THE FATE OF ANALYSIS, #3–The Rise and Fall of Frege.

By Robert Hanna

***

TABLE OF CONTENTS

II. Classical Analytic Philosophy

II.1 What Classical Analytic Philosophy Is: Two Basic Theses

II.2 What Classical Analytic Philosophy Officially Isn’t: Its Conflicted Anti-Kantianism

II.3 Classical Analytic Philosophy Characterized in Simple, Subtler, and Subtlest Ways

II.4 Three Kinds of Analysis: Decompositional, Transformative, and Conceptual

II.5 Frege, The First Founding Father of Classical Analytic Philosophy

II.6 Frege’s Project of (Transformative or Reductive) Analysis

II.7 Frege’s Dead End

II.8 Frege’s Semantics of Sense and Reference, aka Meaning

II.9 Some Biggish Problems For Frege’s Semantics

II.10 Husserl, Logic, and Logical Psychologism, aka LP

II.11 What LP is, and its Three Cardinal Sins

II.12 Husserl’s Three Basic Arguments Against LP

II.13 Has Husserl Begged the Question Against LP? Enter The Logocentric Predicament, and a Husserlian Way Out

II. Moore, Brentano, Husserl, Judgment, Anti-Idealism, and Meinong’s World

III.1 G.E. Moore, the Second Founding Father of Classical Analytic Philosophy

III.2 Brentano on Phenomenology, Mental Phenomena, and Intentionality

III.3 Husserl on Phenomenology and Intentionality

III.4 Moore and the Nature of Judgment

III.5 Moore and the Refutation of Idealism

III.6 Meinong’s World

IV. Russell, Unlimited Logicism, Acquaintance, and Description

IV.1 Russell Beyond Brentano, Husserl, Moore, and Meinong

IV.2 Russell and Mathematical Logic versus Kant

IV.3 Russell’s Unlimited Logicist Project

IV.4 Pursued by Logical Furies: Russell’s Paradox Again

IV.5 Russell’s ‘Fido’-Fido Theory of Meaning

IV.6 Knowledge-by-Acquaintance and Knowledge-by-Description

IV.7 Russell’s Theory of Descriptions

IV.8 Russell’s Multiple-Relation Theory of Judgment

IV.9 Russellian Analysis, Early Wittgenstein, and Impredicativity Again

IV.10 Russell and The Philosophy of Logical Atomism

V. Wittgenstein and the Tractatus 1: The Title, and Propositions 1–2.063

V.1 A Brief Synopsis of the Tractatus

V.2 The Tractatus in Context

V.3 The Basic Structure of the Tractatus: A Simple Picture

V.4 Tractarian Ontology

V.5 Reconstructing Wittgenstein’s Reasoning

V.6 What Are the Objects or Things?

V.7 The Role of Logic in Tractarian Ontology

V.8 Colorless Objects/Things

V.9 Tractarian Ontology, Necessity, and Contingency

V.10 Some Initial Worries, and Some Possible Wittgensteinian Counter-Moves

VI. Wittgenstein and the Tractatus 2: Propositions 2.013–5.55

VI.1 What is Logical Space? What is Real Space?

VI.2 Atomic Facts Necessarily Are in Manifest or Phenomenal Space, But Objects or Things Themselves Necessarily Aren’t in Manifest or Phenomenal Space

VI.3 Logical Space is Essentially More Comprehensive than Manifest or Phenomenal Space

VI.4 Why There Can’t/Kant Be a Non-Logical World

VI.5 A Worry About Wittgenstein’s Conception of Logic: Non-Classical Logics

VI.6 What is a Tractarian Proposition?

VI.7 Naming Objects or Things, and Picturing Atomic Facts

VI.8 Signs, Symbols, Sense, Truth, and Judgment

VI.9 Propositions Again

VI.10 Language and Thought

VII. Wittgenstein and the Tractatus 3: Propositions 4–5.61

VII.1 The Logocentric Predicament, Version 3.0: Justifying Deduction

VII.2 The Logical Form of Deduction

VII.3 Logic Must Take Care of Itself

VII.4 Tautologies and Contradictions

VII.5 What is Logic?

VII.6 Logic is the A Priori Essence of Language

VII.7 Logic is the A Priori Essence of Thought

VII.8 Logic is the A Priori Essence of the World

VIII. Wittgenstein and the Tractatus 4: Propositions 5.62–7

VIII.1 Tractarian Solipsism and Tractarian Realism

VIII.2 Tractarian Solipsism

VIII.3 Tractarian Realism

VIII.4 Is the Tractatus’s Point an Ethical One?

VIII.5 The Meaning of Life

VIII.6 Three Basic Worries About the Tractatus

VIII.7 Natural Science and the Worry About the Simplicity of the Objects or Things

VIII.8 Natural Science and the Worry About the Logical Independence of Atomic Facts

VIII.9 Tractarian Mysticism and the Worry About Metaphilosophy

IX. Carnap, The Vienna Circle, Logical Empiricism, and The Great Divide

IX.1 Carnap Before and After the Tractatus

IX.2 Carnap, The Vienna Circle, and The Elimination of Metaphysics

IX.3 The Verifiability Principle and Its Fate

IX.4 The Davos Conference and The Great Divide

X. Wittgenstein and the Investigations 1: Preface, and §§1–27

X.1 From the Tractatus to the Investigations

X.2 The Thesis That Meaning Is Use

X.3 A Map of the Investigations

X.4 The Critique of Pure Reference: What the Builders Did

XI. Wittgenstein and the Investigations 2: §§28–242

XI.1 The Picture Theory and the Vices of Simplicity

XI.2 Wittgenstein’s Argument Against The Picture Theory: A Rational Reconstruction

XI.3 Understanding and Rule-Following

XI.4 Wittgenstein’s Rule-Following Paradox: The Basic Rationale

XI.5 Wittgenstein’s Rule-Following Paradox: A Rational Reconstruction

XI.6 Kripkenstein’s Rule-Following Paradox: Why Read Kripke Too?

XI.7 Kripkenstein’s Rule-Following Paradox: A Rational Reconstruction

XI.8 How to Solve The Paradox: Wittgenstein’s Way and Kripkenstein’s Way

XI.8.1 Wittgenstein and The Rule-Following Paradox: A Rational Reconstruction

XI.8.2 Kripkenstein and The Rule-Following Paradox: A Rational Reconstruction

XII. Wittgenstein and the Investigations 3: §§242–315

XII.1 What is a Private Language?

XII.2 The Private Language Argument: A Rational Reconstruction

XII.3 Is Wittgenstein a Behaviorist? No.

XII.4 Wittgenstein on Meanings, Sensations, and Human Mindedness: A Rational Reconstruction

XIII. Wittgenstein and the Investigations 4: §§316–693 & 174e-232e

XIII.1 Linguistic Phenomenology

XIII.2 Two Kinds of Seeing

XIII.3 Experiencing the Meaning of a Word

XIII.4 The Critique of Logical Analysis, and Logic-As-Grammar

XIV. Coda: Wittgenstein and Kantianism

XIV.1 World-Conformity 1: Kant, Transcendental Idealism, and Empirical Realism

XIV.2 World-Conformity 2: Wittgenstein, Transcendental Solipsism, and Pure Realism

XIV.3 World-Conformity 3: To Forms of Life

XIV.4 The Critique of Self-Alienated Philosophy 1: Kant’s Critical Metaphilosophy

XIV.5 The Critique of Self-Alienated Philosophy 2: Wittgensteinian Analysis as Critique

XV. From Quine to Kripke and Analytic Metaphysics: The Adventures of the Analytic-Synthetic Distinction

XV.1 Two Urban Legends of Post-Empiricism

XV.2 A Very Brief History of The Analytic-Synthetic Distinction

XV.3 Why the Analytic-Synthetic Distinction Really Matters

XV.4 Quine’s Critique of the Analytic-Synthetic Distinction, and a Meta-Critique

XV.5 Three Dogmas of Post-Quineanism

XVI. Analytic Philosophy and The Ash-Heap of History

XVI.1 Husserl’s Crisis and Our Crisis

XVI.2 Why Hasn’t Post-Classical Analytic Philosophy Produced Any Important Ideas in the Last Thirty-Five Years?

XVI.3 On Irad Kimhi’s Thinking and Being, Or, It’s The End Of Analytic Philosophy As We Know It (And I Feel Fine)

XVI.4 Thinking Inside and Outside the Fly-Bottle: The New Poverty of Philosophy and Its Second Copernican Revolution

***

This installment contains sections II.5 to II.9.

But you can also read or download a .pdf version of the complete book HERE.

***

Gottlob Frege (1848–1925)

II.5 Frege, The First Founding Father of Classical Analytic Philosophy

(i) that Kant was miserably mistaken in holding that arithmetic truth and knowledge are synthetic a priori, and also

(ii) that arithmetic proof is a fully rigorous scientific enterprise.

According to Frege in the Foundations of Arithmetic, a proposition is analytic if and only if it’s

either (i) a logical truth,

or (ii) provable from general laws of logic alone,

or (iii) provable from general laws of logic plus what he calls “logical definitions.”

One problem with this doctrine is that unless general laws of logic are provable from themselves, then they do not strictly speaking count as analytic.

Another and more serious problem is that the precise semantic and epistemic status of “logical definitions” was never adequately clarified or settled by Frege.[ii]

But the most serious problem is that Frege‘s set theory contains an apparently insoluble contradiction discovered by Russell in 1901, as a direct consequence of the unrestricted set-formation axiom V in Frege‘s Basic Laws of Arithmetic: namely, Russell’s Paradox, which says that the set of all sets not members of themselves is a member of itself if and only if it is not a member of itself.

I’ll say more about that philosophical tragedy in section II.7 below.

In any case, Frege’s work is the beginning of the project of Logicism — i.e., the explanatory and ontological reduction of all or at least some of mathematics to pure logic — which Russell, Whitehead, early Wittgenstein, Carnap, and other members or followers of the Vienna Circle all pursued in the first three decades of the 20th century.

Unlimited logicism provides the first half of modal monism.

And the second half of modal monism is provided by the rejection of the very idea of a synthetic a priori proposition.

This rejection was the unique contribution of Wittgenstein and Carnap, via the Vienna Circle and its logical empiricism, aka logical positivism, in the third and fourth decades of the 20th century.

Indeed, this contribution is so seminal to what we now think of as the mainstream classical Analytic tradition, that it’s often overlooked that its first Founding Father, Frege, always explicitly held, just like Kant, that geometry is synthetic a priori.[iii]

In that special sense, therefore, Frege was always an unreconstructed neo-Kantian.

II.6 Frege’s Project of (Transformative or Reductive) Analysis

This is turn is a limited logicism, since it doesn’t include the reduction of geometry to pure logic.

Pure logic is the science of truth and the a priori necessary rules of how to think and talk, such that one cannot proceed from truth to falsity.

Kant, by contrast, had held that arithmetic and geometric cognition, arithmetic and geometric truth, and arithmetic and geometric proof all depend on a special kind of a priori insight into our forms of intuition–i.e., our formal or structural representations of space and time — that he calls pure intuition.

Frege holds that arithmetic expresses analytic truths, not synthetic a priori truths.

And as I also mentioned above, for Frege, a proposition is analytic if and only if it’s

either (i) a logical truth,

or (ii) provable from general laws of logic alone,

or (iii) provable from general laws of logic plus what he calls “logical definitions.”

Otherwise, a proposition is synthetic and depends on principles derived from either “special sciences” or sense perception.

Logical definitions express transformative or reductive analyses, e.g., of the concept and/or property of being a number.

But here’s a problem: are logical definitions in Frege’s sense analytic, synthetic, or neither?

Frege never tells us, or at least he never tells us definitively, hence the very idea of a logical definition remains unclear and indistinct.

II.7 Frege’s Dead End

The problem was that Frege assumed that sets could formed unrestrictedly by simply describing their membership: that’s the notorious Axiom V of Basic Laws of Arithmetic, aka “the naïve comprehension axiom.”

But what about (e.g.) the set of all sets that aren’t members of themselves?

Is it a member of itself, or not?

Well, if it’s a member of itself, then it isn’t a member of itself, but if it isn’t a member of itself, then it is a member of itself: paradox!

Russell discovered this paradox in 1901, then promptly informed Frege, who wrote back that “logic totters.”

The paradox was a genuine philosophical and personal tragedy for Frege, who never really recovered from its discovery.

Russell attempted to get around the paradox in his Principles of Mathematics; and then when that didn’t work, a few years later, in league with Whitehead, he also attempted to get around it in the first volume of Principia Mathematica: but ultimately that didn’t work either.

My own view is that Frege and Russell, alike, failed to distinguish between two categorically different ways of forming sets in particular, and infinite totalities more generally, by recursive self-inclusion, aka “impredicativity”:

(i) one way that presupposes and is grounded on the phenomenal structure of space and/or time, which is logically benign, and

(ii) another way that transcends and is ungrounded by the phenomenal structure of space and/or time, which is logically vicious.[iv]

Correspondingly, it’s also arguable that the distinction between benign (spatiotemporally grounded) impredicativity and vicious (spatiotemporally ungrounded) impredicativity is a specifically Kantian one, anticipated in the “Transcendental Aesthetic” and “Transcendental Dialectic” sections of the first Critique.

In any case, Frege’s limited logicism was a dead letter by 1903.

Nevertheless, Frege‘s semantics, i.e., his theory of linguistic meaning, which had been specially designed to subserve the project of Frege’s logicism, lived on and on forever, or at least until this morning at 6am.

II.8 Frege’s Semantics of Sense and Reference, aka Meaning[v]

The sense vs. reference/Meaning distinction was introduced to account for a puzzle about true identity statements:

how can true identity statements be cognitively informative, if what they mean is merely that something is identical to itself?

Frege’s solution is that the distinct names in informative true identity statements have different senses, but the same reference/Meaning.

According to Frege, sense is the mode of givenness (Art des Gegebenseins) or mode of presentation — that is, a description — of the reference/Meaning of an expression, and in turn, the reference/Meaning is the referent (if any) of the expression.

This leads to a general theory of sense and reference/Meaning, as follows.

1. The sense of a name (e.g., ‘Frege’) is a complete identifying description (e.g., “the philosopher, logician, and mathematician who wrote The Foundations of Arithmetic”) of an individual object.

2. The reference/meaning of a name is the individual object picked out by the sense of that name (e.g., Frege himself).

3. The reference/Meaning of a predicate (e.g., ‘__ is a philosopher’) is a concept, that is, an essentially incomplete entity that’s a function from objects to truth-values, such that “saturating” its incomplete part, or parts, with an individual object (or several such objects) picked out by a name (or names) and thereby providing an input (or inputs) to the truth-function (e.g., ‘Frege is a philosopher’) yields one of the two truth-values, The True or The False, as outputs (e.g., in this case, of course, The True).

Put in terms of classical metaphysics, Fregean concepts are best understood as a properties and relations, i.e., one-place and many-place universals, with the important qualification that for Frege, concepts are essentially abstract, incomplete entities, whereas classical metaphysicians generally think of universals as essentially abstract, complete entities, as opposed to concrete, complete entities, aka “individuals,” in space and/or time.

3. What’s the sense of a predicate?

Frege doesn’t explicitly say, but presumably, just as the concept is an essentially abstract, incomplete entity, then correspondingly the sense of the predicate is an essentially abstract, incomplete sense, perhaps something like a rule specifying the operation of a given concept, insofar as it maps from objects to truth-values.

In that way, there could be different senses for different predicates (say, the sense expressed by the predicate-expression ‘__ is an oculist’ and the sense expressed by the predicate-expression ‘__ is an eye doctor’), each of which picks out the same concept.

Relatedly, what’s the sense or reference/Meaning of function-terms in mathematics?, and what’s the sense or reference/Meaning of logical constants in natural or ordinary language and formal logic (e.g., ‘if’, ‘and’, if and only if’, ‘or’, ‘not’, ‘all’, ‘some’, etc.)?

Again Frege doesn’t explicitly say, but I think that we can also plausibly speculate that for him all functions and logical constants are essentially abstract, incomplete entities, and that their senses are essentially abstract, incomplete senses, perhaps something like rules specifying the operations of the corresponding functions and logical constants.

4. The sense of an indicative sentence (e.g., ‘Frege is a philosopher’) is a proposition or thought, that is, a logically-structured description of a truth-value. (e.g., The True).

5. The referent/Meaning of an indicative sentence is its truth-value, The True or The False.

6. Truth-values are what is shared by all sentences that are true of the world or false of the world, hence they can be intersubstituted without going from truth to falsity or from falsity to truth, yet they can also differ in sense.

More precisely, however, what are The True and The False?

I think it’s best to think of them as total states of the world.

The True is how everything in the world actually has to be, such that any given proposition or thought about it is correct.

And The False is every other total state of the world, i.e., the non-actual possibilities.

So interpreted, Frege would be a modal actualist who accounts for all non-actual possibilities in terms of propositional falsity relative to the actual world.

7. Frege‘s theory of reference/Meaning is based on his theory of functions and objects.

Functions are systematic mappings from something (i.e., arguments of the function) to something else (i.e., values of the function).

The total set of arguments is the domain of the function and the total set of values is the range of the function.

Functions can map from objects to objects, e.g.,

x + 2 = y,

or from objects to truth-values, e.g.,

x is a philosopher,

or

x is taller than y.

Concepts for Frege are therefore functions from objects to truth-values.

The collection of objects that map to the truth-value True is the value-range, aka extension, of the concept.

And in turn Frege identified classes or sets (of unordered or ordered objects) with the value-ranges, or extensions, of concepts.

8. There are four basic Fregean principles about sense and reference/Meaning, as follows.

P1. Sense-Determines-Reference: The sense of an expression uniquely determines its reference/Meaning.

P2. Compositionality 1: The sense of a complex expression is a function of the senses of its parts.

P3. Compositionality 2: The reference/Meaning of a complex expression is a function of the references/Meanings of its parts.

P4. The Context Principle: Words have sense and reference/Meaning only in the context of whole sentences, propositions, or thoughts.

II.9 Some Biggish Problems For Frege’s Semantics

According to Sense-Determines-Reference, the senses of names are supposed to uniquely determine their reference, yet some names clearly don‘t refer to anything that exists in the actual world, e.g., ‘Mr Pickwick’.

Moreover, the the occurrence of an empty or non-referring name in a sentence will guarantee that the whole sentence doesn’t have a truth-value, since according to Compositionality 2, the incomplete sense of the whole sentence will fail to deliver a compound reference/Meaning for that whole sentence, if any of its component expressions fails to deliver a referent/Meaning for that component expression.

So such sentences, it seems, can’t be accounted for by pure logic in Frege’s sense, which includes a strong principle of bivalence:

Necessarily, every proposition or thought is either true or false, not neither (aka “truth-value gaps”), and not both (aka “truth-value gluts”).

(A moderate principle of bivalence would hold that necessarily, every proposition or thought is either true or false, or neither, but not both — thereby allowing for truth-value gaps, but not for truth-value gluts.

And a weak principle of bivalence would hold that necessarily, not every proposition or thought is both true and false, and every proposition or thought is either true, false, neither, or both [provided that the logic is also paraconsistent] — thereby allowing for truth-value gaps and truth-value gluts alike, but not for universal glut-ishness, aka explosion, aka logical chaos.)

Corresponding to the problem of empty or non-referring names are two sub-problems.

First, how can we ever determine in advance of actual language–use, which names are going to be empty and which are going to be non-empty?

Frege simply stipulated that all names will have reference/Meaning for the purposes of logical analysis — so, all names will be non-empty — but that seems just to dodge the deeper worry and also avoid facing up to the problem of the semantics of fiction.

In at least one place, Frege did briefly discuss fictional names and fictional sentences, and said that they respectively expressed “mock senses” and “mock propositions” or “mock thoughts,” but that seems obviously insufficient, since isn’t the following sentence obviously and non-mock-ishly true?

In Charles Dickens’s picaresque novel Pickwick Papers, Mr Pickwick is a fat and jolly man who has many amusing adventures.

Or would Frege have held that such sentences also have “mock truth-values”?

But if so, then what in Kant’s name might those be?

Second, how are we to construe the truth of negative existential claims?, e.g.,

Mr Pickwick doesn’t exist.

If the occurrence of an empty or non-referring name in a sentence guarantees that the whole sentence doesn‘t have a truth-value, then the truth of this sentence cannot be explained by Frege‘s theory.

Another biggish problem for Frege’s semantics is Frege’s triadic ontology and platonism.

According to Frege’s triadic ontology and platonism, all senses, all functions (including, of course, concepts), all classes or sets (i.e., the value-ranges or extensions of concepts), and all universal and necessary truths, are neither mental nor physical, but instead exist in a “third realm,” outside of time and space, that’s nevertheless cognitively accessible to all rational cognizers.

The “first realm” is the mental or psychological world, and the “second realm” is the physical world, so that as an ontological triadist Frege seems to be working with a classical Cartesian mental-physical dualism as an ontological starting place, and then adding one more ontological category that’s neither mental (temporal but not spatial) nor physical (spatiotemporal).

Earlier in the 19th century, the Bohemian (as opposed to bohemian) priest, logician, and philosopher Bernard Bolzano had already postulated something similar to the inhabitants of Frege’s third realm, which Bolzano called the “representation in itself” (Vorstellung an sich) or the “objective representation” (objektive Vorstellung).

And an immediate predecessor and partial contemporary of Frege’s, Hermann Lotze, held that in addition to the class of mental entities and the class of physical entities, there’s also a third class of entities, including contents of mental representations, as well as universal and necessary a priori truths, all possessing “validity” (Gültigkeit).

Generalizing, we might say then that the population of Frege’s third realm includes semantic contents (i.e., senses of all kinds), functions, properties, and relations (especially all kinds of “concepts” in Frege’s sense), logical constants, and other abstract entities like logical laws and logical truths.

Nevertheless, The True and The False don’t seem to fit comfortably into either the first realm, or the second realm, or the third realm.

Moreover, according to Frege, in order to understand a word or sentence we must cognitively “grasp” (greifen) its sense.

Notice that the term for “concept” in German is Begriff.

So presumably, if we follow the clue of German etymology, a concept is the cognitively “graspable” sub-part of the complete sense of a sentence that corresponds to its predicate-expression.

Indeed, the sense of the predicate corresponds to what Kant and many or even most other post-Kantian philosophers, e.g., Husserl, would call a “concept.”

But for Frege, the concept is the reference/Meaning of the predicate, and not its sense.

Thus what Kant and many or even most other post-Kantian philosophers are calling a “concept” is not what Frege calls a “concept,” even though for Frege cognitively “grasping” the sense of a predicate is essentially the same as what Kant and many or even most post-Kantian philosophers call conceptualization, or understanding a concept.

So it’s both a big bummer and also consistently confusing that Frege didn’t call his concepts “properties,” “relations,” or “one-place and many-place universals” or whatever.

In any case, belief or judgment is asserting a sentence that expresses a proposition or thought, which we and others thereby “grasp” and understand, so that we and others can “advance” in thinking from the sense of the sentence to the reference/Meaning of that sentence, i.e., to its truth-value.

But precisely how do we cognitively “grasp” senses, if our thinking is mental and therefore in time, and even if, as many materialists or physicalists hold, our minds are identical or otherwise reducible to or anyhow supervenient on our brains, and our brains are physical and exist in space, yet senses exist in the third realm?

This is of course a classical problem for any platonic epistemology, more recently and famously re-formulated as a dilemma about mathematical truth and knowledge by Paul Benacerraf:

On the one hand, we’re committed to a standard realistic semantics of mathematical truth, according to which such truths and their component referring expressions stand for entities and states of affairs to which these truths refer and correspond; but on the other hand, our prima facie best epistemology, which connects knowers causally or at least directly with the objects they know, can’t make any sense of our engagement with the abstract entities and states of affairs picked out by mathematical truths.[vi]

There are, of course, various destructive solutions to the Benacerraf dilemma that involve either rejecting our standard realistic semantics, or rejecting our our prima facie best epistemology, or both.

Nevertheless, I think that there’s at least one constructive solution that accepts both our standard realistic semantics and also our prima facie epistemology, then refines and reformulates the theory of abstract mathematical entities as a structuralist theory, and is broadly Kantian in inspiration.[vii]

But since the Analytic tradition, whether classical or post-classical, is officially anti-Kantian from the get-go, then its card-carrying members are unlikely to pay any attention whatsoever to any Kantian solution to the Benacerraf dilemma or any other fundamental philosophical problem, much less seriously consider it, much less actually adopt such a solution.

Indeed, this huge and philosophically crippling anti-Kantian blindspot is a vocational disease of Analytic philosophy that I’ll look at more closely later.

In any case, without some serious Kantian help, Frege’s triadic ontology and platonic epistemology remain forever impaled on the horns of the Benacerraf dilemma.

NOTES

[ii] See, e.g., P. Benacerraf, “Frege: The Last Logicist,” in P. French et al. (eds.), Foundations of Analytic Philosophy (Midwest Studies in Philosophy 6) (Minneapolis, MN: Univ. of Minnesota Press, 1981), pp. 17–35.

[iii] See Frege, Foundations of Arithmetic, pp. 101–102; and G. Frege, “On the Foundations of Geometry,” in G. Frege, On the Foundations of Geometry and Formal Theories of Arithmetic, trans. E.-H. Kluge (New Haven, CT: Yale Univ. Press, 1971), pp. 22–37 and 49–112, at pp. 22–26.

[iv] See, R. Hanna, Kant, Agnosticism, and Anarchism: A Theological-Political Treatise (New York: Nova Science, 2018) (THE RATIONAL HUMAN CONDITION, Vol. 4), section 3.14, also available online in preview, HERE.

[v] See G. Frege, “On Sense and Meaning,” in G. Frege, Collected Papers on Mathematics, Logic, and Philosophy, trans. M. Black et al. (Oxford: Blackwell, 1984) pp. 157–77; G. Frege, “Thoughts,” in Frege, Collected Papers on Mathematics, Logic, and Philosophy, pp. 351–72.

[vi] P. Benacerraf, “Mathematical Truth,” Journal of Philosophy 70 (1973): 661–680.

[vii] See R. Hanna, Cognition, Content, and the A Priori: A Study in the Philosophy of Mind and Knowledge (Oxford: Oxford Univ. Press, 2015), chs. 6–8,, also available online in preview, HERE.

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