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5 hours ago, merjet said:

Thank you for that, Stephen, especially for the distinction between logical necessity and physical necessity. Also, I liked your comments about John Locke. I have began a series on my blog about inference and necessity. Here is the first: Blanshard on Implication and Necessity #1. More to come. 

My integrative and conceptual powers applied to a lifetime of experience still leave me scratching my head.  Is this distinction an academic and historical characterization of what philosophers thought or think or is a first hand distinction of a proper philosophy?

Is the distinction as follows: physical necessity IS metaphysical i.e. identity, and logical necessity IS epistemological i.e. non-contradiction?

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The weed-vine example strikes home to anyone who has tried it. In the example, a flowering vine, in the garden, pole beans, cucumber, to a lesser extent squash and pumpkin. Pulling the weed-vine meant being able to distinguish without being able to see, or where finding a viewing angle was just downright awkward.

I took the physical-logical necessity as a neat example of ontologically based logic.

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50 minutes ago, StrictlyLogical said:

Is the distinction as follows: physical necessity IS metaphysical i.e. identity, and logical necessity IS epistemological i.e. non-contradiction?

The distinction I have in mind is different, but not incompatible with that. Physical necessity is about physical things. Logical necessity is broader, and includes the sort of necessity one can grasp in, say, higher mathematics. For example, this consequence is logically implied by this theorem. For example, this function is differentiable, therefore continuous. The reverse may be true, too, but not always.

Edited by merjet

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I concur with the distinction Merlin draws between physical and formal necessity in the preceding post. That’s a good example from mathematics, and I should note additionally that (i) it is a fact—ascertained in the way one does for mathematics—that there are some continuous functions that are nowhere differentiable, and it remains a fact even if it is the case that there simply is nothing physical to which some such function applies and that (ii) we find great success in technology and in extending comprehension of the physical by applying many functions, each one both continuous and differentiable, to electricity, to fluids, and to solids, yet understanding perfectly well that such things are discontinuous at small enough scales. 

SL, I should not want to equate the physical with the metaphysical. When Rand claims that only living things can have values or when philosophers from time immemorial say nothing comes from nothing, those claims are consonant with modern physical science, but the claims are made in what I’d call a metaphysical perspective, not a scientific one.

In his 1967 essay “The Analytic-Synthetic Dichotomy” Peikoff has a section on the traditional distinction within metaphysics between necessary and contingent facts (and how this feeds into the A-S distinction). The meaning of metaphysical necessary/contingent has changed over the centuries, but there is family-descendant resemblance under the continuing distinction. Peikoff did not think such a distinction is correct to make within metaphysics. However, he there drew a distinction between the metaphysical and the manmade (in tune with Rand’s later elaboration). Human free will is the root fact for this distinction. Unfortunately, Peikoff and Rand thought that the rule of Identity in metaphysics entailed complete determinism throughout metaphysics as contrasted with the realm of free will. Furthermore, Rand thought that such metaphysics rightly constrains (a bit) what physical science might find, but that the reverse flow does not soundly occur. That is, she thought metaphysical fundamentals could not be changed in light of advances in science. So for example, the development of chaos theory in the classical regime of physics (starting in the 1970’s as I recall) and the distinction within physics between a classical system in its regular regime as opposed to being in its chaotic regime could not suggest any reformation of general metaphysics. Really, the total determinism that Rand-Peikoff attached to metaphysics under identity was an inheritance from modern physics (Laplace et al.) and is not properly part of right metaphysics, rather should be left open for physics to settle. In his book OPAR, Peikoff does acknowledge that when it comes to value theory, biology supplies the characterized phenomena, pertinent for philosophical fundamentals concerning value.

In his dissertation, Merlin, Peikoff included Blanshard’s books The Nature of Thoughtand Reason and Analysis. He does not cite the former in his text or notes. He cites and makes specific explicit use of the latter from its pages 252–54 and 271–75. The former stretch lays out the traditional view that necessity (the one, as it happens, to be most often sainted by philosophers traditionally) arises only at the level of universals and essences; discerned at the level of conception, not perception. The latter stretch concerns conventionalist theories of logic.

Merlin, I’ve inclined to the view of logic put forth by Rand (1957) and Branden (c. 1968) and Peikoff (1967, 1991) in their orientation towards logic as tool for successful thinking. (I reject Rand’s definition of logic in its differentia. I expect she was misled by a remark in Aristotle’s Metaphysics, which seems oblivious to his great achievement, theory of the syllogism, in Prior Analytics.) It has seemed plain that on the Objectivist orientation towards logic, material implication should not be incorporated. A lot of other thinkers have thought material implication off the mark for deficiency in the relevance factor, as had Blanshard. They developed Relevance Logic (also called Relevant Logic) as replacement for classical modern logic, and I think that the way to go and a way consonant with Objectivism also. I have books telling the history, concerns, and purposes that brought on material implication, but I’ll have to open them. I’ll let you know on your blog what I find.

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4 minutes ago, Boydstun said:

SL, I should not want to equate the physical with the metaphysical. 

Just to clarify, when I say physical x is metaphysical, I mean it in the sense I would say a crow is an animal.

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I had meant to mention in the preceding post that Peikoff 1964 notes that not all classical philosophers subscribed to a metaphysical distinction between the necessary and the contingent. He helpfully mentions John Scotus Erigena, Spinoza, and Hegel.


Links to the sections of this essay so far:


Aristotle I  II   

Kant I  II  III

Conventionalism I  II  III

Edited by Boydstun

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There are patterns of exclusion for which one knows that no counterexample could be found. Such excluded combinations are not in existence and cannot be in existence. Examples are:

(A) A counting number cannot be both even and odd.

(B) The sum of interior angles of a triangle in a Euclidean plane cannot be any value but 2R.

(C) An oscillation of inconstant period cannot at the same time be both increasing and decreasing in frequency.

(D) There are various things; a thing cannot be both itself and a thing not itself.

All varieties of logical ontologists (excepting the coarse-weaver Mr. Locke) and all their opponents agree that those patterns of exclusion, as necessarily so, can be seen only through work of the intellect, not seen in simply sensory perception. How the mind renders this intellectual seeing and what is the relation of this seeing to sensory perception are the contested matters.

(B) and (C) are facts entirely independently of the existence of mind in the world. (A) is a fact about facility of mind in its engagement with mind-independent numerosity: I’ve got five-fingered hands, but whether a particular finger is even or odd depends on a count and the order in which I include that finger in the count. Unlike in my post of 26 Feb in this thread, I now understand the second clause in (D), which is PNC, to be like (A), not like (B) or [C]. The first clause in (D) is the mind-independent basis of PNC, and it is discerned by the senses as well as by the intellect. PNC requires that mind-independent fact, but also the mind activity of mapping. Counting and mapping have mind-independent facts to cognize, and these cognitions, of course, do not exist in the world without mind, specifically intellect.

I had not been able to return to completing my discussion of Peikoff’s dissertation since last May, until today. I needed to work on some final developments fulltime in my own fundamental metaphysics during those months, but that much completed (which I expect to see print in 12 or 18 months from now), I hope now to be back on the Peikoff 1964 issues (and his 1967 A/S issues) sustained to completion of this tread. 


Edited by dream_weaver
Corrected referenced article date.

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17 hours ago, Boydstun said:

(A) is a fact about facility of mind in its engagement with mind-independent numerosity: I’ve got five-fingered hands, but whether a particular finger is even or odd depends on a count and the order in which I include that finger in the count.

Fascinating. This is not how I think about this at all 

(btw what do you mean by "mind-independent numerosity"?)


Whole numbers for me count quantities of things even when simply "counting" time, places, relationships... etc  and an even quantity of things has the unique quality of being capable of being separated into two groups of equal quantities (without violating the integrity of any of the members).


Counting fingers one way on the hand, 1, 2, 3, 4, 5, counts quantities of fingers while traversing the hand.  If one starts from the left, then there is one leftmost finger, then two leftmost fingers, then three...  If one starts from the right, then there is one rightmost finger, then there are two, then three...

The facts that the grouping of the leftmost and rightmost fingers are not the same is a statement about the relationships of the fingers in reality, they are spaced at various locations along left-right.

Even if we construe the counting as merely LABELING positions from the right or left... the labels are still statements about direction and spatial relationships to the other fingers in reality as well as the quantity of positions/fingers.  The second finger from the right is the fourth finger from the left... I.e. it IS THE finger which is both second from the right and fourth from the left. 

Is the finger itself ODD or EVEN?  Certainly one could label a finger ODD or EVEN (you can label a spoon a fork) but a "finger" is not something which, as such, possesses oddness or evenness.  Now, what about a label "2 and 4"?  Can a label, as such, be odd or even? "2 and 4" is not "a number", in fact no label (qua label) can "BE" a number, it is a label.  Now a label of course can contain concrete symbols representing numbers, like "2" or "4".  Can any "number" which is generally represented by a concrete symbol in a label labeling a finger be odd or even?  well yes... and because that thing IS a number it must be odd or even.


Imagine I decide to call my cat "three" or "four"... it is still true (in everyday English parlance) that I cannot "name it as a number" which is both odd and even... but this says nothing about cats, it is a fact of reality about what numbers refer to in general... and those groups of things in reality, referred to by any single number, cannot be both even and odd.


English is far from a perfect philosophical language... we do not have pronouns to distinguish between real or imaginary or to color terms as referring to reality or mental contents (I so wish we had such pronouns or logical "genders" like the French "un" or "une" and "la" or "le")

Consequently, I find distinguishing between fact (A) on the one hand and (B) and (C) to be fraught with difficulty... to what fact is one actually referring by the string of words in A?


EDIT1:  Even the terms "odd" and "even" are potentially fraught with difficulty.  IF "EVEN" refers to a property of a referent number, i.e. "2" "12", as such,  and it is defined "those references to quantities of things, such that those quantities can be split into two even quantities", then technically no group of things are "even", the numbers which refer to them are even.  BUT IF "EVEN" refers to a property of the quantities of things themselves referred to by the number "2" "12", and it is defined as "a property of a quantity of things which can be split into two even quantities". then technically "numbers" are not "even" as such, the quantities to which they refer are "even", and for shorthand we can label the number as even based upon the kind of quantity to which it refers....


17 hours ago, Boydstun said:

(B) and (C) are facts entirely independently of the existence of mind in the world.

I tend to suspect (and my apologies for the lack of technical rigor.. I am limited by ignorance/innocence of formal philosophy) that all facts are facts entirely independently of the existence of mind in the world.  In fact that seems to be what distinguishes fact from non-fact ("falsehood" is not quite commensurate... as touched on obliquely below).

The fact that I think, is not metaphysically caused by or metaphysically dependent upon my thinking, in the exact same way that the fact that an electron is, is not metaphysically caused by its being... the electron simply is.

So facts about what people think are in that sense independent of their thinking... now they ARE thinking, and that is a metaphysical fact but it is not a fact which has a priori genesis (primacy of consciousness) in the mind... new thoughts spring into reality and those are new facts (independent of mind) which can be identified.

It seems then that all facts, at any one time, are metaphysical absolutes, which are not metaphysically dependent upon the existence of any mind.


EDIT2: Mappings of any kind are relative, but so are relationships in reality.  What is the significance that one can identify those relationships in multiple ways, which is a direct result of the relative nature of reality?  Now arbitrary ordering (random numerals for fingers) seems of a different category because the labels are not meaningfully related to reality.  And finally a relationship of each and every thing to itself ... this does not seem "relative" or "arbitrary" at all... and can such be called a "mapping" when there are no alternatives?


(Sorry for this added complication... but I have never been comfortable with the word "fact".  Does the concept "fact" refer solely to metaphysical existence, or is it used to signify "statements" about the world, which themselves refer to metaphysical existence... I suppose the simple question is can facts be true or false [now that sounds nutty], or is it that facts simply ARE... now that I think of it a fact is a fact it is something metaphysically absolute...)


I do not want to clutter up your thread, but I would like to understand your thoughts a bit better.  Would you be willing to have a brief discussion in a new thread? 

Can you think of a title which succinctly identifies the distinction between facts (A) on the one hand and facts (B) and (C) on the other?

New thread Dream Weaver?


I might be completely off base here... if I am I apologize... and am here for discussion and illumination.


Edited by StrictlyLogical

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Thanks for the ideas, SL. Let's just leave your post right here for coming back to later (I'll try to set it in perches of Rand's ontology), after I've completed this my festschrift for Leonard Peikoff in theoretical philosophy.

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PNC Ground Shifts to the Side of the Subject – Conventionalism IV

Leonard Peikoff’s dissertation sets out a developmental conceptual story across the centuries of philosophy. His pages present the story of the theories put forth for the place, character, and origin of PNC and the points on which those theories become untenable, suggesting replacement theories.

That contradictions cannot obtain in reality is a necessary truth. The existence of necessary truths poses a problem for philosophers. “One way of putting the problem is as follows: How can man, who has access to only a very limited portion of a universe whose character is independent of what man thinks about it, nevertheless know with certainty that certain truths will obtain throughout all the regions of space and across the whole course of future time? . . . The conventionalists, in principle, find the solution in denying that necessary truths provide information about any world, real or phenomenal, and in construing them instead as expressive of relations between meanings which men themselves have created; i.e. the conventionalists protect necessary truths from the vagaries of an uncertain world, not by claiming a particular insight into the structure of that world nor legislative power over it, but rather by cutting them off from the world and making them independent of what occurs in it.” (Peikoff 1964, 230)

It is only conventionalism that cuts off necessary truths from the world that was or is provocative. The conventionalists Peikoff mentions were as guilty of equivocation in describing their position as “conventionalist” as were their critics. When they pass off the practicality, convenience, or effectiveness of using PNC and other necessary truths in thinking or communication as showing the “conventionalism” of their position, they are being imprecise (possibly for showiness). Ayer is an example of that, and I’ll look at him in a moment and in the next installment. Firstly, I want to point out that when one takes necessary truths to be tools, having (in our supposed choice of them among supposed alternatives) highest practicality, convenience, or effectiveness, one is certainly not cutting them off from the world and making them independent of what occurs in it, even were one to imagine that is what one is doing. Aristotle pointed out that if one wants an artifact to be effective as a saw, making its teeth of iron is the smart choice. Why? Because of the nature of the world!

It is only certain parts of what conventionalists put forth as conventionalism that actually goes to their position of making necessary truths independent of the structure of the world. Any points at which they are speaking of the practicality, convenience, or effectiveness of necessary truths go against, not to, the provocative part in their pronouncements that conventionality makes necessary truths what they are.

At the end of his dissertation, Peikoff sets out some points on which a new sort of ontologism of PNC would need to diverge from previous ones to block the alternative of conventionalism. This is a highly informed set of constraints—shown to be highly informed by the dissertation—to apply for a viable new ontologism, and Peikoff silently knows that Rand’s epistemology to be issued in a couple of years after his dissertation, along with his own addendum (1967) to that epistemology, will be satisfying those constraints.

Peikoff 1964 does not undertake an exposure of the weaknesses in the conventionalists positions he notes as having displaced ontologism to mid-twentieth century. I shall critique those conventionalist approaches. Conventionalisms too, not only previous ontologisms, had their inadequacies, which by now in philosophy have been exposed, opening the area for reformed ontologisms.

For logical empiricist Ayer, the path to an account of logical and mathematical truths and their necessities is the exclusive and exhaustive division of all truths into either empirical ones or analytic ones, where analyticity is conceived in a very thin way. Like logical empiricists Reichenbach, Schlick, and Carnap before him, Ayer rejected Kant’s synthetic class of necessary, a priori truths. The only necessary, a priori truths Ayer acknowledged were analytic ones. Sebastian Rödl observes that this rejection of the existence of synthetic a priori knowledge is of a piece with rejection of the idea of logic as including general forms of right connection of thought to the world (e.g. Kant’s transcendental logic or Rand’s theory of proper concepts and definitions), leaving only right deductive inference (formal calculii) as logic (Rödl 2012, 3, 22–27, 33–39, 43–45).

In the view of Ayer, analytic propositions “are entirely devoid of factual content. And it is for this reason that no experience can confute them.

“When we say that analytic propositions are devoid of factual content, and consequently that they say nothing, we are not suggesting that they are senseless in the way that metaphysical utterances are senseless. For, although they give us no information about any empirical situation they do enlighten us by illustrating the way in which we use certain symbols. Thus if I say, ‘Nothing can be colored in different ways at the same time with respect to the same part of itself’, I am not saying anything about the properties of any actual thing; but I am not talking nonsense. I am expressing an analytic proposition, which records our determination to call a color expanse which differs in quality from a neighboring color expanse a different part of a given thing. In other words, I am simply calling attention to the implications of a certain linguistic usage. Similarly, in saying  that if all Bretons are Frenchmen, and all Frenchmen Europeans, the further statement that all Bretons are Europeans is implicitly contained. And I am thereby indicating the convention with governs our usage of the words ‘if’ and ‘all’.

“We see, then, that there is a sense in which analytic propositions do give us new knowledge. They call attention to linguistic usages, of which we might otherwise not be conscious, and they reveal unsuspected implications in our assertions and beliefs.” (79–80)

What is an analytic proposition (or analytic truth) according to Ayer? What was his definition of analyticity?

Kant had held that all necessary judgments had to be known a priori. What made an analytic judgment a necessary judgment was that its predicate was conceptually contained in its subject. The concept of a body contains the concept that bodies are extended in space. “Bodies are extended” is necessarily so. Kant’s concept of containment is not entirely determinate. He takes “all bodies are extended” as analytic. What about “all smart women are smart”? Is that a variety of analytic containment? If one denies that all smart women are smart, one plainly has contradicted oneself. Kant required analytic statements to be contradictions upon denial, but it is unclear whether that requirement is a fundamental criterion for analyticity or only a variety of his containment criterion for analyticity (Juhl and Loomis 2010, 4–8).

Bolzano tried to formulate a clearer concept of analyticity, one more centered on logic. If a proposition’s truth value remains constant under any substitution of its terms not belonging to logic itself, it is analytic. Also, the the rules of logic are analytic. Frege took up that concept of analyticity, imported it into his logic wider than subject-predicate logic, and argued that, contra Kant, propositions of arithmetic are analytic, not synthetic a priori. Russell identified an important flaw in that Frege program and made innovations to keep the program afloat (ibid., 11–18; further, Burgess 2005, 34–46; Potter 2000).

Logical empiricists such as Ayer were heirs of this logic-centered concept of analyticity and its proposed undergirding of arithmetic. Then too, they took to heart Wittgenstein’s analysis of logical truths (such as PNC) in Tractatus, a work informed by logical ideas of Frege and Russell, but a work crafting “a single, unified relation between language and the world” (Juhl and Loomis 2010, 19). “Unlike Frege, Wittgenstein did not treat logical truths as statements or propositions at all. Rather, he saw such truths as ‘tautologies’ which, while they might show the ‘logical scaffolding of the world’, do not themselves say anything” (ibid.).

“The fact that language, and the world it pictured, possesses certain ‘formal’ features was thought by Wittgenstein to be shown (although not said) in the fact that certain expressions are tautologies. But the Vienna Circle [1926–1938] was dissatisfied with this conception. Wittgenstein’s talk of ‘showing formal properties of the world’ smacked of the metaphysics they, as empiricists, were concerned to avoid. Rather, Circle members (Moritz Schlick and Rudolf Carnap in particular) proposed treating the truths of logic as expressions of the conventions governing a given language. Their role was thus not one of saying anything about the way things—on this point they agreed with Wittgenstein—but rather that of spelling out the relations of implication among statements. And to the extent that mathematics could be reduced to logic following Frege and Russell, a similar account could be given of mathematical truths as well—they too express relations between statements. /There thus emerges a new conception of analytic truths as expressions of the conventions governing language.” (Juhl and Loomis 2010, 20–21)

Ayer’s philosophy tutor at Oxford, beginning in 1929, was Gilbert Ryle. Ryle introduced Ayer to the works of Russell and to Wittgenstein’s Tractatus Logico-Philosophicus. Ayer was captivated by Tractatus. Circumstances converged such that Ayer was given two terms leave of absence from Oxford, and at Ryle’s urging, Ayer headed for Vienna. Carnap was absent from the Circle for that interval, but it was in Vienna that Ayer became impressed with the work of Carnap. In 1933 Ayer lectured at Oxford on Wittgenstein and Carnap. Other influences on Ayer were Popper and C. I. Lewis. Ayer’s Language, Truth and Logic was published in 1936. It was to become one of the most famous English-language philosophy books of the twentieth century.

The edition of it used by Peikoff in his dissertation and in “The Analytic-Synthetic Dichotomy” was the same as ours today: the second edition, 1948.

(Continuation with Ayer in the next installment.)


Ayer, Alfred Jules [1936] 1948. Language, Truth and Logic. New York: Dover.

Burgess, John P. 2005. Fixing Frege. Princeton: Princeton University Press.

Juhl, Cory and Eric Loomis 2010. Analyticity. New York: Routledge.

Peikoff, Leonard 1964. The Status of the Law of Contradiction in Classical Ontologism. Ph.D. dissertation, New York University.

Potter, Michael 2000. Reason’s Nearest Kin – Philosophies of Arithmetic from Kant to Carnap. New York: Oxford University Press.

Rödl, Sebastian 2012. Categories of the Temporal – An Inquiry into the Forms of the Finite Intellect. Translated by Sibylle Salewski. Cambridge, Massachusetts: Harvard University Press.

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And I am thereby indicating the convention with which governs our usage of the words 'if' and 'all'.

. . . saying anything about the way things are---on this point they agreed with Wittgenstein . . .

1948 1946

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PNC Ground Shifts to the Side of the Subject – Conventionalism V

For the second edition of his book (1946), Ayer did not revise the text, but he wrote an Introduction in which he clarified some points and stated his revised positions on other points. Ayer had taken the usual view contra Mill that necessary propositions or necessary truths had to be a priori, not derived from experience (1946, 74–85). That view and Ayer’s view that only analytic propositions were such a priori ones had not changed. He clarified his view that a priori propositions do not describe any facts, saying he means facts that verify those propositions, as empirical propositions can be verified.

“I now think that it is a mistake to say that they [a priori propositions] are themselves linguistic rules. For apart from the fact that they can properly be said to be true, which linguistic rules cannot, they are distinguished also by being necessary, whereas linguistic rules are arbitrary. At the same time, if they are necessary it is only because the relevant linguistic rules are presupposed. Thus it is a contingent, empirical fact that the word ‘earlier’ is used in English to mean earlier, and it is an arbitrary, though convenient, rule of language that words that stand for temporal relations are to be used transitively; but, given this rule, the proposition that, if A is earlier than B and B is earlier than C, A is earlier than C becomes a necessary truth.” (1946, 17)

How in the world could he avoid seeing the feebleness of claiming necessities of transitivity for these propositions comes from acceptance of an arbitrary rule of language? Is it not straightforward to discern that the necessity in the transitivity of these propositions reflects merely a necessity in temporal relations which are topic of these propositions? Ayer had great intellectual sympathy with Hume, and one might wonder if Ayer was at this stage holding on to a view of temporality as contingent, close to Hume’s in the Treatise (which view Kant thoroughly demolished; see Rödl 2012, 111–68; see also Honderich 1987). Be that as it may, it is clear Ayer recognized the conventions taken as source of logical necessity must be arbitrary if they are to serve in maintaining a chasm between logic and empirical facts.

Even if Kant’s reason for thinking arithmetic to be not analytic is off the mark, Ayer did not make a good case that arithmetic is analytic in his sense (expression of convention governing language), thence necessarily true regardless of the constitution of the physical world empirically present. Among Wittgenstein’s scribbles circa 1937–1938, he observed that various sorts of items, such as apples or beans, are such that by counting portions of a collection of those items and counting totals of that collection, one can demonstrate the correctness of summation in arithmetic.

“This is how children learn sums; for one makes them put down three beans and then another three beans and then count what is there. If the result at one time were 5, another 7 (say because, as we should now say, one sometimes got added, and one sometimes vanished of itself), then the first thing we said would be that beans were no good for teaching sums. But if the same thing happened with sticks, fingers, lines and most other things, that would be the end of all sums. / But shouldn’t we then still have 2+2 = 4? —This sentence would have become unusable.” (Wittgenstein 1978, 51–52)

Late in life, Ayer wrote “I am more baffled than enlightened by his [Wittgenstein’s] Remarks on the Foundations of Mathematics. Nevertheless he does make one point of the utmost importance: he calls attention to the dependence of arithmetic upon contingent matters of fact.” Ayer then rehearsed Wittgenstein’s fancy of a world without exemplification of our arithmetic sums, and Ayer asked: “Would the proposition 7+5 = 12 be false in such a world? No. It would still be valid within the framework of standard arithmetic. It would have become not false but useless. . . . it would, says Wittgenstein, be the end of all sums. I am not convinced that this would be so. . . . I do not find it inconceivable that someone would devise an arithmetic which was adapted to such natural facts” (1989, 484–85). Talk of such deliberate adaptedness seems to forget about the supposed chasm between the analytic and the world.

Ayer held out to the end the hope that arithmetic, being not from empirical generalization, is purely analytic (there being no other sort of necessary truths in the view of Ayer and other logical empiricists). It was not a tenable hope. As a contrast to sums among countable things like beans or like words in a sentence, we need not turn to a radical fancy such as Wittgenstein’s. We can turn to contrasting the sorts of sums in the actual world. A positive road grade of 153 yards per mile added to a positive road grade of 153 yards per mile does not sum to a grade of 306 yards per mile, but to 310. 3 yards per mile. Road grades are just as physically real as beans. Different sorts of summations are right for things different in certain of their magnitude-characters. There is no place, I maintain, indifferent to physical reality for pure mathematical summation to abide in “pure analyticity” nor to be based most fundamentally on arbitrary convention.

It was not only of the a priori, necessary principles constituting mathematics, but of such principles constituting logic, Ayer had maintained: “If they are necessary it is only because the relevant linguistic rules are presupposed” (1946, 17). Linguistic rules concerning whether nouns shall have genders with which varieties of definite article must agree (as in German) surely would not be the relevant sort of presupposition—relevant sort of acceptance by one as rule for one’s language—to source the necessity in logic we all put to work in our discursive thought. A linguistic source plausible enough to get off the ground would have to be some deeper aspect of  grammar, an aspect common to all languages.

Recall the picture by Kant that I recorded in Kant I (this picture omitted from the English translation available for Peikoff 1964 and Ayer 1946): “The science which contains these universal and necessary laws [of thought] is simply a science of the form of thought. And we can form a conception of the possibility of such a science, just as a universal grammar which contains nothing beyond the mere form of language, without words, which belong to the matter of language.” That remark engages an analogy in which grammatical form is to grammar-slots filled with particular nouns, verbs, and so forth, as logical form is to its occasions of use in particular topics. Beyond analogy, Sebastian Rödl plumbs deep grammar of thought, along lines of the first Critique (Rödl 2012, 4, 22–25). Deeper plumbs, whether of grammar of language or grammar of thought, pull logical empiricists such as Ayer willy nilly hopelessly far from arbitrary convention as base for logic and its necessities. Wittgenstein evidently suffered that same pull (Ben-Menahem 2006, 265–69). “To collapse correctness into propriety is to obliterate the essential character of thought” (Haugeland 1998, 317; further, 325–43; see also Rasmussen 1982; 2014, 337–41; Rand 1966–67, 47–48; Peikoff 1967, 104; 1991, 143–44).

(To be continued, with Ernest Nagel and Arthur Pap next.)


Ayer, Alfred Jules. [1936] 1948. Language, Truth and Logic. New York: Dover.

——. 1989. Reply to F. Miró Quesada. In Hahn 1992, 478–88.

Ben-Menahem, Yemina. 2006. Conventionalism. Cambridge: Cambridge University Press.

Biondi, Paolo C. and Louis F. Groake, eds. 2014. Shifting the Paradigm – Alternative Perspectives on Induction. Berlin: De Gruyter.

Hahn, Lewis Edwin. 1992. The Philosophy of A.J. Ayer. La Salle, Illinois: Open Court.

Haugeland, John. 1998. Truth and Rule-Following. In Having Thought – Essays in the Metaphysics of Mind. Cambridge, Massachusetts: Harvard University Press.

Honderich, Ted. 1989. Causation: One Thing Just Happens after Another. In Hahn 1992, 243–70.

Peikoff, Leonard 1964. The Status of the Law of Contradiction in Classical Ontologism. Ph.D. dissertation, New York University.

——. 1967. The Analytic-Synthetic Dichotomy. In Rand 1966–67, 88–121.

——. 1991. Objectivism: The Philosophy of Ayn Rand. New York: Dutton.

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PNC Ground Shifts to the Side of the Subject – Conventionalism VI

(Nagel and Pap will not be reached in this installment after all.)

In the Preface to the first edition of his Language, Truth, and Logic (1936), Ayer writes: “Like Hume, I divide all genuine propositions into two classes: those which, in his terminology, concern ‘relations of ideas’, and those which concern ‘matters of fact’” (31). The propositions of logic and mathematics belong to the former, and they are necessarily true, as Hume would have it. That is clear from his texts. But Hume could not concur with Ayer’s further characterization of such propositions as a priori, where an a priori proposition is one known not only independently of this or that perceptual experience, but independently of each and every one of them. Ayer erred, as so many before him and after him, in identifying such a radical sense of the a priori with Hume’s notion of the a priori for propositions of pure mathematics and logic (including PNC) understood as concerning only relations of ideas.

Hume, like Berkeley and Locke before him, was a full-blown empiricist. Ayer’s route to holding forth the logical empiricist view as an empiricist view was to shrink fact to only facts ascertained by the methods of the empirical sciences, to characterize all necessary propositions as radically a priori, which class coincides with a logic-centered analytic, whose truth and necessity are not from relations to the world, but from arbitrary convention hand-waved (in Ayer’s case) from grammar. Then, Ayer proclaims his view as empirical: all knowledge derives from experience because his genre of analytic propositions do not state facts; they are not knowledge. (See also Friedman 1999, 5–6, 9, 18–19, 32–33, 116–18.)

Graciela De Pierris 2015 concludes of Hume:

“The addition in the Enquiry of a seemingly logical criterion for identifying ‘relations of ideas’ does not mean a departure from the sensible phenomenological model of apprehension and ultimate evidence. Hume does not uphold the logical law of non-contradiction in its own right, prior to and independently of phenomenological sensible factors. On the contrary, the status of non-contradiction as a criterion for the acceptance of propositions based on ‘relations of ideas’ depends solely on the phenomenological inspection of intrinsic characteristics of particular items ostensively present before the mind. Necessary a priori methods, in both the Treatise and Enquiry, are ultimately grounded on nothing but the sensible phenomenological model.” (102)

In consistency with their rejection of Kant’s containment-of-predicate-in-subject criterion for seeing which concepts stand in a logically necessary relation each to the other (see Convention IV), the logical empiricists should have spurned Hume’s model of pure relations of ideas (or thoughts). That is to say, they should have spurned his explication of the a priori character of arithmetic, geometry, and logic. Had Ayer understood Hume on this point in Enquiry as harmonious with Hume’s treatment in Treatise, perhaps Ayer would not have set up Hume for worship on analyticity.

In Hume’s picture, discernment of necessary union of items forming a sum or the union of triangularity with the 2R sum of those three angles stands us in high certainty, indeed our highest certainty. We stand on PNC with that highest certainty, and for that reason, PNC can have normative force. “Hume uses his version of the presentational-phenomenological model of apprehension, just as the tradition before him had done, to give a verdict about ultimate evidence” (De Pierris 2015,  228; see also 220–22).

Locke had taken reason as “the discovery of the certainty or probability of such propositions or truths, which the mind arrives at by deduction made from such ideas, which it has got by the use of its natural faculties; viz. by sensation or reflection” (EU IV.xviii.2). Reason investigates ideas and their degrees of certainty or probability of being true, insofar as said ideas are grounded in sensory experience of outer things or in reflection, where reflection means our notice of our own inner, mental operations (EU II.i.3–4, II.v).

I mentioned in Aristotle II that Peikoff argued the unravelling of Platonic logical ontologism into logical conventionalism (as of mid-twentieth century) to have been mediated significantly by Kant. He concluded the unraveling of Aristotelian logical ontologism had been significantly mediated by Locke (Peikoff 1964, 212–35).

I had written in Conventionalism III:

“Logical necessity holds unconditionally and in all contexts. What I’ve called physical necessity is traditionally taken to be necessity under some sort of limiting conditions, and this necessity has been called a contingent connection, reserving necessary connection for logical (and other formal) necessity. . . .

“Peikoff 1964 points out that Locke avoided the contingent/necessary terminology. Locke instead applied probable/certain to the division. We have seen in my section Aristotle II that Locke maintained we have by sensory perception instances of the general fact that different things are not same things and that a thing is never both A and not A at the same time and in the same respect. Philosophers, including Peikoff in 1964, are correct to fault Locke’s blurring under probable/certain a clear understanding that ampliative inductive generalizations over perceived instances do not suffice to land the absolute necessity in general principles of logic or pure mathematics. . . .

“Locke was not really of one mind in this. Peikoff lays out an opposite strand also in An Essay Concerning Human Understanding: IV 3.31, 4.6, 4.8, 9.1, 11.13–14. ‘What is Locke doing in such passages as these? He is now contrasting eternal truths and existential truths. The former are to be discovered only by “the examining of our own ideas,” and “concern not existence” . . .’ (222). . . . For an empiricist such as Locke, . . . [one] joining considerable nominalism (the conceptualist wing of nominalism) concerning universal ideas to the empiricism, the divide between matters of fact and the eternal, formal truths can make conventionalism concerning the ground of logic ‘almost inevitable’ (223).”

(That is not to say, at least in my view, that it makes almost inevitable a root conventionalism that is arbitrary. It need not make almost inevitable a conventionalism making logical principles entirely independent of constraints from the world and from our physical operations in the world.)

I now think, having studied De Pierris 2015, that Locke’s two sorts of distinctions are harmonious. Whether some knowledge is gotten from “examining our own ideas” or from empirical generalization (or from some combination of the two), the knowledge has placement with us as to its certainty or probability. If, as with Descartes before him and Hume after him, degree of certainty is the fundamental index to truth, index to reality won, then Locke was innocent of fundamental severance of formal truths from empirical truths. We of last century and this have been guilty of writing too much of the divides of the logical empiricists back into the distinctions made by Locke and those by Hume.

Locke can be wrong or vague about how we derive PNC from sensory experience, yet right in his view that sensory experience is found to always conform to PNC. That PNC and mathematics are found as well and with highest possible certainty in ‘reflection’ (even reflection “concerning not existence”) need not somehow compete with or belie one’s finding PNC to hold in sensory experience, just as Euclidean geometry is conformed to in carpentry or in Newton’s system of the world (see also Franklin 2014, 95–100). Insofar as Locke is presented with high, highest certainty that PNC is true of the real in every nook and cranny, then PNC can serve as a norm to conform to for correct thinking. Even were the incorrect nominalist leanings of Locke and the incorrect voiding of natural necessitation of Hume not yet diagnosed and dispelled, PNC (and other formal relations) with its mark of certainty and generality can be given to, not cast by, the human faculties envisioned by Locke or by Hume. Then it is not those empiricists who are rightly regarded as intellectual tributaries into logical empiricism, notwithstanding the distant lineage claims declared by the latter. (Further on Locke: chapter VI “Ideas and Psychology” in Yolton 1984.)

What about kinship of the logical empiricists to the classical British empiricist George Berkeley? Kenneth Pearce 2017 argues that for Berkeley words get to be meaningful by being used in pubic discourse, following conventional rules, but for practical ends. The reason a system of language can be effective practically is because with language we capture the grammar of nature as created by God. There is under this conception, I notice, no severance of our minds, including our mathematical minds, from the world due to conventions in language (Pearce 2017, 45–50, 80–83, 112, 151–52, 158–62, 195, 204). Berkeley’s view is contrary the logical empiricist thesis of arbitrariness of linguistic convention sufficiently deep in any formal truths to decouple them from the world.


De Pierris, G. 2015. Ideas, Evidence, & Method – Hume’s Skepticism & Naturalism Concerning Knowledge & Causation. New York: Oxford University Press.

Franklin, J. 2014. An Aristotelian Realist Philosophy of Mathematics – Mathematics as the Science of Quantity and Structure. New York: Palgrave Macmillan.

Friedman, M. 1999. Reconsidering Logical Positivism. New York: Cambridge University Press.

Pearce, K. L. 2017. Language and the Structure of Berkeley’s World. New York: Oxford University Press.

Yolton, J. W. 1984. Perceptual Acquaintance from Descartes to Reid. Minneapolis: University of Minnesota Press.

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