Priest (12.1) Beyond the Limits of Thought, ‘Frege, Concept and Object,’ summary

 

by Corry Shores

 

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[The following is summary. Boldface and bracketed commentary are my own. You will probably encounter typos and other distracting errors, because proofreading is incomplete. I apologize in advance.]

 

 

 

 

Graham Priest

 

Beyond the Limits of Thought

 

Part 4 Language and Its Limits

 

Ch.12 The Unity of Thought

 

12.1 Frege, Concept and Object

 

 

 

Brief summary:

There is an aporia in Frege’s groundbreaking theory of meaning, namely, that the sense and reference of larger structures is compositionally based on that of its parts, but precisely how to conceptualize the way this compositional relation works is not entirely apparent. Frege’s notion of the unsaturated function structure is our best means for understanding this compositionality. For, the sense and reference of the whole does not result simply from the bare addition of the sentence’s parts. Rather, they must be combined and related by means of the operation of the function upon the terms in its argument place.

 

 

 

Summary

 

Priest explains that Frege’s major philosophical project was “to demonstrate that mathematics (or, at least, analysis) was logic” (Priest 197). [Nolt in section 14.1 of Logics mentions this and notes that it is called “logicism”.] The discovery of Russell’s paradox ruined hopes in this project. But there was a “subsidiary” part of Frege’s project, namely, a formulation for a systematic philosophy of language (Priest 197). Priest explains that “The clarity and power of its structure | had never before been achieved” (Priest 198). So even though this was just a side-project, it became “one of the major influences on twentieth-century philosophy” (198).

 

Priest’s main concern in Frege’s philosophy of meaning will be a particular aporia found especially in “Function and Concept,” “On Sense and Reference,” and “On Concept and Object” (198).

 

Priest then explains Frege’s important innovations. [It seems that before Frege there was a certain understanding of the subject-predicate relation, but Frege modifies it by replacing the category of subject with that of name. This means that a predicate can contain within it these names which are of a different category. So this also means that the category of name is wider than that of subject (since subjects cannot be parts of predicates). But also note that a subject can be an expression like “all men” with the predicate being something like “are mortal.” However, Frege’s category of name does not include such quantified expressions as “all men”, so in that sense the category of name is more limited than that of subject. (Recall from “On Concept and Object” Frege’s explanation for this. He shows that the proper negation of a quantified expression would belong to the quantified expression (that is, it would be positioned in front of the quantified expression), which actually means it belongs to the predicate. I am still not exactly sure what the reasoning is, but what I proposed in that summary as a possible explanation for the reasoning is that the negation term should always be affixed to the predicate (since we are denying the predication to some object), and since the proper place to put it is next to the quantifier in such quantified expressions as “all mammals are land dwellers”, thereby giving us, “not all mammals are land dwellers”, that means the “all” must belong to the predicate and not to the noun coming after it. He wrote:

It must here be remarked that the words ‘all,’ ‘any,’ ‘no,’ ‘some,’ are prefixed to concept-words. In universal and particular affirmative and negative sentences, we are expressing relations between concepts; we use these words to indicate the special kind of relation. They are thus, logically speaking, not to be more closely associated with the concept-words that follow them, but are to be related to the sentence as a whole. It is easy to see this in the case of negation. If in the sentence

‘all mammals are land-dwellers’

the phrase ‘all mammals’ expressed the logical subject of the predicate are land- dwellers, then in order to negate the whole sentence we should have to negate the predicate: ‘are not land-dwellers.’ Instead, we must put the ‘not’ in front of ‘all’; from which it follows that ‘all’ logically belongs with the predicate. On the other hand, we do negate the sentence ‘The concept mammal is subordinate to the concept land-dweller' by negating the predicate: ‘is not subordinate to the concept land-dweller.’

(Frege, “On Concept and Object”, 48)

But this is not an essential point in our current context.) Priest then says that names include proper names and definite descriptions. And Frege also will offer a fresh conception of the predicate. Priest says that instead we are to think of the concept-expression. [From the way Priest describes them, they seem to be the part of the function minus the argument, or the function properly speaking, or the unsaturated part of the function. I am not sure how to conceive it. Frege describes this structure in “Function and Concept”. Let us look first at a mathematical structure and then relate it to a sentential structure. In this text, Frege says that a function can be understood as having two main parts: a variable part and a non-variable part. He illustrates with a function that when its variable part is assigned three different values it produces these formulations:

‘2∙1³ + 1,’

‘2∙2³ + 2,’

‘2∙4³ + 4,’

The first stands for 3, the second, 18, and the third, 132. We ask, what part is variant and what part is not? We might think of the variant part as being like an opening or gap in the structure where the variant argument inputs can be placed. So we might display the structure as:

‘2∙(   )³ +(   )’

More conventionally we use letter variable symbols for those gaps. So we are more accustomed to seeing:

‘2∙x³ + x’

(See Frege, “Function and Concept”, pp.21-24)

If the argument is not specified, as when there is just a gap or variable in the structure, then it is said to be ‘unsaturated’, Frege says. But when instead a specific argument is placed into the gap where it arguments are supposed to be placed, then we say that the function is ‘saturated’. This structure also applies to concepts which are expressed using sentence formulations. We can have the function (or as Priest will call it, the ‘concept-expression’) “... is the capital of England” (Frege will just write it as “the capital of England” but it is to be understood as a function and not as a name). Here we have marked the gap with an ellipsis. When we fill that argument place with ‘Sydney’ it is false, but it is true if the argument is ‘London’. For Priest it seems that the ‘name’ is to be understood as the argument in this function structure, and he calls the function part the ‘concept-expression’. But I might have this wrong. He says that the concept-expression is everything minus the names. But I wonder what that means for the ‘London is the capital of England’ example. ‘London’ and ‘England’ would seem to be names, and then “... is the capital of ...” would then seem to be the expression. Or is ‘capital’ also a name, and thus the concept expression would be ‘... is the ... of ...”? Or is ‘the capital of England’ all one name, and thus the concept-express would be ‘... is ...’? In other words, I am not sure how what Priest is calling the category of name corresponds to the ‘argument’ in a function, since I would think that names could appear in concept-expressions without being variable parts of the expression. The point here might be that “...is the capital of England” is all one concept, but it is itself a function composed of the concept/function “is the capital of ...”, which is a function/concept composed of “is the ... of ...”. I am just wondering. But that would make Priest’s definition of ‘name’ fit how I understand what Frege considers the argument of a function or concept.]

Frege takes the traditional distinction between subject and predicate, and refashions it for his own ends. Instead of the category of subject, Frege proposes the category name. This is wider than the traditional category, since it includes those noun-phrases that occur within the predicate as a grammatical object. But it is also narrower than the traditional category, since it excludes quantifier phrases such as ‘all men’. We are left with proper names and definite descriptions. Instead of the category of predicate, Frege proposes the category concept-expression. A concept-expression is what is left when names are deleted from a sentence. Thus, in ‘Oswald was framed for the murder of Kennedy’, ‘Oswald’ and ‘the murder of Kennedy’ are names and ‘was framed for’ is a concept- expression.

(Priest 198)

 

Priest next addresses another of Frege’s innovations, namely, his new way of conceiving the distinction between connotation and denotation, which are two notions for meaning. Frege’s terms are sense and reference (or Sinn and Bedeutung). The reference is the denotation. For a name the denotation is the object it refers to, and for concept-expression it is a concept. [Let me review the understanding we came to when summarizing Frege’s “On Sense and Reference.” The main distinctions we made there were between a name and a sentence and between sense and reference. The reference of a sign is the thing it designates, and its sense is the contextual conceptual contents that are evoked by the sign in its designation of some object. We can also think of it as the mode by which that object is presented by means of the sign. To illustrate how sense can vary for the same object, Frege offers three useful examples. The first in fact comes from the Begriffsschrift section 8, and it perhaps is not the ideal example for this; but let us begin with it to see what insights it provides in this regard. The basic idea is that we will have two points that begin by not coinciding, and then after a movement takes place, they later will coincide. Frege’s main point will be that when the points coincide, insofar as they occupy that place, they have no need of two different names. But, he argues, that does not mean we have reason to drop one of the two names. For, the way this point of correspondence is designated requires first a distinction not just of the names but also of the things they designate. So, it seems he is saying, when they do coincide, we should still distinguish them, because the mode by which that coincidence is presented requires separate names and entities. (Again, see Begriffsschrift section 8 for a fuller account of this example, along with illustration.) The two other examples come from “On Sense and Reference,” and they perhaps are more obviously illustrative of the difference between the sense and reference of a sign. In the first one, we are to think of a triangle where we might draw lines from any of its vertices to the opposite side’s midpoint.

The point of intersection at the center can be designated with just two of the lines. But the sense will differ depending on which two lines are chosen. He has us consider the point designated as being the intersection of lines a and b.

or as being the point of intersection between lines b and c.

So the same point can be designated by means of two different modes of presentation, each with its own sense. This is why I defined the sense of a sign as being the contextual conceptual material involved in its conception. And by this I do not mean to confuse it with what Frege calls an ‘idea’, which would be the subject associative mental content involved when one is conceptualizing something. The contextual conceptual material I refer to are objective in the sense that they are specified explicitly and would be common to all people conceiving the sense by means of that particular mode of designating the object. Different lines are used for each manner of designating the point of intersection, and so were we to form a concept of it by means of any of the various modes of presenting the object, there would be different contextual conceptual material in each case even though those different sets of contents would designate exactly the same object. The other example is the famous case of ‘the morning star’ and ‘the evening star’. Here, they have different senses, even though they designate the same object. Both of course are Venus, the third brightest heavenly body. But to conceive of Venus with the sense of it being the evening star means to conceive it for example as being the first star to appear at dusk. And to conceive it with the sense of it being the morning star means to conceive it for example as being the last star to disappear in the morning. One reason this example is excellent is that it highlights the connotational component of sense. And in this case of ‘the morning star’ and ‘the evening star’, that connotational difference takes on a poetic quality that is hard to miss. So Priest notes that the denotation (reference) of a name is an object, and the denotation of a concept-expression is a concept. Here is something in addition to what we determined, for we said that the denotation (reference) of a sentence is a truth value., but we did not in our summary mention the reference of the concept-expression. The difference here seems to be that the concept-expression is not a full sentence but is rather the sentence minus the name(s). Priest then says that the sense of a linguistic unit is in general what determines which thing (be it an object, concept, or truth value) is the correct referent. This is not something that I was able to discern in the texts, but I have come across similar such interpretations. For example, in Roger Vergauwen’s A Metalogical Theory of Reference he writes, “Informally stated, an intension is something which makes it possible, in any ‘possible world’, to recognize or determine the extension of a specific element” (Vergauwen 30). Also on this point are also course recordings of John Campbell’s “Theory of Meaning” class at UC Berkeley. In the first lecture he discusses Frege’s “On Sense and Reference”. At around 33 minutes he begins discussing this role of sense in connecting signs with things in the world (and the part we discuss in more detail comes at around 36 minutes). He makes the point that sense explains informativeness. So we know that “the morning star” and “the evening star” are two names for the same object. But ‘the morning star is the evening star’ is informative, because they have different senses. Suppose we say that another name for point A is point alpha (α), such that when we say point A we can if we want instead say or think point α. This means that, so far as we can tell, there is no difference in sense between the terms, and so there is nothing informative in making formulating the equality ‘a = α’. But the next step in Campbell’s reasoning, which will make this point that sense is what serves to pick out the reference, I do not follow very well (again see around 36 minutes). He says that because sense explains informativeness, sense fixes the reference of the sign, and in fact the sense uniquely determines the sign’s reference. He explains that sameness of sense makes the identity uninformative, therefore sameness of sense must guarantee sameness of reference. This should be easy to grasp, but I do not follow really. So in our example, we have ‘a’ and ‘α’. We assumed that both signs use the same conceptual contents when designating their objects. So they are both determined by the same means and with the same conceptual components (unlike the point determined by lines a and b and by b and c in the triangle example). The fact that they have the same sense means that all of these conceptual determinations that are bound up in its designation of the object are identical, in which case it would seem to be that they would have to designate the same object. So sameness of sense must guarantee sameness of reference. But still I am not sure that I followed that right. The next inference, which is based on this prior one, is that we can then conclude that sense determines reference. But I do not see exactly how we are supposed to make that inference, because two senses can pick out the same reference, as the morning star and evening star example illustrates. (Basically the difficulty in my understanding is that if you told me that two senses can pick out the same reference, then I would be inclined to think that we need instead to find something they have in common to explain why they pick out a common reference, and not use something that makes them distinct. This of course is just a failure in my logic abilities, so I would need a simplified elaboration that makes the steps of reasoning more explicit.) I would have thought that instead the reasoning for why the sense determines the reference is because the sense determines what qualifies as such a thing under this designation. So the reason ‘the morning star’ designates Venus is because the sense of ‘the morning star’ is being the thing that is the last star to disappear in the morning (or being the brightest star in the morning) which is Venus, and the reason that ‘the evening star’ designates Venus is because the sense of ‘the evening star’ is being the first star to appear at dusk (or being the brightest star at dusk) which is Venus. In other words, the sense of ‘the morning star’ picks out Venus in the night sky because it directs our attention to the appropriate thing by means of properties and determinations that are properties and determinations that belong to Venus. And depending on its temporal (and spatial) context, Venus has different properties and determinations, meaning that different senses can correspond to or belong to it (or to names designating it). At any rate, my point here in these comments is that I do not find Frege directly making this point that sense is what determines the correct referent of a sign or concept-expression. But I can see how it is implied by the way he defines sense in terms of mode of presentation of the designated thing. For, the mode of presentation is the means by which some sign is correlated to the determining features of some thing. Priest’s last point in this paragraph is that the objective thought expressed by a statement, that is, the proposition it expresses, is its sense.]

Frege also reshaped the traditional distinction between two notions of meaning: connotation and denotation. He distinguished between the sense (sinn), of a linguistic unit and its referent (bedeutung). According to Frege, all linguistic units have both a sense and a reference (denotation). The denotation of a name is an object; the denotation of a concept-expression is a concept. The denotation of a statement is a truth value (true or false). The sense of a linguistic unit is, in general, that which determines which object/concept/truth value is the correct referent. In the case of a statement, this is the (objective) thought expressed by it (the proposition expressed by it).

(Priest 198)

 

Priest’s next point about Frege’s theory of meaning is that it involves compositionality in that

the meaning of a compound linguistic expression is, in some sense, a function of the meanings of its parts. Frege thought that, by and large, the referent of an expression was a function of the referents of its parts, and the sense of an expression was a function of the senses of its parts. 

(Priest 198)

 

Priest now wonders, given that meaning in larger complex structures operates compositionally, how do the senses or the referents of the parts contribute to or produce the sense or references of the whole? Priest observes a problem with forming this conception. The meaning of a whole, like a whole sentence, is a unity of sorts. So for instance, “the thought that Brutus killed Caesar, for example, is a single thought” (Priest 199). This means that the meaning of a whole sentence is not the bare combination of the parts. “It is not, therefore, a mere congeries of the meanings of its parts: <the sense of ‘Brutus’, the sense of ‘killed’, the sense of ‘Caesar’>” (Priest 199). Frege explains the way this compositionality works with respect to reference, so Priest will give that account, but Priest says that “a parallel story is to be told for sense” (199).

 

Priest notes how for Frege, a concept can be understood as a function “that maps an object to a truth value”. [I do not follow these points very well. Priest is discussing a “problem”, which as I understand it is the problem of explaining how the sense or reference of the parts of a sentence compositionally contribute to the sense or reference of the whole. He notes that a mathematical function, like sin, maps a number to another number. For example, sin maps the  the value π to the value zero (see Suppes’ explanation of functions in terms of relations and ‘mapping’ in his Intro section 11.1). And as we noted, Priest says that a concept is a function that maps an object to a truth value. I am not exactly sure what the “object” is in this case. I suppose it would be the argument. So for Frege’s example of the concept-function “... is the capital of England”, it maps “London” to the True and “Sydney” to the false. Another interpretation I suppose could be that the function maps the whole sentence to a truth value, but that would seem strange that the function would include or reference itself in that way. But I am not sure. I then do not understand so well the next point. Priest says, “This does not solve the problem, since exactly the same problem arises with respect to a function and its arguments: ‘sin(π)’ is an expression referring to a single entity (the number zero); it is quite different from <sin, π>” (Priest 199). In the Suppes section I mentioned, we have a function also described as a binary relation whose extension is a set of ordered couples, where the first member of a couple is like the argument of the function and the second member of the couple is like the output value of the function. So for example, the function f(x) = x² can be understood as relation whose extension is the set including {<1,1>, <2,4>, <3,9>...} and so on. So I was a little confused at first by Priest’s formulation <sin, π>. I think now that he is not using the ordered n-tuple structure, but rather I think he means, like he seemed to above with <the sense of ‘Brutus’, the sense of ‘killed’, the sense of ‘Caesar’> to be listing the bare parts of the function sin(π), which would be <sin, π>. With that in mind, I think Priest’s point is that if we only think in terms of the composition in this simple manner of raw combination or concatenation, we still cannot explain how on the basis of combining the parts {sin, π} that we can explain why the function’s value is zero. We need something in addition to that. Frege’s answer is that the function is unsaturated, in that it has gaps. I am not sure how this concept of saturation explains the compositionality, however. So we have the parts {sin, π}. But this is not a function. It is just a set of parts which could make up a function. Instead, a function has more of a structure of something like: value gap, and operation on the value in that gap. I still am not sure how to articulate how this explains compositionality. As I understood from Frege’s analysis in “On Sense and Reference”, the compositionality was a matter of sentences with multiple clauses, and he showed how in the qualifying cases, the value of the whole was determined by the value of the component clauses. I am not sure how this would work for saying that a singular function’s sense or reference is somehow based on the sense and reference of its argument and its functional component. I wonder if perhaps one way to understand this would be the following. We have the function-concept, “the capital of England”. It has an extension (a reference), which includes just one member, namely, the city London. Then we consider two names, “London”  and “Sydney”. In Sydney’s extension (its reference) is the city Sydney, and in London’s extension (its reference) is the city London. Now, when we combine the references of “Sydney” and “capital of England” in a raw way, we get, <{Sydney}, {London}> or something like that. But when we combine it for “London”  and “capital of England”, we get <{London}, {London}>. We say that “London is the capital of England” has as its reference the True, and as its sense, the concept of London being the capital of England. So perhaps we might say that the reference of the whole saturated concept-function is built compositionally on the basis of the references of the parts by means of some sort of additional evaluative function which somehow assigns to couplings where there values are identical (or where the first term is found in the set coming in the second place), like <{London}, {London}>, the value true and ones where they are not identical (or where the first term is not found in the set of members in the set comprising the second term) like <{Sydney}, {London}> the value false. But of course we are not saying that the falsity of the whole sentence “Sydney is the capital of England” is somehow based on the truth values of the parts, because they have no truth values. Rather, the references (or extensions) of the parts are still determinative of the reference of the whole sentence, if we can apply an additional evaluative procedure which assigns truth when the argument is included in the relation’s set. But what about sense? We said that the sense of a term is a matter of its mode of presentation, which we specified as the conceptual determinations (some of which being contextual and arbitrarily related, like the contextual elements bound up in the sense of “the evening star”) that pick out the proper reference to the term. And the sense of a sentence is the concept it expresses, often taking the form of predication. We also had the idea of the sense of a concept-expression (a function), which I think is a more general sort of conception, like “being the capital of England” or just “being the capital of”.  So how might the sense of the whole expression be composed of the sense of its parts? I can only guess; I am sorry. But perhaps the idea is like the following. The argument term as a name has a sense, which is the conceptual material by which the reference is determined as the sign’s proper object. So the sense of “London” is whatever conceptual determinations being used to adequately point us to the city in question. Perhaps this includes geographical co-ordinates, or descriptions of distinguishing features of the city, or something like that. And we also in our example have the concept of “being the capital of England”. I would suspect that contained somehow in this sense are implied criteria for what would qualify as a proper argument. So for example, part of what is implied in the sense of that expression is that the city is geographically located somewhere within English territory. And also implied in its sense is that the city functions the way a capital city functions in a nation of the sort that England is. And so on. So I am just making wild guesses, but perhaps we can say that the sense of the whole sentence is based compositionally on the parts, when we again have some evaluative function or operation which detects whether the determining conceptual material that picks out the city London also picks out an object that fulfills all the criteria implied in the concept. At any rate, somehow the notions of gaps and saturation in the function explain how the sense and reference of the whole structure is based on the sense and reference of the parts. Priest then says that these terms Frege is using are just metaphors, but they are the best means we have at the moment for conceptualizing this notion. And Priest further says that we have reached “bedrock”. Perhaps he means that since we have no better means for understanding the compositionality, that we can really not go too much further in our analysis and understanding. Or perhaps he means something like the point Frege makes about the fact that the argument and the concept cannot be defined, because they are logically simple in that they do not contain any parts (see “On Concept and Object” pp.42-43).]

According to Frege, a concept is a function, like the mathematical function, sin, which maps a number to another number. A concept is a function that maps an object to a truth value. This does not solve the problem, since exactly the same problem arises with respect to a function and its arguments: ‘sin(π)’ is an expression referring to a single entity (the number zero); it is quite different from <sin, π>. Frege's solution to the problem is that a function is, in some sense, inherently ‘gappy’. Objects (the arguments of the function) may fill those gaps, giving completion. As he puts it (p. 24):

The argument does not belong with the function, but goes together with the function to make up a complete whole; for the function by itself must be called incomplete, in need of supplementation, or ‘unsaturated’. And in this respect functions differ fundamentally from numbers [i.e. , objects].

The words ‘incomplete’, ‘unsaturated’, etc. are, of course, metaphors. Frege realised this, but could do no better; neither can I. At this point we seem to have reached bedrock.

(199)

 

 

 

 

 

From:

 

Graham Priest. Beyond the Limits of Thought. Cambridge: Cambridge University, 1995.

 

 

Or if otherwise noted:

 

Gottlob Frege. “Begriffsschrift (Chapter 1)”. Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).

 

Gottlob Frege. “Function and Concept.” Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).

 

Gottlob Frege. “On Concept and Object”. Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).

 

Gottlob Frege. “On Sense and Reference”. Transl. P.T. Geach. In Translations from the Philosophical Writings of Gottlob Frege. Eds. P.T. Geach and Max Black. Oxford: Basil Blackwell, 1960, second edition (1952 first edition).

 

 

Roger Vergauwen. A Metalogical Theory of Reference: Realism and Essentialism in Semantics. Lanham / New York / London: University Press of America, 1993.

 

John Campbell. “Lecture 1” of Philosophy 135: Theory of Meaning. Recorded course of UC Berkeley. On youtube at:

https://youtu.be/vN_yGxabNFw

Course listed at:

http://www.openculture.com/freeonlinecourses

 

 

.

Nolt (11.4) Logics, ‘Inference in Leibnizian Logic,’ summary

 

by Corry Shores

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[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own.]

 

 

 

Summary of

 

John Nolt

 

Logics

 

Part 4: Extensions of Classical Logic

 

Chapter 11: Leibnizian Modal Logic

 

11.4 Inference in Leibnizian Logic

 

 

 

 

Brief summary:

Making proofs using Leibnizian modal logic will involve all the rules from propositional logic along with the identity rules and the following seven additional rules.

DUAL	Duality	From either ◊Φ and ~□~Φ, infer the other; from either □Φ and ~◊~Φ, infer the other.
K	K rule	From □(Φ → Ψ), infer (□Φ → □Ψ).
T	T rule	From □Φ, infer Φ.
S4	S4 rule	From □Φ, infer □□Φ
B	Brouwer rule	From Φ, infer □◊Φ.
N	Necessitation	If Φ has previously been proven as a theorem, then any formula of the form □Φ may be introduced at any line of a proof.
□=	Necessity of identity	From α = β, infer □α = β.

(Nolt 328)

 

We do not use quantification in our proofs, because we will want to avoid the problem of predicating for non-existing objects.

 

 

Summary

 

Nolt will explain how proofs are made in Leibnizian modal propositional logic. We keep all the inference rules from classical propositional logic. But we add some new rules that will allow us to process modal operators. In this section we will not work with quantifiers, although we will look at inferences involving identity. The reason we will avoid quantifiers is that “the quantifier rules depend on how we resolve the question of predication for nonexisting objects” (Nolt 328). Later we will examine one system for doing this, namely free logic. It is called “free” because it is “free of the presupposition that every name always names some existing thing” (328).

 

So we keep the rules for classical propositional logic and also the classical rules for identity [We have not yet summarized Nolt’s exposition of the rules for propositional logic. As far as I can tell right now, they seem for the most part the same as the ones that Agler gives in his Symbolic Logic book. The rules and strategies from Agler’s  and Nolt’s books are compiled here.] We then add to them seven new inference rules.

 

DUAL	Duality	From either ◊Φ and ~□~Φ, infer the other; from either □Φ and ~◊~Φ, infer the other.
K	K rule	From □(Φ → Ψ), infer (□Φ → □Ψ).
T	T rule	From □Φ, infer Φ.
S4	S4 rule	From □Φ, infer □□Φ
B	Brouwer rule	From Φ, infer □◊Φ.
N	Necessitation	If Φ has previously been proven as a theorem, then any formula of the form □Φ may be introduced at any line of a proof.
□=	Necessity of identity	From α = β, infer □α = β.

(Nolt 328)

 

Nolt reminds us that we proved a number of these in section 11.2.1 and section 11.2.2, and the others can easily be shown to be valid as well (Nolt 328).

 

[Nolt’s next point seems to be that when a theorem is a valid formula, then it is true in all worlds on all valuations. He defined theorem previously (in section 4.4) in the following way: “Conclusions provable without assumptions are called theorems,” and he noted that “all the theorems of propositional logic are tautologies, and vice versa” (103). So if something is a tautology, it would seem obvious that it would be true in all worlds and on all valuations, as it cannot possibly be untrue no matter what truth values we assign the terms. This then explains the rule of necessitation. So when we use the rule of necessitation, we do not refer to other premises within the same proof but rather to theorems established in a previous proof. When we notate it in the justification, we just put the prior proved theorem without mentioning line numbers. Nolt, however, it seems will often footnote the page where the theorem was previously proven.]

The necessitation rule differs from the others in that it uses no premises but refers, rather, to theorems established by previous proofs. A theorem is a valid formula, a formula true in all worlds on all valuations. Therefore, if Φ is a theorem, □Φ and any formula of the form □Φ may be asserted anywhere in a proof without further assumptions. When we use the rule of necessitation, we annotate it by writing its abbreviation ‘N’ to the right of the introduced formula, followed by the previously proved theorem or axiom schema employed.

(328)

 

Nolt then notes that “These seven inference rules, together with the rules of classical propositional logic and the identity rules =I and =E, constitute a system of inference that is sound and complete with respect to a Leibnizian semantics for the modal propositional logic with identity”  (328). [We discussed =I and =E in section 8.5.] The rules other than =I, =E, and □= (that is, the set of purely propositional rules) makes up a logic called S5. Nolt then says that in this section he will for the most part explore “the valid inferential patterns of S5” (329).

 

Nolt begins by proving the sequent:

P ⊢ ⋄P

(Nolt 329)

[As we can see, we have an atomic formula. In Nolt’s strategic rules, he has it in negated form, but no matter, we just consider it the same way. He has:

If the conclusion or subconclusion you are trying to prove is of the form:	Then try this strategy:
~Φ	Hypothesize Φ and work toward a subconclusion of the form Ψ & ~Ψ in order to obtain ~Φ by ~I.

 

Agler formulates this rule as: “SA#1(P,¬Q): If the conclusion is an atomic proposition (or a negated proposition), assume the negation of the proposition (or the non-negated form of the negated proposition), derive a contradiction, and then use ‘¬I’ or ‘¬E.’” So the first thing we will do after setting it up is to hypothesize the negated form of the conclusion. But recall from section 11.1 and from the Duality rule above that ◊Φ↔~□~Φ. So Agler will formulate the hypothesis as □~P, with that being equivalent to the negation of the conclusion ⋄P.

 

1.	P	A
2.	|    □~P	H (for ~I)

 

So we will want to use negation introduction. Recall the rule is: “Given a hypothetical derivation of any formula of the form (Ψ & ~Ψ) from Φ, end the derivation and infer ~Φ.” So we will want a formula of the form P & ~P. We have P already in the main proof. Now we want to derive ~P. We have our new rule T, “From □Φ, infer Φ.” So we will use that, which will allow us to make the contradictory conjunction P & ~P.

 

1.	P	A
2.	|    □~P	H (for ~I)
3.	|    ~P	2 T
4.	|    P & ~P	1,2 &I

 

Now we can use negation introduction to bring out ~□~P into the main proof, and we can then apply Duality to it to get our goal proposition, the original conclusion.]

 

1.	P	A
2.	|    □~P	H (for ~I)
3.	|    ~P	2 T
4.	|    P & ~P	1,2 &I
5.	~□~P	2-4 ~I
6.	⋄P	5 DUAL

(Nolt 329)

 

Nolt explains the reasoning behind this proof in the following passage:

The strategy is an indirect proof. Recognizing initially that ‘⋄P’ is interchangeable with ‘~□~P’, we hypothesize ‘□~P’ for reductio. Using the T rule, the contradiction is obtained almost immediately. This yields‘~□~P’, which is converted into ‘⋄P’ by DUAL at line 6.

(329)

 

Nolt then turns to rules N and K. He says they are often used in conjunction with one another in order “to obtain modalized versions of various theorems and rules.” He shows this with the following example.

□(P & Q) ⊢ □P

(Nolt 329)

[This proof will use rule N (necessitation), and it will take a theorem that was proven in a prior chapter. So it is hard to figure this one out on our own. He will import that theorem, which is a conditional, and he will place the necessity operator in front of the whole theorem, using the necessitation rule. Then each part of the conditional will receive the necessity operator, by means of rule K, which opens the door to us using conditional elimination to derive the goal proposition.]

 

1.	□(P & Q)	A
2.	□((P & Q) → P)	N ((P & Q) → P)
3.	□(P & Q) → □P)	2 K
4.	□P	1, 3 →E

(Nolt 329)

 

Nolt will now show a more sophisticated case using rules N and K. [We note that rules N and K deal with the necessity operator. Nolt will show how they can be used to derive a formula with a possibility operator. It will involve using the duality rule to render the possibility operator into a form with the necessity operator, which will allow us to operate on the necessity-operated formulas obtained from N and K, then also to convert them back to possibility.]

⋄P ⊢ ⋄(P ∨ Q)

(Nolt 329)

 

1.	⋄P	A
2.	□(~(P ∨ Q) → ~P)	N (~(P ∨ Q) → ~P)
3.	□~(P ∨ Q)  → □~P)	2 K
4.	~□~P	1 DUAL
5.	~□~(P ∨ Q)	3, 4 MT
6.	⋄(P ∨ Q)	5 DUAL

(Nolt 329)

 

Nolt again demonstrates the combined use of rules N and K in the following proof.

⊢ ⋄~P → ~□P

(Nolt 329)

[Recall that the strategic rule for conditionals is: “Hypothesize Φ and work toward the subconclusion Ψ in order to obtain the conditional by →I.” Nolt will import a theorem, affix to it the necessity operator using N, then distribute that operator using K. This will allow him to find the consequent of the goal proposition. So he can use conditional introduction to derive the conclusion.]

 

1.	|    ⋄~P	H (for →I)
2.	|    ~□~~P	1 DUAL
3.	|    □(P → ~~P)	N (P → ~~P)
4.	|    □P → □~~P	3 K
5.	|    ~□P	2, 4 MT
6.	⋄~P → ~□P	1-5 →I

(Nolt 330)

 

But Nolt says that we might use a different strategy when proving the similar theorem:

⊢ □~P → ~⋄P

(Nolt 330)

[So we recall again that to prove conditionals, we begin by hypothesizing the antecedent in hopes of deriving the consequent, so that the whole goal conditional can be derived in the main proof.

 

1.

|    □~P

H (for →I)

 

This means that we want to derive ~⋄P in this subproof. Since that is an atomic formula, we hypothesize its negation and look for a contradiction. We will find that formulation by using duality to create a form that contradicts the first hypothesized formula, and the rest follows as planned.]

 

1.	|    □~P	H (for →I)
2.	|    |    ⋄P	H (for ~I)
3.	|    |    ~□~P	2 DUAL
4.	|    |    □~P & ~□~P	1,3 &I
5.	|    ~⋄P	2, 4 MT
6.	□~P → ~⋄P	1-5 →I

(Nolt 330)

 

Nolt now shows again how we can use N and K together, with a new sequent.

□(P → Q) ⊢ □~Q → □~P

(Nolt 330)

[As he will import a theorem, it will be hard to figure this one out on our own. The basic idea is that by applying N and K to this theorem, we will be able to then use conditional elimination (modus ponens) to obtain a formula that will become the goal proposition after we apply rule K to it.]

 

1.	□(P → Q)	A
2.	□((P → Q) → (~Q → ~P))	N ((P → Q) → (~Q → ~P))
3.	□(P → Q) → □(~Q → ~P)	2 K
4.	□(~Q → ~P)	1,3 →E
5.	□~Q → □~P	4 K

(Nolt 330)

 

Nolt now will show us how the B rule (Brouwer Rule) works. [Recall that it is: “From Φ, infer □◊Φ.”] Nolt will use this rule in the following proof.

⋄□P ⊢ P

(Nolt 330)

[Since we have an atomic formula as our goal proposition, we will hypothesize its negation. This means we also need to find a contradiction. This will be accomplished in a fairly complicated way. First Nolt will use the Brouwer rule to obtain a form of the hypothesis where there is both a necessity and possibility operator. He then will import a theorem of a conditional form, using N and K to introduce and distribute the necessity operator. Then by using duality and modus tollens, he will produce the parts we need for a contradiction. This will allow us to derive our goal proposition in the main proof.

 

1.	⋄□P	A
2.	|    ~P	H (for ~I)
3.	|    □⋄P	2 B
4.	|    □(⋄~P → ~□P)	N (⋄~P → ~□P)
5.	|    □⋄~P → □~□P	4 K
6.	|    ~□~□P	1 DUAL
7.	|    ~□⋄~P	5, 6 MT
8.	|    □⋄~P & ~□⋄~P	3, 7 &I
9.	~~P	2-8 ~I
10.	P	9 ~E

(Nolt 330)

 

[Recall the S4 rule: From □Φ, infer □□Φ.] Nolt will now use the S4 rule to prove:

⊢ ⋄⋄P → ⋄P

(Nolt 330)

[Since we have a conditional structure, our strategy will be to hypothesize the antecedent and derive the conditional. So we hypothesize ⋄⋄P, with the aim of deriving ⋄P. Since ⋄P is atomic, that means we hypothesize its negation, and we try to find a contradiction. Finding that contradiction will not be easy. Nolt in the first place will not directly hypothesize ~⋄P. Recall that ⋄P is equivalent to ~□~P. So Nolt will hypothesize □~P. He will then use S4 to get □□~P. He will also use N to import a theorem and add the necessity operator, then use K to distribute it. This will allow us to derive a formulation that is contradictory with the first hypothesis we made. And the rest follows as planned.]

 

1.	|   ⋄⋄P	H (for →I)
2.	|    | □~P	H (for ~I)
3.	|    | □□~P	2 S4
4.	|    | □(□~P → ~⋄P)	N (□~P → ~⋄P)
5.	|    | □□~P → □~⋄P	4 K
6.	|    | □~⋄P	3,5 → E
7.	|    | ~□~⋄P	1 DUAL
8.	|    | □~⋄P & ~□~⋄P	6,7 &I
9.	|    ~□~P	2-8 ~I
10.	|    ⋄P	9 DUAL
11.	⋄⋄P → ⋄P	1-10 →I

(Nolt 331)

 

Nolt says that just as with propositional and predicate logic, we can use derived rules by listing “the previously proved sequent to the right, together with the line numbers of the premises ( if any) that are instances of the previously proved sequent's premises. (Rules derived from theorems have no premises, and we cite no lines for them)” (Nolt 331). [What Nolt will do it seems is use a previously proved sequent, which has a premise and conclusion already in the sequent. In the proof at hand, there will be a formula that is the same as the full premise of the previously proved sequent. So on that basis, we can introduce the conclusion of the previously proved sequent, it seems. This happens in line 4.]

□(P → Q) ⊢ ⋄P → ⋄Q

1.	□(P → Q)	A
2.	|    ⋄P	H (for →I)
3.	|    |    □~Q	H (for ~I)
4.	|    |    □~Q → □~P |    |	1□(P → Q) ⊢ □~Q → □~P
5.	|    |    □~P	3,4 →E
6.	|    |    ~□~P	1 DUAL
7.	|    |    □~P & ~□~P	5,6 &I
8.	|    ~□~Q	3-7 ~I
9.	|    ⋄Q	8 DUAL
10.	⋄P → ⋄Q	2-9 →I

(Nolt 331)

 

Nolt’s next point is that whenever we prove a sequent, we have in a sense proven the validity of its structure, and it does not matter which terms are used in that formula. This includes even substituting an expression of the form ‘a = b’ for a letter ‘P’, as we will see  in lines 5 and 7 of the following proof.

~a = b ⊢ □~a=b

1.	~a=b	A
2.	|    ~□~a=b	H (for ~I)
3.	|    ⋄a=b	2 DUAL
4.	|    □(a=b → □a=b)	N (a=b → □a=b)
5.	|    ⋄a=b → ⋄□a=b |	4 □(P → Q) ⊢ ⋄P → ⋄Q
6.	|    ⋄□a=b	3,5 →E
7.	|    a=b	6 ⋄□P ⊢ P
8.	|    a=b & ~a=b	1,7 &I
9.	~~□~a=b	2-8 ~I
10.	□~a=b	9 ~E

(Nolt 332)

 

Nolt then explains the important thing that this proof establishes, namely that rigid designation holds, because nonidentity is necessary [just as identity is necessary. In other words, I think the idea is that a name always has its designation, and in no world does it take a different designation.]

This proof shows that not only is identity necessary as the □= axiom schema asserts, but also nonidentity is necessary – a result fully appropriate in light of the semantics of rigid designation.

(Nolt 332)

 

The next proof will show that whatever is possible is necessarily possible, or that:

⋄P ⊢ □⋄P

(Nolt 332)

[To do this, Nolt will import a theorem and add the necessity operator in line 3, then distribute the operator using K. For this proof, also recall the Brouwer rule: From Φ, infer □◊Φ.]

 

1.	⋄P	A
2.	□⋄⋄P	1B
3.	□(⋄⋄P → ⋄P)	N (⋄⋄P → ⋄P)
4.	□⋄⋄P → □⋄P	3 K
5.	□⋄P	2,4 →E

(Nolt 332)

 

The next thing that Nolt proves is interesting, namely that whatever is possibly necessary is in fact necessary too. [I am not sure how to conceive this, but perhaps the idea is the following. If something is possible, that it is so on at least one world. This means that on at least one world something is necessary. But if it is necessary on one world, then it is the case on all worlds. Thus possible necessity implies unqualified necessity. But the proof works differently than that reasoning.]

 

1.	⋄□P	A
2.	□(⋄□P → P)	N (⋄□P → P)
3.	□⋄□P → □P	2 K
4.	□⋄□P	1 ⋄P ⊢ □⋄P
5.	□P	3,4 →E

(Nolt 332)

 

 

 

 

From:

 

Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.

 

 

.

Priest (1.1) Doubt Truth To Be a Liar, ‘Introduction [to Ch.1],’ summary

by Corry Shores

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Graham Priest

 

Doubt Truth To Be a Liar

 

Part 1 Truth

 

Ch.1 Aristotle on the Law of Non-Contradiction

 

1.1 Introduction

 

 

Brief summary:

Some Presocratic philosophers held views that were self-contradictory. Aristotle argued against such contradictions in book Γ of the Metaphysics where he defended his Law of Non-Contradiction. Since then Aristotle’s view has prevailed. However, in the last century, logic developed in certain ways that have allowed for the legitimate view that certain contradictions do exist. This means we should evaluate Aristotle’s claims to see if they really do hold up. So far, all other commentators on the matter have taken Aristotle’s side in the debate, even if some think his arguments have flaws. Priest is unique in that he will argue Aristotle was ultimately wrong in his defense of the law of non-contradiction.

Summary

Priest explains that there were Presocratic philosophers who “endorsed explicitly contradictory views” (Priest 7), and against them, Aristotle defended the law of non-contradiction (LNC), which has prevailed since then.

A number of the Presocratic philosophers endorsed explicitly contradictory views. In book Γ of the Metaphysics, Aristotle took these in his sights, and defended what was to become known as the Law of Non-Contradiction. This was a crucial moment in the history of philosophy. With the exception of Hegel and his fellow-travellers, and whilst Aristotle’s opinion on nearly every other matter has been overturned—or at least challenged—nearly every Western philosopher and logician has accepted the authority of Aristotle on this matter. There is hardly a defence of the Law since Aristotle’s, worth mentioning.

(Priest 7)

But recent advances in logic allow us to seriously consider the possibility that there are true contradictions. In this chapter Priest will examine whether Aristotle’s defense is conclusive (7).

 

Many commentators in the last hundred years have addressed this issue. All of them think that Aristotle was correct, although some may still say that although Aristotle was ultimately right, his arguments were still flawed. Priest is unique in that he does not think that Aristotle’s conclusion was correct (7).

 

Priest will analyze the relevant parts of Aristotle’s text. But his aim is not a purely scholastic sort. For that kind of a treatment he refers us to other commentaries (see p.7).

What interests me is not so much the niceties of exegesis as whether there is any interpretation of what Aristotle says that will establish what he wishes.

(Priest 7)

 

Graham Priest. Doubt Truth To Be a Liar. Oxford: Oxford University, 2006.

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Doubt Truth To Be a Liar

Part 1 Truth

Ch.1 Aristotle on the Law of Non-Contradiction

1.1 Introduction

1.2 The Law of Non-Contradiction (5^b 18-22)


1.13 Some Modern Variations III: Negation as Cancellation 



Ch.4 Contradiction

4.1 Introduction [to Ch.4 Contradiction] 

Graham Priest. Doubt Truth To Be a Liar. Oxford: Oxford University, 2006.

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Peirce (CP1.332-1.334) Collected Papers of Charles Sanders Peirce, Vol1/Bk3/Ch2/B/§6, "Ego and Non-Ego", summary

 

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[The following is summary. Boldface and bracketed commentary are mine. Proofreading is incomplete, so please forgive my typos.]

 

 

Summary of

 

Charles Sanders Peirce

 

Collected Papers of Charles Sanders Peirce

 

Volume 1: Principles of Philosophy

 

Book 3: Phenomenology

 

Chapter 2: The Categories in Detail

 

B: Secondness

 

§6: Ego and Non-Ego [1.332-1.334]

 

 

Brief summary:

Volition involves dyadic relations. We have the will to exert our forces upon the world to influence it, and we have the will not to have ourselves become influenced by the external world impinging upon us in a disruptive way. So we have an active volition of “reform” and a passive, inertial volition of “conservatism”. Our passive volition, which strives for an inner homeostasis and a protection from exterior influence, is able to maintain stasis when perceptions are relatively expectable. Even though our perceptions come from outside, there is a feeling of control, because those perceptions do not correspond to disruptions in our inner stasis. However, unexpected perceptions change our inner feelings and states, which means our passive, innertial volition of “conservatism” is checked. The degree to which our passive volition is checked is the power of the shock we feel in those cases. We have a sensation whenever the unexpected perceptions initiate in us a new feeling. This pure qualitative feeling persists even as the intensity of the sensation dies down, and the feeling is only changed to another one when we are shocked by a new sensation from another unexpected perception. Pleasure and pain are not essential to feeling. Rather, we volitionally strive to attain pleasure and avoid pain, and to these volitions we have the feeling of volitionally striving for pleasure or pain.

 

 

 

 

Summary

 

1.332

[There are phenomenological differences between feeling, volition, and cognition. Feeling has no parts. Volition however is always a matter of opposing relations. There is an active volition that is related with a volition of reform, and there is a passive volition (a volitional inertia) that is related with a volition of conservatism. We have sensations whenever we are shocked by unexpected perceptions. In these cases, our inertial volition loses its capacity to maintain a homeostasis of passive volitional expenditure, and the degree to which its energies are checked is the strength of the shock. All unexpected perceptions have some degree of shock, and they involve the sensing of an external non-ego. Sensations are also the initiations of feeling.  The pure feeling persists even as the shock of the sensation dies down, and it is only changed to another feeling when we are shocked by a new sensation from another unexpected perception].

 

[Peirce says that he performed careful phenomenological analysis of his own acts of consciousness, and he has found that feeling, volition, and cognition involve three very different modes of awareness. There are psychological differences between these three, but Peirce is interested more in their phenomenological differences, namely, he wants to examine “the differences between that of which we are aware in feeling, volition, and cognition” (emphasis mine). A feeling is a quality, but purely as such, it does not involve some particular subject (see section 1.303 and section 1.305). (I am not sure about his next idea. It seems to be that a feeling is not grasped by means of our reason. And somehow the expression “to have a jaundiced mind” “well expresses feeling without reason”. But I do not understand what he means by this. He discuses how feelings cannot be conceived in sections 1.306 and 1.310.) (Feeling does not have component parts), and thus it is “unanalyzed” (see section 1.303, and sections 1.306 and 1.307). However, unlike feeling, which has no parts, “Volition is through and through dual”. It involves such dualities as between agent and patient, effort and resistance, active effort and inhibition, and of acting on self and acting on external objects. Peirce then distinguishes types of volitions which are duals: active volition and passive volition (or inertia), and (these following might be synonymous with the prior pair, but I am not sure) the volition of reform and the volition of conservatism. Peirce then notes how volition is involved in experiences of shock. We experience shock when we are unexpectedly forced to recognize something with the pressing need to provide some explanation for it. The shock itself is our sense of volitional inertia of expectation that is being checked and thus strikes us like the “blow of a water-hammer”. (I am not entirely sure what is meant. The volitional inertia is being checked. What does this mean? He distinguished active volition from passive volition, and the second one he said was inertial. So although it is inertial, it is still volitional. Given how he defined volition in section 1.331, this means that it involves our efforts and intentions having effect in the world. But the fact that it is passive suggests that they are in action in an automatic, perhaps unconscious, sort of way. So we are acting volitionally in the world, perhaps having some relative sense of control over the environment as a result of our volitional interference in it. And then this inertial passive volition is suddenly checked when we no longer feel that control over the situation, and we instead are forced to try to understand a situation that previously posed us no such critical problems. Now we bring into this the notion of the volition of conservatism. I am not sure what this is (and it might even be synonymous with inertial or passive volition) but it perhaps is our efforts to not alter our patterns of volitional expenditure and to maintain a sort of automatic volitional status quo or homeostasis. It was contrasted to the volition of reform. So this is further reason to suppose that we might think then that what is being checked is our efforts to prevent changes in how we exert our efforts. Peirce says that the energy of the shock can be measured as the amount of energy of the conservative volition that is being checked. So perhaps something is very shocking if we cease investing much energy in our efforts to maintain a status quo, or rather, if that energy is prevented from making that investment. I wonder if we might simply say that when we are shocked by something, we lose our feeling of control or mastery in the situation and over our own abilities to maintain stability or stasis in our functioning. Peirce then writes, “Low grades of this shock doubtless accompany all unexpected perceptions; and every perception is more or less unexpected”. Lower levels of shock are the result of our awareness of externality, which is also the presence of a non-ego. This non-ego, or perhaps the shock of our awareness of it, is what helps us distinguish waking perceptions from acts of dreaming. (Peirce discussed non-ego in section 1.325. Here it seemed to be understood in terms of the otherness of the world that resists our willful exertions.) (For the next point, recall Peirce’s use of this example in prior sections of this text. In section 1.304, he used the sound of a train whistle is an example of something (of a phaneron) with a pure qualitative feeling that can be understood apart from the actual experience of it. He wrote: “Among phanerons there are certain qualities of feeling, such as the color of magenta, the odor of attar, the sound of a railway whistle, the taste of quinine, the quality of the emotion upon contemplating a fine mathematical demonstration, the quality of feeling of love, etc. I do not mean the sense of actually experiencing these feelings, whether primarily or in any memory or imagination. That is something that involves these qualities as an element of it. But I mean the qualities themselves which, in themselves, are mere may-bes, not necessarily realized” (150). In section 1.305, he uses the example of a train whistle again to make roughly the same point, namely, that we are to conceive of the quality of feeling apart from the experience of it and from the many sorts of conditions surrounding that experience. But here he also has us think of the train whistle sound as going on eternally and unvarying. This is because in order to conceive it as a pure qualitative feeling, we cannot think of it as having temporal determinations. He writes, “Suppose I begin by inquiring of you, Reader, in what particulars a feeling of redness or of purple without beginning, end, or change; or an eternally sounding and unvarying railway whistle; or a sempiterne thrill of joyous delight – or rather, such as would afford us delight, but supposed to be in that respect quite neutral – that should constitute the entire universe, would differ from a substance?” (151).) Peirce then defines sensation and feeling, but his manner of presenting these related concepts makes it a bit hard to clearly distinguish and adequately characterize them. Let us start with his example of hearing a train whistle. He has us consider it as having three main parts. There is a sensation at the very beginning when it first sounds. And this sensation ends after it has been “going on for any considerable fraction of a minute”. So it ends after it sustains for some noticeable extent of time. The third part is the second sensation that appears as soon as the sound stops. (This part is ambiguous. He says “at the instant it stops there is a second sensation”, but it is not clear if the “it” refers to the sound or to the sensation. They might be identical or at least cotemporaneous, but I am not sure.) In between the two sensations is a state of feeling. Before giving this example, Peirce defined feeling as “nothing but sensation minus the attribution of it to any particular subject”. So the feeling is the sensation as a pure quality. He defined sensation as the “initiation of a state of feeling.” What I find unclear is how we are to understand the whistle example in relation to these definitions. One interpretation would say that when the whistle starts, we have a sensation as the initiation of the feeling corresponding to the experience of the whistle, with this feeling being the sensation minus an attribution of it as being our own sensation. Then, before we have another sensation, there is a state of feeling. And then when the sounds stops, we have a second sensation, which would have to be the initiation of another state of feeling, either from the experience of some other sound or other stimulus, or from the experience of silence or of the transition to silence. What is unclear to me here is that we are saying that in between the two sensations is a state of feeling, but we are defining feeling as a sensation (minus the attribution to a subject). So on the one hand, we seem to be saying that between the sensations there is a lack of sensation, but on the other hand we seem to be saying that there is a sustained sensation of some kind. So perhaps when Peirce defined sensation, he could have also said that sensation is “the initiation or continuation of a state of feeling.” But given the context of shock, another interpretation could be that there is only sensation when there is the experience of unexpected variation. This would bring it more in lines with Deleuze’s account of sensation. A third interpretation would take issue with the “between” in “Between them (the two sensations) there is a state of feeling.” Here we would not actually conceive there being some duration between one sensation and the next, as if there is a period of no sensation. Rather, we would say that one follows immediately after the other, and by “between them” we mean something like “among them” or “during them”. Yet one more interpretation to consider is that sensations can have temporal limits and durations, while states of feeling cannot. So to say “between them there is a state of feeling” would not mean that the feeling as a pure quality has some duration lying between limits. It might work like the following. We have sensations whenever we are shocked by unexpected perceptions, but these shocks can be relatively minor ones. Yet our experiences on this sensory level have two aspects. On the one hand, we experience the sensation as our own, that is, as attributed to us as the one having them. At the same time, we experience the firstness of the phenomenon. Here the sensations have a certain qualitative feel to them that we do not experience as belonging to us at that moment but that rather have some sort of non-durational quality to them. Now, with this in mind, we will say that the physical, self-attributable sensation can cease even as the stimulus continues. So at first the loud whistle gives us a sensation. But that sensation diminishes as we become accustomed to hearing the noise. However, all the while, even as the sensation diminishes, we also experience the qualitative feel of that whistle sound, which was initiated by the onset of the sensation, and which continues undiminished until another sensation brings about another feeling. Peirce said in section 1.305 that qualities of feeling cannot be understood as having certain qualitative and quantitative determinations, including it having certain intensities, for this involves comparison and thus secondness. So it is conceivable that although the feeling is sensation minus the attribution to a subject, nonetheless that feeling maintains itself even as the sensation causing it dies away. This is how I propose we understand the difficult passages at the end of this paragraph. Note by the way that so far this interpretation still remains in line with Deleuze’s account of sensation, because it is still understood as existing only for as long as it is unexpected and shocking. One more important thing to note with Peirce’s account in relation to Deleuze is the possibility that Peirce understands the sensation not to be the sensation of some particular stimulus but rather as the sensation of a difference in what stimulates our senses. This is suggested especially by the fact that in the example, the second sensation comes when the whistle sound ceases. So we might on the one hand say that the sensation of silence is the second sensation. But this can be an odd thing to say, namely that we are sensing a lack of stimulus (at least if we regard sensations to arise from stimuli). It would seem on the other hand more accurate to say that we sense a change in the stimulus, that is to say, that we notice it has stopped, rather than saying we notice its lack. But as we will see in the next main section entitled “Shock and the Sense of Change”, he will use the example of the Doppler effect altering the train whistle’s sound. He says specifically that we do not sense the change. Rather, we just sense the lower note, but not the change in note. We will examine this next in greater detail. But he will say that we cognize the change, and this cognition is a matter of experiencing the change rather than sensing it. The shock of this experience is caused by the fact that we volitionally make efforts to continue our perception of the sound as it is, such that when it changes, it is met with our own volitional resistance. So he is in fact saying that we do not sense the change in notes. However, he is saying that we have a new sensation when the notes  change, which initiates a new feeling. So Peirce seems to have the following view. We are always having sensations while we are having perceptions. We directly sense the stimulus. So our sensations change as the stimulus changes. But we do not directly sense the change in the stimulus itself. This happens when our internal systems volitionally make efforts to maintain our state of experience, but these efforts are thwarted when some perception forces us to change our state of experience. Our experiential awareness of our internal resistance to the forces disturbing our homeostasis is the source of our shock, and the energy of that shock is as great as the volitional effort that is being thwarted by the disruptive force of the unexpected perception. Thus we would have to say that Peirce’s model differs from Deleuze’s account. For Peirce, we directly sense stimuli of different kinds, but we do not sense the differences between them. For Deleuze, as I read him, we do not directly sense stimuli as whole constituted things but rather we directly sense variations in stimuli, and only on higher orders of perception do we artificially constitute unified coherent perceptions.]

The triad, feeling, volition, cognition, is usually regarded as a purely psychological division. Long series of carefully planned self-experiments, persistent and much varied, though only qualitative, have left me little doubt, if any, that there are in those elements three quite disparate modes of awareness. That is a psychological proposition; but that which | now concerns us is not psychological, particularly; namely the differences between that of which we are aware in feeling, volition, and cognition. Feeling is a quality, but so far as there is mere feeling, the quality is not limited to any definite subject. We hear of a man whose mind is jaundiced. That phrase well expresses feeling without reason. Feeling also as such is unanalyzed. Volition is through and through dual. There is the duality of agent and patient, of effort and resistance, of active effort and inhibition, of acting on self and on external objects. Moreover, there is active volition and passive volition, or inertia, the volition of reform and the volition of conservatism. That shock which we experience when anything particularly unexpected forces itself upon our recognition (which has a cognitive utility as being a call for explanation of the presentment), is simply the sense of the volitional inertia of expectation, which strikes a blow like a water-hammer when it is checked; and the force of this blow, if one could measure it, would be the measure of the energy of the conservative volition that gets checked. Low grades of this shock doubtless accompany all unexpected perceptions; and every perception is more or less unexpected. Its lower grades are, as I opine, not without experimental tests of the hypothesis, that sense of externality, of the presence of a non-ego, which accompanies perception generally and helps to distinguish it from dreaming. This is present in all sensation, meaning by sensation the initiation of a state of feeling; – for by feeling I mean nothing but sensation minus the attribution of it to any particular subject. In my use of words, when an earsplitting, soul-bursting locomotive whistle starts, there is a sensation, which ceases when the screech has been going on for any considerable fraction of a minute; and at the instant it stops there is a second sensation. Between them there is a state of feeling.

(166-167)

 

 

1.333

 

[Pleasure and pain are not essential to feeling. Rather, we volitionally strive to attain pleasure and avoid pain, and to these volitions we have the feeling of volitionally striving for pleasure or pain.]

 

[Kant and other thinkers consider pleasure and pain as being essential to feeling. One reason they may think this is because they apply the word feeling to different modifications of awareness (I am not sure how this is reasoning for considering pleasure and pain to be of the essence of feeling. Perhaps the idea is that all modifications of our awareness are accompanied by some pleasure or pain even if slight.) Another reason they may think this is because they have wrongly analyzed feeling or pain and pleasure. Peirce however thinks that pure unadulterated feeling bears not relation to pain and pleasure. He says that instead pleasure and pain are related to our volition, as we volitionally make efforts to avoid pain and seek pleasure. So if there is any way that feeling is related to pain and pleasure, it is not a direct relation. Rather, it would be the feeling of volitionally trying to avoid pain and the feeling of volitionally trying to seek pleasure. But to be clear, this does not mean that there is a feeling common to all pleasures and one common to all pains. Peirce then elaborates this insight by discussing a flaw in hedonist thinking. The hedonist regards this feeling of volitionally trying to seek pleasure (and avoid pain) as being like guiding forces or principles for our behavior (as if they are like desire or drives, or as he puts it, as if they are “active agencies”). But these feelings are really just indicators of the workings of our unconscious volitional behavior (he says they are “conscious indications of real determinations of our subconscious volitional beings”. It seems he is saying that it is not that we have feelings that drive our volitions to make us seek pleasure but rather that our volitions are already structured such that they drive us to seek pleasure, and the feelings we have in this regard is merely the feeling of volitionally trying to seek that pleasure. Or perhaps it would be better to say that our volitions are structured such that they aim for certain things, and whenever we attain those aims, we have pleasure (and whenever we fail to, we have pain). And so it is not that our volitions are shaped by our drives for pleasure and pain but rather that our volitions create the conditions for us having pleasure or pain, depending on how successful those volitions are.) Peirce also does not think that pain is merely a privation of pleasure. However, he does acknowledge that pain indicates an active determination of our volitional being and pleasure indicates a passive determination of our volitional being. (I am not sure what he means by that however.)]

As for pleasure and pain, which Kant and others have represented to be of the essence of feeling, whether it be merely because they and the section of the psychological world for which at this moment I have the presumption to speak apply the word feeling to different modifications of awareness, or whether there be a faulty analysis on the one part or the other, we certainly do not think that unadulterated feeling, if that element could be isolated, would have any relation to pain or | to pleasure. For in our opinion if there be any quality of feeling common to all pleasurable experiences or components of experience, and another one quality of feeling common to all that is painful (which we are inclined to doubt, to say the least), then we hold the opinion that the one is the feeling of being attracted, the other that of being repelled, by the present state of experience. If there be two such feelings, they are feelings of states of volition. But perhaps pleasure and pain are nothing more than names for the state of being attracted and that of being repelled by present experience. Of course, feelings accompany them, but under the latter hypothesis no feeling would be common to all pleasures, and none to all pains. If we are right, the position of the hedonists is preposterous, in that they make mere feelings to be active agencies, instead of being merely conscious indications of real determinations of our subconscious volitional beings. [I may mention that their talk (however it may be with their thought) is further preposterous as seeming to make pain a mere privation of pleasure, although it is plain that it is pain that indicates an active, and pleasure only a passive, determination of our volitional being.]

(167-168)

 

 

 

1.334

[Volition is “the momentary direct dyadic consciousness of an ego and a non-ego.” As such, there are two sides to this one unified mode of consciousness. The ego side: there is the active and intentional volition of our muscular contractions as we try to exert our influence on the world around us. The non-ego side: there is the passive and unintentional volition that gives us feelings of shock when it is checked by external forces acting on us (which we detect perceptually) and also this side of the awareness gives us our sense of externality (because it involves our awareness of the external world impinging on us and affecting our inner workings). ]

 

[Peirce now explains that he considers volition to be no more than the “momentary direct dyadic consciousness of an ego and a non-ego then and there present and reacting each upon the other”. (Peirce says also that he will expand the meaning of volition in another way, but I could not discern what that is.) I am not certain, but Peirce’s next point seems to be that in the ego the activity (of the volition) is more active, and in the non-ego it is more passive. He writes specifically, “I would limit it to the momentary direct dyadic consciousness of an ego and a non-ego then and there present and reacting each upon the other. In one, the action is generally more active, in the other more passive; but precisely what this difference consists in I do not feel sure. I think, however, that the will to produce a change is active, the will to resist a change is passive.” The next point is a little confusing, because we might think that sensation is passive, as we seem to be receiving stimuli. However, he says that not only is sensation active, it is so by its very definition. [Recall from above that he defined sensation as the initiation of a state of feeling. Since something is active when it produces a change, and since sensation produces a change in states of feeling, then perhaps that is why it is by definition active. But it does not seem to be active in the sense that we actively enact our sensations and control them somehow.] He then addresses an objection. [I do not follow this part very well, and I am not even sure I know what the objection is against. If it is against the idea that sensation is by definition active, then I do not know how to make sense of what follows. If the objection is against the idea that the will to produce a change is active and the will to resist a change is passive, then perhaps I can work through the objection. So the next point might be that according to this view just stated, were we to resist a change, it should not involve a sense of effort (since we said it is passive). The next point I do not follow, but it seems to have the following idea. We can distinguish between the sense of externality in willing and in perception. In willing, perhaps, our sense of externality is one of us acting upon it, and in perception, perhaps, our sense of externality is one of it acting upon us. But how all this constitutes as an objection I do not know, which means I misinterpreted it. The text is quoted below for your own interpretation. Then Peirce makes a point that seems to be his reply to the objection. It at any rate seems to be a statement of what he is arguing, and it is an interesting point. He says that “the sense of externality in perception consists in a sense of powerlessness before the overwhelming force of perception.” I am not sure about the next point, because he says something about learning forces, but I do not know what it means to learn a force (unless it means to learn about a force or to learn that a force is present). His overall point seems to be that we have two classes of volition. One is the active and intentional volition of muscular contraction (our acting willfully to influence the external world) and the other is the passive unintentional volition that is the source of shock when we are surprised by unexpected perceptions and that give us a sense of externality (as it acts upon and influences us). Peirce’s main idea here is that there are not two distinct modes of consciousness. Rather, they are one act of awareness, that is, of an ego/non-ego dyad, which has two sides to that awareness, namely the active side of us exerting our volition on the world and the passive side of it affecting us and blocking our volitions.]

As for volition, I would limit the term in one way and extend it in another. I would limit it to the momentary direct dyadic consciousness of an ego and a non-ego then and there present and reacting each upon the other. In one, the action is generally more active, in the other more passive; but precisely what this difference consists in I do not feel sure. I think, however, that the will to produce a change is active, the will to resist a change is passive. All sensation is essentially, by its very definition, active. The objection to this is that, according to it, the voluntary inhibition of a reflex should not give a sense of effort; and probably the definition of the distinction between the sense of externality in willing and in perception requires a supplement or other slight modification on this account. But the important point [is] that the sense of externality in perception consists in a sense of powerlessness before the overwhelming force of perception. Now the only way in which any force can be learned is by something like trying to oppose it. That we do something like this is shown by the shock we receive from any unexpected experience. It is the inertia of the mind, which tends to remain in the state in which it is. No doubt | there is a marked difference between the active and intentional volition of muscular contraction and the passive and unintentional volition that gives the shock of surprise and the sense of externality. But the two are to be classed together as alike modes of double consciousness, that is, of awareness, at once and in the same awareness, of an ego and a non-ego. . . .

(168-169)

 

 

 

 

 

 

Peirce, C.S. Collected Papers of Charles Sanders Peirce, Vol 1: Principles of Philosophy.  In Collected Papers of Charles Sanders Peirce [Two Volumes in One], Vols. 1 and 2. Edited by Charles Hartshorne and Paul Weiss. Cambridge, Massachusetts: 1965 [1931].

 

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