## 1. Antoine Arnauld, Jansenism, and the Port-Royal Abbey



Antoine Arnauld, the primary author of the *Logic*, was born in
Paris, on 8 February 1612, to Antoine and Catherine Arnauld. His
father was one of the most famous lawyers of his time. The son
Antoine, the youngest of their 20 children, originally wanted to study
law, but because his father had died in 1619, he decided to honor his
mother's wish that he study theology. He entered the Sorbonne,
becoming a disciple of Lescot, the confessor of Cardinal Richelieu. In
addition to the *Port-Royal Grammar* and the *Logic*,
Arnauld is best known as the author of the Fourth Objections to
Descartes' *Meditations*. He also engaged in lengthy
correspondence with Leibniz, carried on a polemic against Malebranche
in the *Treatise on True and False Ideas*, and wrote several
theological essays, including *The Perpetuity of the Faith*.
Pierre Nicole, the secondary author, was born at Chartres in 1625. His
father was also a prominent lawyer, with ties to literary circles in
Paris. Nicole studied theology at the Sorbonne, where he came into
contact with teachers inclined towards Jansenism. When Jansenism came
under attack at the Sorbonne, he withdrew and went to the abbey at
Port-Royal-des-Champs. He eventually became one of the most prominent
Jansenist writers of the seventeenth century; his *Moral
Essays* (1671-7) was at that time his most famous work.



Jansenism was a radical reform movement within French Catholicism
based on Augustine's views of the relation between free will and
the efficacy of grace. The movement was named after Cornelis Jansen
(or Cornelius Jansenius), a Dutch theologian born in 1585 who studied
at the Sorbonne. He became the Bishop of Ypres in the Spanish
Netherlands in 1636 and died two years later. His major work
*Augustinus* was published posthumously in 1640. A second
figure in Jansenism was the Abbot of Saint-Cyran, born Jean Duvergier
de Hauranne in 1581. He received his M.A. in theology at the Sorbonne
in 1600, where he met Jansen. The two worked together from
1611-1617 on scriptural questions. The issues bringing Jansenism
into conflict with Catholic orthodoxy concerned the efficacy of grace,
the role of free will in salvation, and the nature of penitence. The
attack on Jansen began with Isaac Habert's sermons and writings
during 1643-44. By 1653 Pope Innocent X issued an encyclical,
*Cum occasione*, declaring five propositions in
*Augustinus* to be heretical. They expressed the views that a
just person who wishes to obey God's commandments cannot do so
without the necessary grace to carry them out; that in the state of
corrupt nature one can never resist interior grace; and that
meritorious actions require only a liberty exempt from constraint
rather than one exempt from necessity.



Although he did not agree with all of Jansen's views on grace
and free will, Arnauld devoted several major works to defending
aspects of Jansenism, including *On Frequent Communion*, the
*Defense of Monsieur Jansenius* and a *Second Defense*.
These writings resulted in a trial and his expulsion from the Sorbonne
in 1656. In 1661 the Council of State decreed that all churchmen must
sign a formulary drawn up in 1657, condemning the heretical
propositions in Jansen's work *Augustinus*. Arnauld and
Nicole had taken the position that the propositions were heretical but
did not appear in the *Augustinus*. From 1669 to the late 1670s
there was a truce between the Catholic Church and the Jansenists. But
in 1679, after attacks had resumed, Arnauld went into exile in the
Netherlands; he died at Brussels on 8 August 1694. Nicole had joined
Arnauld in exile, but he returned to Paris in 1683 where he reconciled
with the authorities. He died in Paris in 1695.


## 2. The Cartesian Background



Although Augustine shaped the theology of Jansenism, Rene
Descartes' influence predominates in the philosophical aspects
of *the Logic*. Arnauld's insightful and respectful set
of "Objections" to Descartes' *Meditations*
along with Descartes' "Replies" were first published
in 1641 (see Descartes 1984 for English translation, abbreviated CSM).
This exchange confirmed Arnauld in his belief that Descartes'
system of philosophy was very well suited to furthering his own
version of the Port-Royal theological program. He was impressed by
Descartes' frequent insistence that natural science,
metaphysical speculation, and even mathematics and logic were to be
subordinated to ethical and religious ends (Lachterman 1989, Jones
2006). Especially important was Descartes' project of
demonstrating that the foundations of natural science are compatible
with, and in fact depend on God. The dualism of mental and corporeal
created substances made it possible to conceive the natural world
mechanically without recourse to any of the additional quasi-spiritual
entities posited by Aristotelians. Geometrically construed matter
requires only God's creative activity to be the subject matter
of Cartesian physics. The *Logic'*s introductory
"First Discourse" announces the intention to choose
examples throughout the book that promote both correct religious
morality and the best (i.e., Cartesian) science. Examples of how
reason favors Cartesian science over its competitors are accordingly
sprinkled throughout the *Logic*.



The resources of Cartesian philosophy for repudiating skepticism also
appealed to Arnauld. Cartesian knowledge is grounded in clear and
distinct perceptions of reason alone. Truths delivered by reason alone
cannot be doubted unless they are not clear and distinct and thus
"confused" in Descartes' terminology. For Descartes
and Arnauld this means that skeptical denials of fundamental truths
are merely verbal (*Logic*, First Discourse 7; also see Lennon
2008: 62-77). The soul surrenders to the truth whenever it is
strongly penetrated by it just as eyes see whenever they are open and
struck by sunlight (*Logic*, First Discourse, 8). For example,
one who affirms that "It is doubtful that God exists" must
fail to clearly and distinctly perceive God. The *Logic* sets
itself against Montaigne and Pyrrhonian skepticism in particular, but
also takes frequent opportunities to undercut skeptical threats from
materialism and irreligion. Elsewhere, Arnauld wanted to emphasize
along with Descartes himself that the latter's "method of
doubt" as presented in the *Meditations* "should be
studied only by very intelligent and well-educated readers"
(Replies to fourth set of objections [CSM II, 172]). This restriction
was meant to avoid encouraging the very doubts that Descartes
introduces as part of the specialized method for attaining certain
knowledge.



Even more important for the *Logic* is Descartes'
attitude toward formal logic. In his early, incomplete *Rules for
the Direction of the Native Intellect* (c. 1628) and the
well-known *Discourse on Method* (1637), Descartes eschewed
traditional logic in favor of techniques for acquiring knowledge in
the first place. Syllogisms rarely accomplish that; they serve instead
as a means of presenting what is already known. His view of deductive
inference was accordingly based on his theory of the mind's
perceptions. From our contemporary perspective, it could be said that
Descartes' philosophical system replaced formal logic with
epistemology and the philosophy of mind. The *Logic* does
devote considerable space to a treatment of formal techniques, but the
reader is repeatedly reminded that this is of secondary importance.
The full title of the book is *Logic, or the Art of Thinking*,
not the art of reasoning or the art of inferring. The book also
contains a wealth of examples meant to illustrate moral and religious
norms and



> 
> 
> ...almost as much metaphysics as one ought to know, although we
> do not claim not to have borrowed from anyone. (*Logic*, First
> Discourse, 12)
> 
> 
> 



Because the *Logic*'s digressions into the non-logical
territories of natural science and religion are often lengthy, they
influence the character of the book. The authors claim that these
features add to the book's readability and that they are
concrete applications of its general lessons to present knowledge
about actual things.


## 3. Overview of the Structure of the *Port-Royal Logic*



The text of the *Logic* is organized around the four mental
operations important to the "art of thinking":
*conceiving* ideas, affirming and denying propositions
(*judging*), affirming and denying propositions on the basis of
other propositions (*reasoning*, i.e., inferring), and
*ordering* results of good reasoning. Logic is the sequential
application of these operations. Conceiving ideas is the "first
action" of the mind, propositions are formed from ideas, and
reasoning involves propositions. Ordering these products of thought
constitutes *method*. This organization reflects the point that
logic is taken to be primarily a science of the activity of thinking
in the pursuit of knowledge. This framework for understanding logic is
now regarded as "psychologistic" in opposition to both
earlier and post-Fregean approaches that focus on methods for dealing
with formal aspects of reasoning.



Introductory material to the final (1683) edition includes a Preface
(added 1683), a Foreword and First Discourse (1662), and a Second
Discourse (added 1664). The First Discourse lays out the plan of the
*Logic*, explaining that its main purpose is to educate
judgment to make it more precise. This is seen as instrumental for
making the speculative sciences more useful. Thus the *Logic*
contains not only a treatment of correct reasoning, but also examples
of how reasoning can go wrong. The Second Discourse offers a reply to
objections made to the first edition. The main point is to justify the
critical treatment of Aristotle on the ground that knowing how a great
mind can err can help others avoid making the same mistakes. But it
also takes pains to point out that the *Logic* owes some things
to Aristotle's *Analytics* and other works. The main text
that then follows is about the four mental operations with one Part
devoted to each.



Part I on ideas consists of 15 chapters devoted to five topics: the
nature and origin of ideas (chapter 1); the objects ideas represent
(chapters 2-4); simple versus compound ideas (chapter 5); the
logical analysis of universal, particular, and singular ideas and the
extension and comprehension of terms (chapters 6-8); and clear
and distinct versus obscure and confused ideas, including a discussion
of types of definition (chapters 9-15).



Part II on judgment consists of 20 chapters. Recognizing that the mind
closely links ideas with the words that express them, their discussion
begins with an analysis of parts of speech in chapters 1 and 2.
Chapters 3 and 4 present a version of the Aristotelian theory of
categorical proposition and the square of opposition. Chapters
5-14 treat the properties of simple, compound, and complex
propositions, including how to identify and classify them. This
section contains the famous distinction between restrictive
("determinative") and non-restrictive
("explicative") subordinate clauses (chapter 6), as well
as a discussion of what we would now call logical connectives and
non-truth-functional propositions (chapter 9). The theory of
definition returns in chapters 15 and 16 in the context of scientific
applications. Finally, chapters 17-20 on the conversion of
propositions contain part of the Port-Royal version of the medieval
doctrine of distribution.



Part III focuses on rules for syllogistic reasoning, and is divided
into 20 chapters. Although the authors insist that most erroneous
reasoning is based on false premises rather than incorrect inferences
(*Logic*, 9), they believe the study of syllogistic forms can
be helpful for exercising the mind. The *Logic* classifies
syllogisms into simple and conjunctive, and simple syllogisms into
complex and noncomplex (despite the awkward terminology). After
defining terms in chapters 1 and 2, they present general rules for
simple, noncomplex syllogisms in chapter 3. This chapter completes
their reworked version of the theory of the distribution of terms,
begun in the last four chapters of Part II. Chapters 4 through 8
explain with innovative compactness (albeit still with tedious detail)
the figures and moods of simple syllogisms, again reproducing
traditional Aristotelian and medieval results. In chapters 9 through
13 the *Logic* treats in a less formal way principles for
recognizing validity in syllogisms that are not simple. Chapters 14,
15, and 16 discuss respectively enthymemes, sorites (syllogisms with
more than three propositions) and dilemmas. Despite their view that
the traditional theory of topics (the method for finding arguments) is
especially lacking utility, the authors treat it in chapters 17 and
18. "Those who object to this can spare themselves by not
reading them" (*Logic*, 11). Here they criticize
Aristotle, Ramus, and the Scholastics. Finally, chapters 19 and 20
discuss sophisms and fallacies.



The *Logic* ends with Part IV on method. The first three Parts
of the book correspond to the tripartite layout of traditional logic
texts. Part IV, however, constitutes an innovation reflecting the
authors' enthusiasm for the philosophy of Descartes and Pascal.
Chapter 1 lays the groundwork in Descartes' and
Augustine's rationalism, criticizing the role of the senses in
providing knowledge, as well as the claims of Academic and Pyrrhonian
skeptics. After spelling out the methods of analysis and synthesis in
chapter 2, chapters 3 through 10 focus on the methods of geometry,
including rules of definitions, axioms, and demonstrations. Chapter 11
then recapituates eight rules of scientific method. Finally, chapters
12 through 16 contrast the nature of rationally obtained knowledge
with faith or belief.


## 4. Ideas



In the *Logic*, ideas are the basic terms, or elements, of the
art of thinking,



> 
> 
> ...the reflections we can make on our ideas are perhaps the most
> important part of logic, since they are the foundation for everything
> else. (I.1, 25)
> 
> 
> 



All operations of the mind involve ideas. The *Logic*
characterizes ideas in the Cartesian way. They are, in the first
place, individuated by their objects. Ideas are "as it were the
images of things" as Descartes wrote (*Meditations*,
Third meditation [CSM II, 25]). The notion of an image is taken
generally; some very important ideas, the idea of God for example, are
not acquired by sensory perception. The *Logic* takes pains to
emphasize this aspect of rationalist philosophy Thinking about a thing
requires an idea that represents the thing. Because ideas are in this
sense objective, i.e., of objects, they can be classified according to
the metaphysical classification of their objects. In the Cartesian
metaphysics of the *Logic*, there are three fundamental kinds
of objects, substances, unchanging aspects of substances called
"attributes", and features that change over time called
"modes". Although Arnauld elsewhere takes up metaphysical
questions about the relation between ideas and their objects, then are
not taken up here (see Nadler 1989).



Descartes' contrast between ideas that are clear and distinct as
opposed to ideas that are obscure and confused is central to the
*Logic*. It is primarily concerned with the ways in which
thought arrives at the truth; the epistemic status of the ideas with
which one thinks is more basic than rules for proceeding from one
thought to another (Gaukroger 1989, Nelson 2017). Although Descartes
is usually read as considering the clarity and distinctness of an idea
to be a psychological criterion external to the idea itself (but see
Lennon 2008: 137-48), the *Logic* agrees with Spinoza
that perceptions of unconfused, true ideas are known as such without
recourse to a criterion of the sort sought by ancient sceptics. A good
example of the employment of clarity and distinctness is provided by
the characterization of the *attributes* and *modes* of
substances. Again following Descartes, according to the *Logic*
an attribute is recognized by one's inability to distinctly
perceive a substance if one of its attributes is excluded from it.
Extension, for example, is an attribute of bodies. Bodies cannot be
distinctly perceived except as extended. A mode cannot be distinctly
perceived except in relation to the substance of which it is a mode,
but substances can be distinctly perceived while some particular modes
are excluded from them. A mind can be distinctly perceived without the
thought of a particular pain, but that particular thought is only
confusedly perceived unless it is perceived as that mind's
thought.



Communication with others requires that ideas be expressed with words.
Complex thought and the accumulation of personal knowledge are also
hardly possible without recourse to words. Thinking and expressing
thought with words with insufficient care, however, are the main
sources of error. Much thinking takes place with exclusive attention
to words without regard for the ideas signified by those words. It is
also common for the association between words and ideas to drift in
ways that damage the coherence of thought. The priority of thought to
linguistic expressions is put in extreme terms.



> 
> 
> If reflection on our thoughts never concerned anyone but ourselves, it
> would be enough to examine them in themselves, unclothed in words or
> other signs. (*Logic*, 23)
> 
> 
> 



This has the practical result that most disputes are verbal.
Disagreement about clearly and distinctly ideas themselves is
virtually impossible. Careful assignment of words to ideas making use
of explicit definitions when necessary minimizes disputes. This
stresses the importance of precisely tracking how words are associated
with ideas (I.12, 62).



Nevertheless, it is practically impossible to do all one's
thinking without words and other symbolic representations. And it
would also be tiresome in a philosophical treatise to repeatedly
explain whether the author is referring to words, or to the ideas for
which they stand, or to both at once. The *Logic* depends on
the reader to use context to make the appropriate disambiguations.



Ideas fall into classes that parallel the Cartesian ontology. Ideas of
substances are conventionally associated with ideas of nouns and thus
further associated with written and spoken words, which are themselves
objects of the word-ideas. Ideas of attributes and modes are
associated with abstract nouns such as "hardness" and
"justice", as well as by adjectives such as
"hard" and "just" (I.2, 30-31). Words
themselves signify objects only transitively, as it were, via ideas of
words and then ideas of objects. The *Logic* often takes the
shortcut of writing as though words directly signify things. It
depends on the reader to keep the full chain of associations in mind.
The logical distinction between general terms and particular or
singular ones emerges from ideas of attributes and modes. As noted
above, these ideas are most distinct when conceived along with the
idea of the substance on which they depend. When they are conceived
without attention to their substance idea, they can represent
additional objects. This is illustrated with a geometrical
example.



General or abstract ideas are an important class of ideas are derived
from ideas of particular things. Although everything that exists is
singular, ideas can represent more than one thing. The general idea of
a triangle represents all particular triangles. Next, proper nouns
which indicate single individuals, such as "Socrates",
"Rome", "Bucephalus", are distinguished from
common or appellative nouns such as "man",
"city", "horse", which can indicate more than
one thing. Throughout the text, the authors call both universal ideas
and common nouns "general terms." The mental process
yielding the ideas associated with general terms is explained with a
geometrical example.



> 
> 
> ... if I draw an equilateral triangle on a piece of paper, and if
> I concentrate on examining it on this paper along with all the
> accidental circumstances surrounding it, I shall have an idea of only
> a single triangle. But if I ignore all the particular circumstances
> and focus on the thought that the triangle is a figure bounded by
> three equal lines, the idea I form will, on the one hand, represent
> more clearly the equality of the lines and, on the other, be able to
> represent all equilateral triangles. (I.5, 38)
> 
> 
> 



Similarly, one can ignore the equality of the lines and have a more
abstract idea representing all triangles. Ignoring the number of lines
enables one to represent all plane figures, and so on. It is easy to
see how this works for ideas of modes such as red, loud, painful, etc.
It is noteworthy that the authors affirm that all existences are
particular. Their construal of abstract ideas and the words associated
with them constitutes a thoroughgoing nominalism about the
signification of abstract ideas and the words associated with
them.



Medieval philosophers explained the significance of general terms by a
complex theory of supposition (see Spade 1982). The *Logic*
condenses this framework so that the significance of general ideas has
two aspects: the *comprehension* and the *extension*.
The comprehension consists in the set of attributes essential to the
idea. For example, the comprehension of the idea
"triangle" includes the attributes extension, shape, three
lines, three angles, etc. The extension of the idea consists in the
particular objects to which it applies, which includes "all the
different species of triangles" (I.6, 39-40). Here the
authors confuse matters by trying to incorporate the traditional terms
"inferiors" and "species", taking the latter
for individual objects. Usually, however, they straightforwardly take
the extension of a general idea to be the individuals possessing the
attributes in its comprehension. There are three key features of this
theory of signification. First, the comprehension rather than the
extension is essential to the function of a general idea: one cannot
remove an attribute without destroying it--making it into a
different idea, whereas one can restrict its extension by applying it
to only some of the objects that fall under it. Second, the
comprehension governs the extension: the set of attributes determines
the individuals in its extension. Finally, comprehensions and
extensions are inversely related. In adding attributes to the
comprehension of an idea one restricts its extension. For example, the
comprehension of the idea "mammal" includes that of the
idea "animal" so the extension of "animal"
includes that of "mammal." The *Logic* implicitly
assumes this principle throughout the text. In recognizing these two
modes of the signification of words--both the comprehension of
the associated idea and the objects in the extension--it replaces
the traditional distinction between incomplete and complete entities
in the treatment of the signification of general terms.



It is tempting to interpret the *Logic* as providing a
precursor of contemporary semantic theories in which comprehension is
like meaning or Fregean sense and extension is like reference. The
comprehension of a term does usually determine the objects in its
extension. It must be remembered, however that it is psychological
ideas and not words that have comprehensions. No ideas are identically
shared among minds, so only in the case of innate ideas or complexes
strictly defined in terms of innate ideas can communication result in
something like transmission of meanings in the modern sense. Hacking
(1975) argues that seventeenth-century texts, including the
*Logic*, are primarily concerned with mental discourse and do
not include a philosophy of language at all.



Moving to linguistic terms and their association with ideas, the
authors first take up the noun system. This is mostly drawn from Part
II of the *Port-Royal Grammar*. As explained in Part II of the
*Logic*, nouns are names of objects, that is, substances and
attributes. Substantive nouns such as "earth" and
"sun" signify substances, and adjectival nouns such as
"good" and "just" signify attributes,
"indicating at the same time the subject to which they
apply...". Just as substances are ontologically prior to
their manners or modes, nouns are prior to adjectives in the structure
of language. From the adjective one then creates a secondary
substantive, an abstract noun:



> 
> 
> after having formed the adjective *human* from the substantive
> word *man*, we form the substantive *humanity* from the
> adjective *human*. (II.1, 74)
> 
> 
> 



Thus there are three kinds of nouns: concrete substantives,
adjectives, and abstract substantives. The *Logic* says that
adjectives have two significations: a distinct signification of the
mode or manner, and a confused signification of the subject. Although
the signification of the mode is more distinct, it is indirect; by
contrast the confused signification of the subject is direct (II.1,
74-75). This is because the idea of a mode is distinctly
perceived only when its substance is perceived along with it. So every
noun picks out or *distinctly signifies* one thing, either an
individual, a collection of individuals, or an attribute. Concrete
substantives distinctly signify complete objects, that is, individual
substances: "man" distinctly signifies human beings. The
adjective "human" distinctly signifies the attribute of
being human. And abstract substantives such as "humanity"
also pick out this attribute distinctly. But the adjective
"human", unlike the concrete substantive
"man", is linguistically incomplete, since it means
"a human [being]". Linguistically, adjectives are gappy
and require completion by a substantive to signify. The *Logic*
identifies this incomplete signification as the *connotation*
or confused (but direct) signification of an adjective. So adjectives
signify ideas of substances *directly and confusedly*, and
ideas of attributes *indirectly and distinctly*. Because
substantive nouns of both kinds are linguistically complete, they lack
connotation and have only distinct and direct signification to the
individual substances or attributes they name.



The production of general ideas from less general ones is, as we have
seen, effected by the mind separating or deleting some components of
the less general idea. The general idea of triangle could be produced
by deleting equilaterality from the idea of an equilateral triangle.
Similarly, ideas of greater complexity can be formed by combining less
complex ideas. If, for example, one imagines a pink elephant, that
idea is formed by combining an idea of an elephant with an idea of
pink. The mental act of affirming or denying one idea of another is
called judging. Mental operations of special importance to logic are
those that result in propositions. This is the focus of the next part
of the *Logic*.


## 5. Judgment and Propositions



The mental act of judging includes both affirming one idea agrees with
another and denying such agreement. In affirmation, the result is a
joining of the ideas; they are judged to agree with each other. In
denial, the result is a separation of the ideas; they are judged to be
"repugnant" to each other. "Agreement" and
"repugnance" are unpacked in terms of the crucial notions
of comprehension and extension. When one judges that "*s*
is *P*", affirms that the comprehension of *s*
includes the comprehension of *P* and that *s* is
included in the extension of *P*. Because ideas are
characterized by their objects, a judgment is true when the ideas are
related as the objects are related. As usual, this works
straightforwardly only when the ideas are sufficiently clear and
distinct. If one's idea of Arnauld confusedly contains the
attribute of being married, then one might erroneously judge that the
idea of Arnauld agrees with the idea of being married.



Judgments are also called propositions, so propositions are mental
entities. The priority of thought to language thus includes the point
that linguistic expressions of mental propositions are propositions in
a derivative, verbal sense. Verbal propositions are usually simply
called "propositions" and the reader is expected to sort
this out by context. Additionally supposing that the reader will not
forget the distinction between thought and language, the authors use
the terms "subject" and "predicate" to refer
both to the ideas making up a judgment and to the linguistic
expressions of the subject and predicate ideas. Because every simple
judgment is composed of a subject idea and a predicate idea that are
related by an act of affirmation or denial, the subject and predicate
ideas are expressed linguistically in the simplest case by a proper or
substantive noun and a common noun or adjective. The affirmation is
expressed by a verb; expressions of denial add a negative word (II.3,
82-3). Arnauld and Nicole criticize Aristotle and other
philosophers who combine the copula with features of the predicate
(time) and subject (person); in a language mirroring Cartesian
thought, there would be only one *substantive verb*, namely
*to be*. In fact, natural languages often combine the predicate
with the verb, as in "Peter lives", and Latin verbs
sometimes express all three elements of judgment in one word, as in
*cogito* and *sum.* Such verbs as "lives"
can always be expressed with verb phrases including "is"
and a noun phrase. The proposition "Peter lives" for
example, can be linguistically rendered as "Peter *is*
living."



The doctrine that judgment arises exclusively from mental acts of
affirmation and denial raises some important philosophical questions.
First, if it is an act of *affirmation* or *denial* that
results in a proposition, there seems to be no room for doubting or
suspending judgment on a proposition. The *Logic* proposes a
"one act" theory of affirmation and denial. Another kind
of theory would require two acts: first, the formulation of a
judgment, and second, affirming or denying it. In much post-Fregean
philosophy it is a requirement that propositions can be thought
without asserting them and that mental "attitudes" such as
believing, hoping, fearing, etc. can be taken regarding a proposition
that is neither affirmed nor denied.



In the *Logic*, however, things are different because of the
oft-cited sense in which thought is prior to language. Well formulated
mental propositions can, in principle, be thought wordlessly--a
linguistic sentence will rarely mirror the form of a thought with
perfect accuracy. This means that to affirm a verbal proposition is
not always to have a clear and distinct perception of it. In a
generally Cartesian theory of the mind, what is plausibly called
"entertaining" or "suspending judgment" on a
proposition does not require a mental act apart from affirming or
denying. Suppose one thinks the proposition,
***P***, but simultaneously thinks much more
clearly the proposition, "I should investigate the truth of
***P***", or the proposition
***Q*** where ***Q*** is
incompatible with***P***. Those states of mind
seems to answer at least as well to the description "merely
entertaining ***P***" as does a special
mental act of *entertaining* or "suspending judgment
on" ***P***. The authors could even claim
that one never does merely entertain a proposition
***P*** without the further concomitant judgment
that ***P*** is to some extent doubtful, or that
it would be rash to act on ***P*** without further
investigating whether ***P*** is true, etc.



A second set of related questions is raised by the treatment of
logical operations in modern quantificational logic.



> 
> 
> Indeed, a loose chronicle of the role of the proposition in modern
> philosophy runs as follows: Most of the important Early Moderns
> conflated the formation of a thought that has propositional content
> with endorsement of that content and simply failed to see the
> disastrous logical and epistemological implications of this
> conflation. Part of Frege's genius, the story goes, was to see
> through this mess and clearly distinguish propositional content from
> judgment from assertoric force. (Marusic 2014: 255)
> 
> 
> 



Complex propositions are subject to a generalized version of what is
now called the Frege-Geach problem. In a conditional proposition, for
example, it might seem that one could affirm the conditional without
separately affirming its components. But in the *Logic*, the
propositions constituting the antecedent and consequent are formed by
an act of affirmation (or denial). The proposition, "If the
Earth is flat, then astronomy needs revision" should be
affirmable without one's accepting the antecedent as true, i.e.,
without affirming it.



The authors take no notice of this problem and do not directly address
it. When discussing complex propositions, the *Logic* shifts
from explicit considerations of comprehension and extension to truth
conditions. A conditional is false when the antecedent is true and the
consequent false and true when both are true; a conjunction is true
when both conjuncts are true and false otherwise, and so on in the
familiar way. Regarding conditional propositions, the *Logic*
additionally relates them to inferences. To think "If
***P*** then ***Q***" is
to think of deducing the consequent from the antecedent (II.9, 100).
From the perspective of Kant (for example), these problems stem from
supposing that mental propositions are formed by merely combining or
"fusing" elements together. If conditionalizing,
disjoining, predicating, etc. are taken to be mental operations
distinct from affirmation and denial, a closer connection between
mental and verbal propositions becomes possible (see Section 1 of
 Kant's theory of judgment).



A different device comes into play for modal judgments of possibility
and necessity (II.8). Verbal propositions including modal terms such
as "possible" and "necessary" express mental
propositions in which the basic acts of affirmation and denial are
modified. The ideas entering into modal propositions are not
themselves modified (Van der Scharr 2008: 334-335;
Marusic 2014: 277-278). The authors label such acts
"tacit" or "virtual" affirmations (II.7, 93).
It seems open to the authors to regard such complex thoughts as
conditionals, conjunctions, and disjunctions as similarly involving
modified affirmations, but they do not explicitly endorse this
strategy.



Negation also raises interesting questions for the theory in the
*Logic*. The act of denial extends to the entire judgment; this
is expressed by the linguistic "not" attaching to the
verb. As Frege points out in his essay "Negation" (1918),
this account does not neatly fit with the requirement that a false
thought be recognized as false. It seems that propositions must be
affirmed (as true) or denied (as a true negation). For example, to
recognize that "3 is greater than 5" is false requires
having a complete thought, and not merely fragments of a thought.
Similarly, there seems to be no room for true thoughts that have false
thoughts as their components, such as true conditionals with false
antecedents. In addition, the *Logic* seems to have trouble
with double negation. If denying separates the thought into its parts,
then double negation would function as a sword that magically unites
the parts it had sundered (Frege 1918 [1952: 122-129]). From a
modern perspective, the root problem in treating negation as denial is
once again the failure to distinguish the thought or proposition that
is grasped from the act of judging it.



Frege's problems here stem from his requirement that the
structure of thought conform perfectly with a notation for expressing
the thought. The *Logic* "psychologistically"
prioritizes the theory of thought over the theory of symbolization. If
the authors had faced Frege's objections, they could have
insisted on inherent limitations in verbal expressions of thought.
When one clearly and distinctly perceives that "5 is greater
than 3" it is always affirmed. It is psychologically impossible
to affirm "3 is greater than 5" unless the ideas are badly
confused. If someone were to verbally assert "3 is greater than
5", it would be a mystery *what* they were thinking
unless it was a proposition about the words. Asserting a well-formed
verbal expression corresponds to a well-formed thought only insofar as
the thought is clear and distinct. Double negations are common in
complex verbal propositions, but the *Logic* is not committed
to these words signifying iterated "separations" that
result in rejoinings. They can simply stand for affirmations. This
means that when denial is described as "separating" ideas,
that should be understood metaphorically. Denial in the *Logic*
is perhaps better described as the recognition that the idea of the
predicate is not among those included in the comprehension of the idea
of the subject. And as usual, thoughts that are confused need to be
sorted out before they can be precisely expressed. The same holds for
triple negations and denials. (For a further discussion, see section
2.7.1 of the entry on
 Frege.)



The *Logic*'s view of proposition formation also suggests
a problem for embedded generality (Buroker 1994). Subordinate clauses
must be located in either the subject or the predicate. But some
embedded clauses make assertions and some do not. Despite the two
verbs in the complex proposition "Men who are pious are
charitable", for example, it is clear that one is not affirming
of all men or even some men that they are pious. On the other hand,
"God who is invisible created the world which is visible"
permits three assertions, "God is invisible", "The
world is visible", and "God created the world"
(II.5, 87). The *Logic* explains the difference between these
two kinds of embedding in terms of "determinative" and
"explicative" subordinate clauses (or, as they say,
relative pronouns, see II.6-8). Determinative subordinate
clauses restrict the signification of the antecedent of the relative
pronoun (e.g., "men who are pious") whereas explicative
clauses do not (e.g., "God who is invisible"). In fact
both determinations and explications can be carried out as well
without embedded or subordinate clauses, as in the sentences
"Pious men are charitable" and "The invisible God
created the visible world." When working these cases out, as
usual in the *Logic*,



> 
> 
> it is often necessary to pay more attention to the meaning and the
> speaker's intention than to the expression alone. (II.6, 90)
> 
> 
> 



Part II on judgment concludes in Chapters 17-20 with rules for
the conversion of verbal propositions, which is the exchanging of
subject and predicate terms. These operations on propositions are the
*Logic*'s reworking of the traditional theory of the
distribution of terms which in turn depended on a complicated theory
of "supposition" (Spade 1982, Parsons 2006, Martin 2020).
In traditional logic, elaborate rules of conversion were necessary to
relate the premises of a syllogism to each other and to the
conclusion. This is mostly unnecessary in the *Logic*'s
treatment of syllogistic inference in Part III, but the authors take
the opportunity to display simplified conversion rules developed on
the basis of a few "axioms" regarding the containment
relationships between the comprehensions and extensions of terms.


## 6. Reasoning and Formal Methods



Part III begins with a preamble warning the reader not to accord too
much importance to formal rules of reasoning.



> 
> 
> The majority of people's errors, as we have already said
> elsewhere [i.e., on p. 9], depend more on reasoning based on false
> principles, than from reasoning incorrectly from their principles. We
> rarely allow ourselves to be misled by arguments that are defective
> merely because the conclusion is badly drawn. And those who could not
> recognize a fallacy by the light of reason alone would usually not be
> able to understand the rules behind it, much less to apply them.
> (*Logic*, 135; also see III.9, 157 and IV.20, 203)
> 
> 
> 



Here the authors stress that a good logic is a means for acquiring
knowledge rather than a tool for organizing it into premises and
conclusion. The final point is that when ideas are made properly clear
and distinct, the natural light of reason requires no assistance from
rules. We had been previously warned that primary utility of formal
rules consists in the exercise of the natural light that might come
from working through them. The authors continued that evaluation by
pointing out that the chapter titles can guide readers in omitting
this relatively unimportant "exercise" if they wish
(*Logic*, 10). They thus regard Part IV on method as more
important than Part III on formal reasoning.



The classification of syllogisms for formal treatment depends on the
classification of propositions although the latter is in the service
of the former. The *Logic* accordingly classifies simple
categorical propositions into four forms: universal affirmatives (All
*S* is *P*), universal negatives (No *S* is
*P*), particular affirmatives (Some *S* is *P*)
and particular negatives (Some *S* is not *P*). Simple
categorical syllogisms are arguments with a major premise, a minor
premise, and a conclusion. They contain only three terms. The major
premise has the "middle" term and the attribute of the
conclusion (the "major" term), while the minor premise has
the middle term and the subject of the conclusion (the
"minor" term). In Part II, the *Logic* prepares for
this by adopting the theory of categorical propositions, classifying
them in terms of quantity as universal, particular, or singular, and
in terms of quality as affirmative or negative. It takes the
traditional stand that singular propositions function logically like
universals, so all simple propositions have one of the following four
forms:



A:
All *S* is *P*
E:
No *S* is *P*
I:
Some *S* is *P*
O:
Some *S* is not *P*


Also following the tradition, the *Logic* treats the
quantifiers "all" and "some" as part of the
subject, so that "all men" and "some men" are
logically significant units. In explaining the rules of conversion for
subject and predicate terms, in Part II, chapter 17 they argue that
predicates are implicitly quantified: when one says "All lions
are animals", one does not normally think that all lions are
*all* the animals, but only *some* of the animals. So
"All *S* is *P*" in general means "All
*S* is (some) *P*" (II.17, 130).



Consider, for example this syllogism.



* Every good prince is loved by his subjects.
* Every pious king is a good prince.
* Therefore every pious king is loved by his subjects. (III.2,
137)




Here, the term "good prince" is the middle term, because
it appears in both premises but not in the conclusion. Traditional
logic categorizes forms of syllogisms according to their
"mood" and their "figure". The mood is the
arrangement of the three propositions in the
 A E I O forms.
 The figure is the arrangement of the major, minor, and middle terms
in the three propositions.



Most propositions, however, are more complex than this classification
suggests, for subjects and predicates need not be simple. In the
proposition "God who is invisible created the world which is
visible", both the subject and predicate include subordinate
clauses that appear to contain propositions (II.5-8). Because of
the overall subject-predicate structure of all propositions, embedded
propositions must be located in the subject or predicate. Because
propositions can be complex, so can syllogisms. That becomes important
in Part III when rules of inference are discussed, because all
propositions, including conditionals and disjunctives, are forced into
standard categorical forms. The treatment of verbal propositions,
then, requires that subjects and predicates have unlimited complexity.
There is, therefore, no basic inventory of simple linguistic parts
permitting a recursive analysis, as in the modern classification of
variables, function or predicate symbols, and logical symbols. On the
classical view the proposition has a simple organic unity from the
outside and a reiterable complexity from the inside. This again shows
the importance of the authors' commitment to the priority of
thought over linguistic expressions of thought. Complexity is to be
resolved in mental propositions by careful and attentive thinking
instead of by perspicuous verbal notation.



With this treatment of classification in hand, the *Logic*
proceeds to examine rules of formal inference. In traditional logic,
the rules for syllogism are connected with rules for the conversion of
propositions which in turn are connected with the distribution of
terms. A term is *distributed* if it is "taken
universally" or refers to all the individuals it denotes;
otherwise it is undistributed (Kneale & Kneale 1962 [1984: 272]).
As Parsons (2006: 61) puts it, the denotation of a term is its
extension on its own, whereas its reference is its extension in a
proposition. Thus the term "prince" denotes (i.e., is
predicable of) all princes, but in the proposition "Some princes
are just", the quantifier "Some" restricts its
extension in the proposition (its reference) to a subset of its
extension on its own (denotation). According to the rules for
distribution stated by medieval logicians, the subject terms of
universal propositions and the predicates of negative propositions are
distributed or taken universally throughout their extension; all other
terms are undistributed. The *Logic* subsumes a streamlined
version of these rules in chapters 17-20 of Part II, in its
explanation of the conversion of propositions, accounting for them in
terms of the comprehensions and extensions of the terms (II.17,
130-31).



Chapter 3 of Part III gives six rules for valid simple categorical
syllogisms. These rules conform to standard medieval logic, but they
are derived from the Part II rules mentioned above for converting
propositions. The first two rules for syllogisms govern the
distribution of terms.



* Rule 1: *The middle term cannot be taken particularly twice,
but must be taken universally at least once.*
* Rule 2: *The terms of the conclusion cannot be taken more
universally in the conclusion than in the premises*. (III.3,
139-40)




The remaining four rules express the standard views that


1. at least one premise must be affirmative,
2. the conclusion must be affirmative if both premises are
affirmative,
3. if one premise is negative the conclusion must be negative,
and
4. nothing follows from two particular premises (III.3,
141-42).



Chapter 4 begins a series of chapters that develops more specific
rules for correctly drawing inferences. The three propositions of a
syllogism can be in one of the
 four A E I O categories.
 This mathematically yields sixty-four arrangements into syllogisms:
AAA, AAE, AAI, AAO, AEA, ... OOO. Of these, only fifty-four are
compatible with the six general rules for syllogisms. When these moods
are duly arranged into figures, further rules are required to
determine which have true premises and conclusions (III.4). The
*Logic* thus emphasizes soundness over
validity--presumably because arguments with false conclusions are
"useless" in application. Much of the
*Logic*'s subsequent importance as a work in formal logic
is owing to what Kneale and Kneale (1962 [1984: 319]) refer to as the
"quasi-mathematical" rigor with which these rules are
derived and stated.



Chapter 9 moves on to complex syllogisms. These involve a complex
conclusion in which the terms "are not always joined in their
entirety to the middle term in each of the premises, but only to a
part of one of these terms". This example is provided in which
the attribute of the conclusion is split across the attributes of the
premises:



* The sun is an insensible thing.
* The Persians worshipped the sun.
* Therefore the Persians worshipped an insensible thing.
(III.9,158)




Traditional logic texts (and the first edition of the *Logic*)
methodically discuss how to reduce complex syllogisms to simple ones.
Later editions of the *Logic* abbreviate this
"pedantry" in favor of a general principle for reducing
complex syllogisms, or actually, to evaluate them directly without an
explicit reduction. The general principle is that *the conclusion
must be contained in the premises*. The principle is applied by
making a characteristic Cartesian move. One is to come up with a
"better known" proposition from which the conclusion
follows--the "containing" proposition (III.10,163).
Then one needs another proposition that shows that the conclusion is
indeed contained in the containing proposition--this is the
"applicative" proposition. Recall that the comprehension
of a term contains all the attributes constituting it while the
extension contains the objects that have those attributes.



The application of the general principle is not governed by any
further rules; that would defeat its purpose. Instead, in Chapter 11 a
series of examples is provided showing the general principle in
action. Evidently, working through the examples is meant to provide an
exercise in the direct deployment of their faculty for rationally
"seeing" which inferences are valid. It is noteworthy that
the examples proceed by taking the conclusion, the
"question", to be true. The authors are thus more
interested in the soundness of useful arguments than they are in the
more theoretical notion of validity. For a detailed examination of the
general principle in action see Pariente (1985) and Wahl (2018).



The next chapters proceed to consider "conjunctive"
syllogisms. Such syllogisms include, for example, a conditional
proposition in a premise or the conclusion or a disjunction in the
major premise. No new formal rules are introduced to deal with these.
Instead, the *Logic* again runs through many examples which are
treated by inspection (as we now say) making use of "the natural
light of reason" (III.9, 158, see also p. 23). This is
understandable given the scope of the general principle established in
the preceding chapters.



Chapters 19 and 20 conclude Part III with an extended treatment of
fallacious reasoning in learned contexts as well as in everyday life.
Sophisms and fallacies are taxonomized, copiously illustrated, and
attributed to specific flaws of intellectual character in those who
propagate them. These chapters occupy nearly as many pages as the
material on formal reasoning. With updated examples, these chapters
could compete with textbooks currently in use in college courses on
"informal logic".


## 7. Method, Science, and Faith



Part IV begins by rehearsing the defense of metaphysical and
scientific knowledge against skeptical philosophy. Following
Descartes, it is argued that some items can be so distinctly perceived
that they cannot be sincerely doubted. That one must exist in order to
think, credited to Augustine rather than Descartes, is cited as an
example. "The whole is greater than the part" and similar
principles constitute another set of examples which were not much
disputed. The same holds for geometrical propositions, but when
Cartesians took these as being true of the essence of matter,
sceptical resistance was plentiful. Against this, the authors'
only counter is that careful attention reveals the truth as they see
it.



Chapter 2 introduces a version of the crucial distinction between the
methods of *analysis* (sometimes called resolution) and
*synthesis* (sometimes called composition). The authors regard
analysis as more important method because it is a method for
discovering truths. Synthesis is the presentation of knowledge as
following from given premises. The material might be presented as
formal deductions as in logic texts including Part III of the
*Logic*. An alternative kind of synthetic presentation finds a
conclusion in geometrical style as following from definitions and
axioms. Descartes argued that the main function of synthesis is to
instruct others in what is already known--especially to compel
their assent if they are "argumentative or stubborn"
(*Meditations*, Replies to second set of objections [CSM II,
110-111]).



An analytic treatment begins with the question under
examination--the item that is to be analytically demonstrated. A
successful analysis separates out what is contained in the question
until some previously known items are found. One example the authors
provide is a sketch of an analysis of magnetic phenomena, summarizing
a case that Descartes dealt with in some detail (IV.2, 236; Descartes
*Rules*, rule 12 [CSM I, 49-50] and Descartes 1644, Part
4 [CSM I, 276-279]. Carefully attending to a lodestone, the goal
is to



> 
> 
> distinctly conceive a certain combination of beings and natures that
> are known to us so as to produce all the effects we see in
> magnets.
> 
> 
> 



If the previously known "beings and natures" are revealed
to have been contained in the question and thus to analytically follow
from it, the analysis is complete. If instead some absurdity or
impossibility is contained in the question, the analysis shows it is
false. Because analysis depends on this kind of careful attention to
what is contained, the method "consists more in judgment and
mental skill than in particular rules" (IV.2, 238).



This description is closely related to the general principle that the
*Logic* laid down for the evaluation of syllogisms. That
principle required that one take the conclusion of a syllogism as true
and then examine or inspect what is contained in the conclusion in
search of a more general truth from which the conclusion follows.
Although the *Logic* does not explicitly state that formal
rules for syllogisms should be valued less than the analytic method,
it is clear that this is what the authors had in mind when saying that
the only point in learning formal rules for syllogisms is obtaining
mental exercise. Synthetic demonstrations are cheap because whenever
one has successfully performed an analysis, one is trivially in
possession of a synthesis as well. The methods of analysis and
synthesis,



differ only as the route one takes in climbing a mountain from a
valley differs from the route taken in descending from the mountain to
the valley, or as the two ways differ that are used to prove that a
person is descended from St. Louis. One way is to show that this
person had a certain man for a father who was the son of a certain
man, and that man was the son of another, and so on up to St. Louis.
The other way is to begin with St. Louis and show that he had a
certain child, and this child had others, thereby descending to the
person in question (IV.2, 238)



The analogy is that if a proposition (or a genealogy) is unknown, then
one ascends to more general truths (the ancestor), but if the
proposition (or the genealogy) is known, then one descends to the
conclusion (the descendants). In other words, the analysis and the
synthesis of an item is just the same demonstration, the only
difference being the route or direction, i.e., whether one begins or
ends with the proposition in question (Rogers & Nelson 2015). This
also makes clear the sense in which method is a matter of ordering
knowledge. In analysis one puts the conclusion first in the order and
proceeds to find the truths that it contains. Synthesis reverses this
beginning with propositions that are already known and then
understanding that the conclusion follows.



Chapters 3-10 are concerned in one way or another with axiomatic
presentations of geometry. This is understandable given the reputation
of Euclid's *Elements* as a model of rigorous reasoning
and certain results. It is interesting therefore that the
*Logic* also, while acknowledging the virtues of the work of
Euclid and other geometers, offers some pointed criticisms. Applying
logical method to geometry was among Arnauld's many special
interests. He wrote a book entitled, *New Elements of Geometry*
(1667) in which he attempted to remedy the faults discussed here in
Part IV of the *Logic*. The authors treatment begins with some
rules for synthetic demonstrations, especially in geometry. There are
no surprises here. Terms must be scrupulously defined; propositions
that are taken to be axiomatic must be completely certain, and so on.
As stated, the rules might appear obvious, but they are elaborated in
a decidedly Cartesian way. The point of rules for definitions, for
example, is to ensure that linguistic terms become precisely
associated with distinct *ideas*. It follows that definitions
will be useless for those who do not have prior access to the ideas in
question. Regarding rules for axioms, the main point that emerges is
that they must be distinctly perceived in the Cartesian sense put
forward in Part I on ideas and Part II on propositions. This means
that axioms derive from innate resources and not from sensory
experience and that attributes must be clearly and distinctly
perceived as included in subjects. It also means that no sincere
skeptical questions can arise from properly chosen axioms. A good deal
of Part IV recapitulates Descartes' unfinished manuscript,
*Rules for the Direction of the Native Intellect* (c. 1628).
This was unpublished in Latin until after the fifth edition of the
*Logic*, but Arnauld had access to at least part of some
version of the manuscript. Chapter 7 offers as examples of "some
important axioms" others provided by Descartes in his replies to
the second set of objections to the *Meditations* (CSM II,
116-117).



The rules for synthetic demonstrations lead to a list of criticisms of
"geometers". Chapter 4 notes some equivocations in
Euclid's use of "angle", "ratio", and
"proportion." Later mathematicians are criticizes for not
respecting the distinction between nominal and real definitions. The
former should be stipulations about how words are associated with
ideas. The latter should characterize the nature of the object being
defined. Examples are given about disputes among prominent
mathematicians over nominal definitions as if they were real (IV.5).
Another mistake the authors find geometers making is "proving
things that do not need to be proved" (IV.9, 254). This mistake
is attributed to inattention to the connection between truth and clear
and distinct perception. Euclid proves that that any two sides of a
triangle are greater than the third side. Because this is immediately
evident, the proof violates method by mis-ordering what is known.
Another mistake Euclid makes in ordering is tacking back and forth
from more general to more specific cases; a number of examples are
provided. Proper order requires always moving from more simple items
to more complex ones. A more significant disagreement with geometrical
practice is about indirect proofs by *reductio ad absurdum.*
Because the denial of a true proposition is always confused, the
derivation of a contradiction will usually offer no insight into why
the conclusion is true (IV.9, 255-56). It is conceded, however,
that indirect proofs might be necessary when no direct proof can be
found, and they can be useful when a result is already obtained
directly.



The *Logic* ends in a way befitting a treatise associated with
the Port-Royal abbey. The last four chapters consider how to fit into
the method the deliverances of knowledge derived from authority
instead of from reason. Knowledge from authority is of two kinds:
knowledge from God, which is called faith, and knowledge from humans
"worthy of credence", which is called belief or opinion
(IV.12, 260). God's deliverances are infallible, but human
authorities differ in their reliability. The best opinions come from
the constant testimony of many sources. These can reach moral
certainty, i.e., all the certainty that is required for practical
purposes. Faith and opinion are, however, underwritten by reason
because, according to the authors, reason reveals that God is
infallible and that skepticism about reliable human authority is
groundless.



There follow some commonsensical rules for handling opinions. For
example, for a proposition to be accepted as true, it need not be
necessary and the possibility of a proposition's truth should
not automatically lead to its acceptance. These rules are applied to
evaluating claims of miracles. One should take care to evaluate all of
the available evidence before accepting a particular claim.
Outstanding authorities should be generally trusted. The miracles
related by Augustine, for example, will be accepted as such by
"a person of good sense" (IV.14, 267). In Chapter 15, the
commonsensical principles for evaluating testimony are applied to some
cases of then current interest regarding the disputed authorship of
various letters and texts. Chapter 16 concludes the *Logic*
with the topic of judgments regarding the future. The authors point
out that it is a common error to overemphasize great future harms or
benefits when these outcomes are very unlikely and to underemphasize
small harms or benefits that are very likely or repeatable. Infinite
benefits require a different strategy. The observation that the
authors view as most important is the case made famous by fellow
Jansenist, Blaise Pascal.



Only infinite things such as eternity and salvation cannot be equalled
by any temporal benefit. Thus we ought never balance them off against
anything worldly. This is why the slightest bit of help for acquiring
salvation is worth more than all the goods of the world taken
together. And the least peril of being lost is more important than all
temporal harms considered merely as harms.



That is enough to make all reasonable people draw this conclusion with
which we will end this *Logic*: that the greatest of all
follies is to use one's time and life for something other than
what may be useful for acquiring a life that will never end, since all
the goods and harms of this life are nothing in comparison to those of
the other life, and the danger of falling into those harms, as well as
the difficulty of acquiring these goods, is very great (IV.16,
275)



The book's conclusion one again stresses that formal logic and
even science are at best instrumental to the goal of living well.