas of the nature of things -- are what we are used to calling "qualities."The latter -- the causes of those ideas -- are the powers of those things to cause the ideas in our minds.   For Locke's chosen terminology, the actual or real "qualities"of things are the causes of those ideas in our minds -- the powers in the objects, not the ideas themselves.

Although Locke here also warns the reader that he will "sometimes"(although in fact he does this more often than not) speak about qualities as if these were the ideas in our minds.   When he does so, he wants the reader to understand that he really means the powers in the objects to cause the relevant ideas in our minds.

[M]ost of the simple ideas that make up our complex ideas of substances, when truly considered, are only powers, however we are apt to take them for positive qualities; v.g. the greatest part of the ideas that make our complex idea of gold are yellowness, great weight, ductility, fusibility, and solubility in aqua regia, &c., all united together in an unknown substratum: all which ideas are nothing else but so many relations to other substances; and are not really in the gold, considered barely in itself, though they depend on those real and primary qualities of its internal constitution, whereby it has a fitness differently to operate, and be operated on by several other substances. [Essay II,xxiii,37]

To Locke, then, "qualities"are powers in objects to cause ideas our minds.   And Locke divides these "powers"into three different categories:

(i)   Primary powers or Primary Qualities consist of those powers of things to cause certain sorts of ideas in minds such as ours.   He lists them as: shape (figure), size (extension), position, number, mobility (motion or rest), solidity, and texture.

(ii)   Secondary powers or Secondary Qualities are powers, resulting from the primary qualities of their corpuscular constituents, to cause certain sorts of ideas in minds such as ours.   He lists them as "all other qualities of things"such as sounds, colours, tastes, smells, etc.

(iii)   Tertiary powers (Locke does not speak of them as Qualities) are powers, resulting from the primary qualities of their corpuscular constituents, to cause certain sorts of changes in other things.

The distinction between primary and secondary qualities is meant by Locke to be a distinction between different kinds of powers.   Not, as is sometimes attributed to him, a distinction between powers and qualities or relations and qualities.

So, given this understanding of what Locke means by a "quality", what does he mean by "Ideas of primary qualities are resemblances; of secondary, not"?   To understand this, we must recognize that Locke maintains a causal theory of perception, not a representational one as is usually attributed to him.   Locke is quite clear that it is the powers in things that cause ideas in minds.   It is the real qualities of things that cause the ideas of qualities in our minds.  

Consider this candidate presentation of Locke's causal theory of perception -

                   (L1)         S perceives that O is Q if and only if
                                              (i)     O is Q; and
                                              (ii)   O has the power to cause (in the normal way) an idea of Q in S.

The key to Locke's Primary / Secondary distinction is the criterion (i).   Because Locke adheres to Boyle's Corpuscular Hypothesis, he maintains that in some cases (the cases of Primary Qualities), O is in fact Q.   But in other cases (the cases of the Secondary Qualities), O is not in fact Q.   We can therefore see that in the case of Primary Qualities, when S has the idea that O is Q, and O is in fact Q, then the idea resembles the facts of the matter.   But in the case of Secondary Qualities, when S has the idea that O is Q, and O is not in fact Q, then the idea does not resemble the facts of the matter.   The words we use to describe the causes will in some cases (the Primary Qualities) resemble the words we use to describe the ideas.   In other cases (the Secondary Qualities), the words we use to describe the causes do not resemble the words we use to describe the idea.  

Basing his analysis on the Corpuscular Hypothesis of Boyle, those qualities that Locke classifies as Primary, are those that he argues are the fundamental properties of the atoms or corpuscles that make up the things we perceive.   Secondary qualities are those that he argues are not properties of the corpuscles themselves, but are caused in us by various configurations and motions of those corpuscles.   Hence, the snowball is round because the configuration of the corpuscles is round.   But the snowball is cold because the motion of the corpuscles is slow.   In the case of shape, the words describing the idea resemble the words describing the configuration of corpuscles.   In the case of warmth, the words describing the idea do not resemble the words describing the motion of the corpuscles.

Does Locke succeed in establishing this principle of resemblance to an acceptable extent?   To a certain extent he does, of course, simply as a matter of definition.   The more fundamental question is whether or not any particular quality should be classed as a Primary Quality -- as being a basic property of the atoms of matter, or a Secondary Quality -- as being the causal result of more basic properties of atoms.   And, of course, the related question of whether, in fact, there are any qualities that should be classed in one category or another.   For these questions, one can either adopt the level of science knowledge available to Locke, or one can adopt the best available science knowledge of today.   On either basis, one can debate whether any given quality should be considered Primary or Secondary.   But these are irrelevant concerns to this particular essay.

Given Locke's distinction between the idea and the power in objects to cause those ideas, and given Boyle's Corpuscular Hypothesis, it is relatively straight forward to demonstrate that Locke's causal theory of perception will inevitably set up a distinction between those cases where O is in fact Q and those cases where O is not in fact Q even when in both cases, S perceives that O is Q.   And given the additional assumption that there is at least one perceived quality that is not in fact a property of the atoms that constitute the material things we perceive, then Locke's claim that ideas of primary qualities resemble the qualities themselves, while ideas of secondary qualities do not, is adequately established.

References

(1)   Locke, John (Author) & Yolton, John W. (Editor);   Essay Concerning Human Understanding, Everyman's Library, Dent Publishing, London England, 1961. ISBN: 0-460-00332-1.   All quotations from the Essay will be cited as [Essay book.chapter.section] of this edition.

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