Digital Dante

Dante’s Canzone Le dolci rime Translated into Formal Logic

(With Interlinear Translation)
Jenny Clark Schiff (2020)

Key
Version Without Interlinear Translation

Introduction to the Project
A Brief Note about First-Order Logic and Modal Logic
Stanza 1
Stanzas 2, 3, and 4
Stanzas 5 and 6
Stanza 7 and Congedo
Acknowledgments

Introduction to the Project

I apply the tool of logic to formalize the arguments related to nobility and virtue in Dante Alighieri’s poem Le dolci rime and, accordingly, am able to assess the validity of such arguments. A “valid” argument is understood to be one in which the conclusion necessarily follows from the premises. The premises need not be true for an argument to be valid. A valid argument with true premises is deemed “sound.” The extent to which Dante presents sound arguments is not the subject of this work. In his canzone, Dante refutes old views about nobility and puts forward views of his own. His arguments are driven by logic and, therefore, may be assessed by use of a formal tool.

The formalizations here are presented in first-order logic and, where appropriate, modal logic (see below for an explanation of these types of logic). In my formalizations, I have attempted to stay as close as possible to Dante’s statements as expressed, or reasonably implied, by the natural language in the poem alone. The question of what kind of logic is adequate to express Dante’s views is a complex one and will not be addressed here. Here I will confine myself to assessing validity with regard to first-order logic including modal operators and, accordingly, the efficacy of the canzone in delivering valid arguments. In any case, however, we ought not presume that only one logic underlies Dante’s claims, inferences, and arguments. For example, given Dante’s interest in the poem in time as it relates to nobility, it may prove worthwhile to express some of Dante’s arguments using temporal logic. Such an approach could allow for areas of further precision in characterizing tense and other temporal expressions.

Thus, with this work, what I offer readers is a translation of Dante’s arguments in the canzone from Italian to the language of first-order logic including modal operators. It is this translation, from Italian to logic, that offers the most effective, clearest, and most beautiful rendition of Dante’s arguments. Indeed, ultimately, the validity of Dante’s arguments— or lack thereof in the case of his argument at the end of the sixth stanza— is evident merely by looking at the logic on its own, without recourse to the poem itself.

The sophistication in Dante’s logical argumentation that my work demonstrates, by showing that Dante’s arguments can be expressed in logical notation, constitutes a significant contribution to our understanding of Dante’s philosophical preparation and ability. It is also thereby a contribution to the history of logic—perhaps indeed Le dolci rime constitutes the first time logic is so fully utilized in a lyric poem. For more context and support of this claim, including the suggestion that Le dolci rime is the first lyric poem to include a translation of Aristotle, see T. Barolini, “Aristotle’s Mezzo, Courtly Misura, and Dante’s Canzone Le dolci rime: Humanism, Ethics, and Social Anxiety,” in Dante and the Greeks, ed. Jan M. Ziolkowski (Washington, 2014) 171.

Pars destruens and pars construens

After formalizing, where appropriate, Dante’s language, it becomes apparent— and as I demonstrate using logical notation— that the pars destruens (where Dante explains and refutes old views on nobility) is more effective in presenting valid arguments than the pars construens (where Dante presents his own views on nobility), if certain expressed or reasonably implied principles are applied. For the purposes of this project, a “principle” can be understood as a universal conditional statement that, although not explicitly stated in his argumentation, Dante applies to a specific case of such a statement. For example, according to one of the principles Dante uses— namely, Principle (4)— for all x, y, and z, if x is a necessary condition for y and y is a necessary condition for z, then x is a necessary condition for z. In the course of his argumentation, Dante applies this to the specific case in which x is time, y is old wealth, and z is nobility.

Dante will “destroy” nearly all that he sets out to destroy in the pars destruens (see Proofs #1-#4) by arguing:

• that the definition of wealth as “antica possession d’avere / con reggimenti belli” (long-standing possession of wealth together with pleasing manners [Le dolci rime, 23-24]), or even as “antica possession d’avere” alone, is not correct; and
• that the opinion of his opponents— namely, the opinion that a man who is not noble cannot become noble and the offspring of a man who is not noble cannot be noble— not only implies a consequence that is inconsistent with the very definitions of nobility that holders of this opinion maintain but also is false because it leads to a contradiction.

The only component of what Dante said he would demonstrate in the pars destruens that he does not adequately show and, accordingly, cannot be expressed in logic without a gap in the argumentation, is that the opinion “che di gentilezza / sia principio ricchezza” (that nobility depends on wealth [16-17]) is “vile” (base [15]). This gap can potentially be filled with recourse to a gloss of the prose in Dante’s Convivio, written approximately ten years after the poem. However, such recourse is beyond the scope of this work, which, instead, focuses on what is expressed, or reasonably implied, in the poem alone. It is also not methodologically appropriate, because my goal here is to show, through the use of formal logic, what Dante was capable of doing circa 1293-1294.

The relation of the Convivio prose to the canzoni it purports to gloss is a complex issue in its own right. I hope one day to continue this project by addressing the question of the validity of the arguments as they are formulated in the Convivio prose.

By contrast, while the pars construens does incorporate some valid arguments (see Proof #5), on the whole, it is lacking. Accordingly, when Dante finally presents his own definition of nobility in the sixth stanza, namely, that it is “’l seme di felicità… / messo da Dio nell’anima ben posta” (the seed of happiness placed by God in a well positioned soul [119-120]), it is abrupt and comes without premises that necessarily support such a definition. That is, his definition comes without logical proof. In addition, “i segni che ’l gentil uom tene” (the distinctive features that a noble person possesses [80]), which are enumerated by Dante in the seventh stanza, are not supported by rigorous logical argumentation in the poem.

Summary of Proofs

Within this work, I include five proofs from the poem based on what is expressed or reasonably implied in the natural language of the third, fourth, fifth, and sixth stanzas.

• Proof #1 shows that the definition of nobility as antica possession d’avere con reggimenti belli is not correct by demonstrating that wealth is not a necessary condition for nobility and, consequently, the part of the definition according to which nobility is antica possession d’avere is not true.
• Proof #2 shows that the definition of nobility as antica possession d’avere con reggimenti belli is not correct because it is not a complete definition given its reliance on reggimenti belli.
• Proof #3 shows that the definition of nobility as antica possession d’avere con reggimenti belli or as antica possession d’avere alone contradicts the very opinion of those who maintain such definitions— namely, the opinion that a man who is not noble cannot become noble and the offspring of a man who is not noble cannot be noble. That is, the definitions require that time be a necessary condition for nobility, but the opinion, which indeed is a conjunction of two separate views, does not.
• Proof #4 shows that the opinion that a man who is not noble cannot become noble and the offspring of a man who is not noble cannot be noble is false because it leads to a contradiction. That is, the opinion requires the following: that either all men are noble or (exclusive) all men are not noble, or (exclusive) else that there was more than one man who was a beginning to mankind. However, such is not the case since there exist both men who are noble and men who are not noble, and there was exactly one man who was a beginning to mankind.
• Finally, Proof #5 shows that virtue, and, more precisely, the “abito eligente / lo qual dimora in mezzo solamente” (habit of choosing which keeps steadily to the mean [86-87]), derives from nobility by demonstrating that nobility comprehends virtue and something else as well.

For the purposes of this work, I take Dante’s references to virtue in the canzone to refer specifically to moral virtue. Accordingly, any time I write “virtue”— in natural language or in the logic I present— it is to be understood as such. A comment on my part is in order here. That there are other types of virtues, such as intellectual virtues, is not made explicit in the poem, as it is in the Convivio. Indeed, in the poem Dante does not accompany vertù or vertute with the adjective morale. Nonetheless, it is reasonable to infer that he is speaking specifically of moral virtue, particularly given what he cites from Aristotle. I have chosen to present a literal representation of Dante’s argumentation by translating each instance of the word “virtue” not as “moral virtue” but as “virtue” alone. I caution the reader, however, not to take this as an indication that, in my natural language and logic, I report Dante to be speaking of all virtues in general. As alluded to above, while it is not the subject of the present work, in the future I plan to examine potential differences between Dante’s arguments in the canzone and in the Convivio. In such a setting, the present comment will be worthy of more serious attention. For further context to my comment, see Barbi and Pernicone’s note on verses 83-84, Rime della maturità e dell’esilio, 426.

Exposition of Gaps in the pars construens

As the logic I present demonstrates, while Dante’s conclusion in the pars construens that virtue, and, more precisely, the abito eligente lo qual dimora in mezzo solamente, derives from nobility is supported, his later claims in the sixth and seventh stanzas are not fully supported. Specifically, in the sixth stanza, his intermediate conclusion that it is not by birth that a man is noble and his final conclusion that nobility is ’l seme di felicità… messo da Dio nell’anima ben posta are not fully supported. This is because Dante seems to rely on missing premises (premises that are not expressed or reasonably implied in the poem alone). Rather than posit what these missing premises might be, or attempt to fill in these gaps with recourse to the Convivio— which, again, is beyond the scope of this project— here I merely point to Dante’s claims in the sixth stanza that are not sufficiently supported by reasoning worked out in the poem alone. Such claims include the following:

1. that nobility is something that must be gifted to an individual’s soul;
2. that only God gifts nobility;
3. that God gifts nobility only to a soul that is “in sua persona / perfettamente” (in its body perfectly [117-118]); and
4. that nobility is ’l seme di felicità.

As is made clear in the logic I present, the first three claims, although not fully supported in the poem, are in turn used to support Dante’s definition of nobility, which incorporates the fourth claim. Although Dante seemingly intends for his (supported) intermediate conclusion that virtue, and, more precisely, the abito eligente lo qual dimora in mezzo solamente, derives from nobility at least to partially account for the four claims above, there is nothing expressed or reasonably implied in the poem alone that works out this reasoning. Consequently, most notably, the inclusion of God in characterizing nobility is abrupt and is not supported by rigorous logical argumentation. In the work that follows, I write “(Missing Premises)” prior to the first instance of a statement in logic that incorporates one or more of the four claims above.

Unlike the sixth stanza, which does indeed present arguments (albeit with missing premises), the seventh stanza is particularly lacking in logical underpinnings. This is because it is not comprised of a complete chain of inferences, or even chain of inferences with missing premises, but instead of a list of declarations. As such, when Dante lists i segni che ’l gentil uom tene, it is not clear why these are necessarily i segni. Given the overall lack of logical form, I do not render any of the verses of the seventh stanza into logic.

A Brief Note about First-Order Logic and Modal Logic

First-order logic, also known as first-order predicate calculus, can be understood as a system of symbolized reasoning in which sentences or statements are represented according to their “objects,” “predicates,” and stated or implied “quantifiers.” An “object” can be considered analogous in grammar to a singular term (a proper name or a definite description). It may refer to a specific person, a specific physical or abstract object, or a specific concept, in which case an individual “constant” is used (symbolized by a non-italicized lower case letter), or may also refer to something less specific, in which case a “variable” is used (symbolized by an italicized lower case letter). A “predicate” can be considered analogous in grammar to an adjective, a common noun, or a verb phrase template. It may refer to a property of an object or a relationship among objects, and is symbolized by a non-italicized upper case letter. Predicates can be unary (one-place), i.e. K(x) to mean “x is kind,” binary (two-place), i.e. L(x, y) to mean “x is larger than y,” ternary (three-place), i.e. C(x, y, z) to mean “x is a child of y and z,” etc. In first-order logic, statements can be further structured by use of a universal quantifier (∀) or an existential quantifier (∃) in order to refer to a property of a group of objects.

Let us consider an example. We can let the predicate R represent “is a rectangle,” the predicate S represent “is a shape,” and the predicate F represent “has four sides.” Accordingly, we can express statements such as the following in first-order logic using the variable x as the object:

∀x [R(x) → F(x)]

∃x [S(x) ∧ F(x)]

The first statement translates to “For all x, if x is a rectangle, then x has four sides.” The second translates to “There exists an x such that x is a shape and x has four sides.”

Put simply, modal logic can be understood as an extension of first-order logic that includes additional operators to capture notions of possibility and necessity. Because Dante theorizes about what is or is not possible on various occasions in the poem, modal logic is, at times, appropriate. The symbol ◊ is used to capture notions of possibility. For example, using modal logic, we can express statements such as the following:

∀x [S(x) → ◊F(x)]

This translates to “For all x, if x is a shape, then it is possible that x has four sides.”

Thus, modal logic allows for further specificity in representing Dante’s logical argumentation.

The Interlinear Translation

You may view this work in two formats. Below is a version “With Interlinear Translation.” This version allows the reader to follow the logic and line of reasoning with the assistance of a direct line-by-line translation into English present under each line of logic. I note that my decision to present the interlinear translation in English, as opposed to in Italian, was merely a matter of convenience for me as a native English speaker. Sometimes angular parentheses are used in the interlinear translation for the sake of clarity. In all interlinear translations, the use of ‘or’ is inclusive unless explicitly stated that it is exclusive, in accordance with the relevant representations in logic. You may also view a version “Without Interlinear Translation” (see HERE) in which the lines of logic are not accompanied by an interlinear translation. Viewing the work in this format allows for a cleaner and more visually appealing representation of the logic and line of reasoning.

The interlinear translations are offered only with a view to giving readers without a background in logic an entry point for understanding how to “read” the logic presented. The core “bilingualism” of my work is the bilingualism that I make visible by translating Dante’s Italian into the language of first-order logic including modal operators. The interlinear translation, which only happens to be in English, is simply an after-the-fact translation, indeed one that was formulated after I had already carried out the primary translation from Italian to logic.

 A Few Notes on Legibility

For a key containing the meaning of all predicates and constants (in Italian and English), principles (numbered as they are used in the poem), and equivalences (in Italian), please see HERE.

Where appropriate, I provide logically equivalent contrapositives to Dante’s conditional statements. A conditional statement “If A, then B” is composed of antecedent A and consequent B. The contrapositive has its antecedent and consequent inverted and flipped. That is, the contrapositive of “If A, then B” is “If not B, then not A.” The two are logically equivalent.

I have attempted to make this work as clear and accessible as possible with regard to how I define my predicates and constants. Although I do present portions of the poem that are rendered into logic in English and include English translations of my predicates and constants, it is indeed the Italian text of the poem that was my primary source for determining how to formulate the logic. Thus, given the large number of predicates and the appearance of multiple words in the poem (in Italian) that begin with the same letter, I have chosen, as much as possible, to define predicates alphabetically as they first appear or are implied in the poem in the original Italian, not in the English translation. After the 26 letters of the English alphabet are used, I then turn to lowercase Greek letters to define predicates. Where appropriate, constants are defined according to the letter of the predicate with which they are associated, i.e. C(x) means ‘x is noble’ and so the constant c means ‘nobility’. The only exceptions are the letters used for the constants vertù, l’abito eligente lo qual dimora in mezzo solamente, tempo, ben, mal, Dio, and ’l seme di felicità (v, v′, t, e, m, g, and s, respectively), since I do not use any predicates with which these constants are associated.

Occasionally I use a smaller font size to accommodate longer lines of logic. As much as possible, the same font size is used for lines of logic within the same proof, or at least the same section of a proof (as in the case of Proof #3).

For the text of Le dolci rime, I cite from the edition of D. De Robertis, Rime, (Florence, 2005). English translations are those of K. Foster and P. Boyde, Dante’s Lyric Poetry, 2 vols. (Oxford, 1967). Occasionally I translate words or selections myself to render the formalizations into logic more explicit.

The work is divided into sections by stanza. In each section, the stanza of the poem is provided in Italian. Portions that are represented in logic are highlighted in yellow so that a contrast can be seen with parts of the poem that are not represented in logic. After the stanza in Italian is cited, where necessary, I summarize a few important notes about the logic with a view to helping the reader understand why it is formulated as such. Sections in Italian that are represented in logic are then repeated in italics, followed by an English translation. After the English translation, I present the logic, below which appears an (optional) interlinear translation into English. The fifth and sixth stanzas in Italian are cited one after the other since the logical argument of Proof #5 spans both stanzas.

Stanza 1

Le dolci rime d’amor ch’io solea
cercar ne’ miei pensieri
convien ch’io lasci; non perch’io non speri
ad esse ritornare,
ma perché gli atti disdegnosi e feri                               5
che nella donna mia
sono appariti, m’han chiusa la via
dell’usato parlare.
E poi che tempo mi par d’aspettare,
diporrò giù lo mio soave stile                                      10
ch’i’ ho tenuto nel trattar d’amore,
e dirò del valore
per lo qual veramente omo è gentile
con rima aspr’e sottile,
riprovando il giudicio falso e vile                                15
di que’ che voglion che di gentilezza
sia principio ricchezza.
E cominciando, chiamo quel signore
ch’a la mia donna negli occhi dimora,
per ch’ella di sé stessa s’innamora.                             20

A Few Notes on the Logic

For Dante, only man can be gentile. That is, ∀x [C(x) → B(x)], which translates to “For all x, if x is noble, then x is a man.” Accordingly, in the logic presented, I do not specify that x is a man when x has the valore (12)— i.e. that B(x) when A(x). I also do not specify that x is a man in each case in which x is gentile— i.e. that B(x) when C(x).

dirò del valore
per lo qual veramente omo è gentile (12-13).

I will speak concerning the quality by which man is truly noble.

∀x [A(x) ↔ C(x)]

For all x, x has the quality by which x is truly noble if and only if x is noble.

riprovando il giudizio falso e vile
di que’ che voglion che di gentilezza
sia principio ricchezza (15-17).

demonstrating the judgment of those who hold that nobility depends on wealth to be false and to be base.

¬∀x [C(x) → H(x)] ∧ F(d) ≡ ¬G(h, c) ∧ F(d)

It is not the case that <for all x, if x is noble, then x is rich>, and the judgment that nobility depends on wealth is base.

is logically equivalent to

It is not the case that wealth is a necessary condition for nobility, and the judgment that nobility depends on wealth is base.

Stanza 2

Tale imperò che gentilezza volse,
secondo il suo parere,
che fosse antica possession d’avere
con reggimenti belli;
ed altri fu di più lieve savere                                        25
che tal detto rivolse
e l’ultima particula ne tolse
che non l’avea fors’elli.
Di rieto da costui van tutti quelli
che fan gentile per ischiatta altrui                              30
che lungamente in gran ricchezza è stata;
ed è tanto durata
la così falsa oppinion tra noi,
che l’uom chiama colui
omo gentil, che può dicere: ‘I’ fui                                35
nepote o figlio di cotal valente’,
bench’e’ sia da nïente.
Ma vilissimo sembra, a chi ’l ver guata,
cui è scorto il cammino e poscia l’erra,
e tocc’a tal, ch’è morto e va per terra.                        40

A Few Notes on the Logic

I have no notes on the logic of the second stanza.

Tale imperò che gentilezza volse,
secondo il suo parere,
che fosse antica possession d’avere
con reggimenti belli (21-24).

There was a ruler of the Empire who maintained that in his view nobility consisted in long-standing possession of wealth together with pleasing manners.

∀x [C(x) ↔ [H′(x) ∧ J(x)]] ≡ ∀x [¬C(x) ↔ ¬[H′(x) ∧ J(x)]] ≡ ∀x [¬C(x) ↔ [¬H′(x) ∨ ¬J(x)]]

For all x, x is noble if and only if <x has been rich for a long time and x has pleasing manners>.

is logically equivalent to

For all x, x is not noble if and only if it is not the case that <x has been rich for a long time and x has pleasing manners>.

is logically equivalent to

For all x, x is not noble if and only if <x has not been rich for a long time or x does not have pleasing manners>.

ed altri fu di più lieve savere
che tal detto rivolse
e l’ultima particula ne tolse (25-27).

and someone else, of shallower wit, reconsidering this dictum, dispensed with the last detail.

∀x [C(x) ↔ H′(x)] ≡ ∀x [¬C(x) ↔ ¬H′(x)]

For all x, x is noble if and only if x has been rich for a long time.

is logically equivalent to

For all x, x is not noble if and only if x has not been rich for a long time.

Di rieto da costui van tutti quelli
che fan gentile per ischiatta altrui
che lungamente in gran ricchezza è stata (29-31).

In his wake follow all those who count a man as noble for belonging to a family which has been very rich for a long time.

∀x [H′(x) → C(x)] ≡ ∀x [¬C(x) → ¬H′(x)]

For all x, if x has been rich for a long time, then x is noble.

is logically equivalent to

For all x, if x is not noble, then x has not been rich for a long time.

Stanza 3

Chi diffinisce: ‘Omo è legno animato’,
prima dice non vero,
e dopo ’l falso parla non intero,
ma più forse non vede.
Similemente fu chi tenne impero                                45
in diffinire errato,
ché prima puose il falso e d’altro lato
con difetto procede;
ché le divizie, sì come si crede,
non posson gentilezza dar né tôrre,                            50
però che vili son da lor natura,
poi chi pinge figura,
se non può esser lei, no·lla può porre,
né·lla diritta torre
fa piegar rivo che da lungi corre.                                  55
Che siano vili appare ed imperfette,
ché, quantunque collette,
non posson quïetar, ma dan più cura;
onde l’animo ch’è dritto e verace
per lor discorrimento non si sface.                             60

A Few Notes on the Logic

In Proof #1:

I use the statement “Che siano vili appare ed imperfette” (56) as the basis for formulating in logic ¬O(h) → F(h), which translates to “If wealth is not perfect, then wealth is base.” (Verses 57-58 actually supply the reason that wealth is not perfect.) The statement ¬O(h) → F(h) is formulated as such based on Foster and Boyde’s comment that ed (56) is to be interpreted as “because,” which I simply render in conditional form as “if.” See Foster and Boyd’s note on verses 56-58, Dante’s Lyric Poetry, 217.

---

In combination with Dante’s statement that wealth cannot take away nobility, I use the statement “né·lla diritta torre / fa piegar rivo che da lungi corre” (54-55) as the basis for formulating the following in logic:

N(c, h) → ∀x [[C(x) ∧ K(x, h)] → C′(x)]

This translates to “If wealth flows at a distance from nobility, then <for all x, if x is noble and x acquires wealth that x did not have before, then x is still noble>.” This formulation is subtly implied by the natural language of the poem with Dante’s use of the example of the river and tower. For support of this formulation, see Barbi and Pernicone’s note on verses 52-55.

---

In Proof #2:

Dante shows that the definition of nobility as antica possession d’avere con reggimenti belli is not correct because it is not a complete definition given its reliance on reggimenti belli. He applies an implied principle (2):

∀x, z [[I(x, z) ∧ ∃u, w ∀y[I'(y, z) → [I_P(u, x) ∧ I_P(u, y) ∧ ¬I_P(w, x) ∧ I_P(w, y)]]]→ ¬I′′(x, z)]

This translates to “For all x and z, if <x is a proposed definition of z and there exist a u and a w such that, for all y, if y is a correct definition of z, then u is a part of x and u is a part of y and w is not a part of x and w is a part of y>, then x is not a complete definition of z.” Interestingly, Dante’s argument can be interpreted in three ways: (1) he is speaking about a part of a definition as incomplete (i.e. in the case that the definition has multiple parts), (2) he is speaking about the whole definition as incomplete (i.e. in the case that the definition consists of only one part, namely, the whole), or (3) either (1) or (2), where ‘or’ is understood to be inclusive. Under any of these interpretations, according to my formalization, i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is not a complete definition. For this reason, I do not indicate in Principle (2) that ¬(u ≈ x), i.e. that a part cannot be equivalent to the whole. Since Dante demonstrates that i_1p1 (the part of i₁ according to which nobility is old wealth) is false, i_1p2 (the part of i₁ according to which pleasing manners are necessary for nobility) effectively becomes the new “whole.” Nonetheless, this once-part-now-whole is still incomplete in defining or characterizing nobility and, accordingly, i₁, by application of Principle (2), is not a complete definition.

Chi diffinisce: ‘Omo è legno animato’,
prima dice non vero,
e dopo ’l falso parla non intero,
ma più forse non vede.
Similemente fu chi tenne impero
in diffinire errato,
ché prima puose il falso e d’altro lato
con difetto procede;
ché le divizie, sì come si crede,
non posson gentilezza dar né tôrre,
però che vili son da lor natura,
poi chi pinge figura,
se non può esser lei, no·lla può porre,
né·lla diritta torre
fa piegar rivo che da lungi corre.
Che siano vili appare ed imperfette,
ché, quantunque collette,
non posson quïetar, ma dan più cura (41-58).

If anyone defines: ‘Man is an animate tree’: first what he says isn’t true, and then, after the falsehood, he is leaving the definition incomplete—but perhaps he can see no further. He who ruled the empire was similarly mistaken in his definition: for, first he has stated a falsehood and then, this apart, what he goes on to say is incomplete. For—contrary to what is generally believed—riches cannot either confer or take away nobility, being themselves base by nature: thus he who paints a form, if he cannot ‘be’ it, cannot set it down; nor is an upright tower made to lean by a stream that flows at a distance. That riches are base is clear because they are imperfect, since in whatever quantity they are amassed, they bring no peace but only increase anxiety.

Proof #1

¬◊P(h) ∧ Q(h)

It is not possible that wealth brings peace, and wealth increases anxiety.

[¬◊P(h) ∧ Q(h)] → ¬O(h)

If <it is not possible that wealth brings peace, and wealth increases anxiety>, then wealth is not perfect.

________________________

¬O(h)

Therefore, wealth is not perfect.

¬O(h) → F(h)

If wealth is not perfect, then wealth is base.

______________

F(h)

Therefore, wealth is base.

F(h) → ∀x [[B(x) ∧ ¬C(x) ∧ K(x, h)] → ¬C′′(x)]

If wealth is base, then <for all x, if <x is a man and x is not noble and x acquires wealth that x did not have before>, then x does not become noble>.

∀x [[B(x) ∧ ¬C(x) ∧ K(x, h)] →  ¬C′′(x)] → ¬◊L(h, c)

If <for all x, if <x is a man and x is not noble and x acquires wealth that x did not have before>, then x does not become noble>, then it is not possible that wealth confers nobility.

_________________________________________________

¬◊L(h, c)

Therefore, it is not possible that wealth confers nobility.

N(c, h)

Wealth flows at a distance from nobility.

N(c, h) → ∀x [[C(x) ∧ K(x, h)] → C′(x)]

If wealth flows at a distance from nobility, then <for all x, if <x is noble and x acquires wealth that x did not have before>, then x is still noble>.

∀x [[C(x) ∧ K(x, h)] → C′(x)] → ¬◊M(h, c)

If <for all x, if <x is noble and x acquires wealth that x did not have before>, then x is still noble>, then it is not possible that wealth takes away nobility.

_______________________________________

¬◊M(h, c)

Therefore, it is not possible that wealth takes away nobility.

[¬◊L(h, c) ∧ ¬◊M(h, c)] → ¬G(h, c)

If <it is not possible that wealth confers nobility and it is not possible that wealth takes away nobility>, then wealth is not a necessary condition for nobility.

________________________________

¬G(h, c)

Therefore, wealth is not a necessary condition for nobility.

¬G(h, c) → ¬E(i_1p1)

If wealth is not a necessary condition for nobility, then i_1p1 (the part of i₁ according to which nobility is old wealth) is not true.

___________________

¬E(i_1p1)

Therefore, i_1p1 (the part of i₁ according to which nobility is old wealth) is not true.

∀x, y [[I(x, y) ∧ ∃z [I_P(z, x) ∧ ¬E(z)]] → ¬I′(x, y)]                    (1)

For all x and y, <if <x is a proposed definition of y and there exists a z such that z is a part of definition x and z is not true>, then x is not a correct definition of y>.

I(i₁, c) ∧ I_P(i_1p1, i₁) ∧ ¬E(i_1p1)

i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is a proposed definition of nobility and i_1p1 (the part of i₁ according to which nobility is old wealth) is a part of i₁ and i_1p1 (the part of i₁ according to which nobility is old wealth) is not true.

____________________________________________

¬I′(i₁, c)

Therefore, i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is not a correct definition of nobility.

Q.E.D. (#1)

Proof #2

I(i₁, c)

i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is a proposed definition of nobility.

∀x, z [[I(x, z) ∧ ∃u, w ∀y[I'(y, z) → [I_P(u, x) ∧ I_P(u, y) ∧ ¬I_P(w, x) ∧ I_P(w, y)]]]→ ¬I′′(x, z)]                    (2)

For all x and z, if <x is a proposed definition of z and there exist a u and a w such that, for all y, if y is a correct definition of z, then u is a part of x and u is a part of y and w is not a part of x and w is a part of y>, then x is not a complete definition of z.

__________________________________________________________________________

[I(i₁, c) ∧ ∃u, w ∀y[I′(y, c) → [I_P(u, i₁) ∧ I_P(u, y) ∧ ¬I_P(w, i₁) ∧ I_P(w, y)]]] → ¬I′′(i₁, c)

Therefore, if <i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is a proposed definition of nobility and there exist a u and a w such that, for all y, <if y is a correct definition of nobility, then <u is a part of i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) and u is a part of y and w is not a part of i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) and w is a part of y>>>, then i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is not a complete definition of nobility.

∃w ∀y [I′(y, c) → [I_P(i_1p2, i₁) ∧ I_P(i_1p2, y) ∧ ¬I_P(w, i₁) ∧ I_P(w, y)]]

There exists a w such that, for all y, <if y is a correct definition of nobility, then <i_1p2  (the part of i₁ according to which pleasing manners are necessary for nobility) is a part of i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) and i_1p2 (the part of i₁ according to which pleasing manners are necessary for nobility) is a part of y and w is not a part of i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) and w is a part of y>>.

__________________________________________________________________________

∃u, w ∀y [I′(y, c) → [I_P(u, i₁) ∧ I_P(u, y) ∧ ¬I_P(w, i₁) ∧ I_P(w, y)]]

Therefore, there exists a u and a w such that, for all y, <if y is a correct definition of nobility, then <u is a part of i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) and u is a part of y and w is not a part of i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) and w is a part of y>>.

______________________________________________________________

I(i₁, c) ∧ ∃u, w ∀y[I′(y, c) → [I_P(u, i₁) ∧ I_P(u, y) ∧ ¬I_P(w, i₁) ∧ I_P(w, y)]]

Therefore, i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is a proposed definition of nobility and there exists a u and a w such that, for all y, <if y is a correct definition of nobility, then <u is a part of i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) and u is a part of y and w is not a part of i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) and w is a part of y>>.

______________________________________________________________

¬I′′(i₁, c)

Therefore, i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is not a complete definition of nobility.

∀x, y [[I(x, y) ∧ ¬I′′(x, y)] → ¬I′(x, y)]                    (3)

For all x and y, <if <x is a proposed definition of y and x is not a complete definition of y>, then x is not a correct definition of y>.

I(i₁, c) ∧ ¬I′′(i₁, c)

i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is a proposed definition of nobility and i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is not a complete definition of nobility.

__________________________________

¬I′(i₁, c)

Therefore, i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is not a correct definition of nobility.

Q.E.D. (#2)

Stanza 4

Né voglion che vil uom gentil divegna
né di vil padre scenda
nazion che per gentil già mai s’intenda:
quest’è da lor confesso.
Onde la lor ragion par che s’offenda                           65
in tanto quanto assegna
che tempo a gentilezza si convegna
diffinendo con esso.
Ancor segue di ciò che ’nnanzi ho messo
che siàn tutti gentili o ver villani                                70
o che non fosse ad uom cominciamento;
ma ciò io non consento,
néd eglino altressì, se son cristiani.
Per ch’a ’ntelletti sani
è manifesto i lor diri esser vani,                                   75
ed io così per falsi li ripruovo
e da lor mi rimuovo,
e dicer voglio omai, sì com’io sento,
che cosa è gentilezza, e da che vene,
e dirò i segni che ’l gentil uom tene.                           80

A Few Notes on the Logic

In Proof #3 and Proof #4:

The fourth stanza consists of two distinct arguments, both in the form of a reductio ad absurdum. Accordingly, I present two distinct proofs. That there are two distinct arguments is consistent with Foster and Boyd’s notes on verses 61-80 and 69-73. Each of the two arguments assumes the truth of the judgment of Dante’s opponents—“Né voglion che vil uom gentil divegna / né di vil padre scenda / nazion che per gentil già mai s’intenda: / quest’è da lor confesso” (Now my opponents maintain that a base man can never himself become noble, and that the offspring of a base father can never be reckoned noble: this is what they say [61-64])— and uses it as an initial premise in order to reach a contradiction and, hence, reject the judgment in question. Given the use of the judgment in each of the two arguments, verses 61-64 appear in both the third and fourth proofs.

---

At various points in the fourth stanza, instances of different forms of vile are to be understood as not gentile. Specifically, my interpretations of vil uom (61) as a man who is not gentile, vil padre (62) as a father who is not gentile, and villani (70) as men who are not gentili are reasonably implied by Dante’s language in the poem. For support of these interpretations, see, for example, Barbi and Pernicone’s comment regarding verses 61-64. Given these interpretations, in Proof #3 and Proof #4, I represent the statement “Né voglion che vil uom gentil divegna / né di vil padre scenda / nazion che per gentil già mai s’intenda” (61-63) as follows:

∀x [[B(x) ∧ ¬C(x)] → ¬◊C′′(x)] ∧ ∀x [[B(x) ∧ R(x, y) ∧ ¬C(y)] → ¬◊C(x)]

This translates to “For all x, if <x is a man and x is not noble>, then it is not possible that x becomes noble, and for all x, if <x is a man and x descends from father y and y is not noble>, then it is not possible that x is noble.” In addition, in Proof #4, I represent the individual statement “siàn tutti gentili o ver villani” (71) as follows:

∀x [B(x) → C(x)] ⊻ ∀x [B(x) → ¬C(x)]

This translates to “Either for all x, if x is a man, then x is noble or (exclusive) for all x, if x is a man, then x is not noble.”

---

In Proof #4:

I represent the statement “che non fosse ad uom cominciamento” (71) as follows:

∃x, y [B(x) ∧ B(y) ∧ S(x, b′) ∧ S(y, b′) ∧ ¬(x ≈ y)]

This translates to “There exist an x and a y such that x is a man and y is a man and x was a beginning to mankind and y was a beginning to mankind and it is not the case that x is equivalent to y.” For support of this formulation, see Barbi and Pernicone’s note on verses 69-73 and note on verse 71.

---

On two occasions, logic meant to be on one line does not fit on one line (that is, without compromising ease of legibility). On such occasions, I break up the logic into two lines, with the main logical connective (i.e. → or ∧) at the end of the first line.

Né voglion che vil uom gentil divegna
né di vil padre scenda
nazion che per gentil già mai s’intenda:
quest’è da lor confesso.
Onde la lor ragion par che s’offenda
in tanto quanto assegna
che tempo a gentilezza si convegna
diffinendo con esso (61-68).

Now my opponents maintain that a base man can never himself become noble, and that the offspring of a base father can never be reckoned noble: this is what they say. Clearly then their position is self-contradictory, inasmuch as they make time a factor in nobility, including it in their definition.

Proof #3

∀x [[B(x) ∧ ¬C(x)] → ¬◊C′′(x)] ∧ ∀x [[B(x) ∧ R(x, y) ∧ ¬C(y)] → ¬◊C(x)]

For all x, if <x is a man and x is not noble>, then it is not possible that x becomes noble, and for all x, if <x is a man and x descends from father y and y is not noble>, then it is not possible that x is noble.

[∀x [[B(x) ∧ ¬C(x)] → ¬◊C′′(x)] ∧ ∀x [[(B(x) ∧ R(x, y) ∧ ¬C(y)] → ¬◊C(x)]] → ¬G(t, c)

If <for all x, <if x is a man and x is not noble>, then it is not possible that x becomes noble, and for all x, if <x is a man and x descends from father y and y is not noble>, then it is not possible that x is noble>, then time is not a necessary condition for nobility.

_______________________________________________________________________________

¬G(t, c)

Therefore, time is not a necessary condition for nobility.

 

I′(i₁) ∨ I′(i₂)

i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is a correct definition of nobility or i₂ (the definition of nobility that says that nobility is old wealth) is a correct definition of nobility.

[I′(i₁) ∨ I′(i₂)] → ∀x [C(x) → H′(x)]

If <i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is a correct definition of nobility or i₂ (the definition of nobility that says that nobility is old wealth) is a correct definition of nobility>, then <for all x, if x is noble, then x has been rich for a long time>.

∀x [C(x) → H′(x)] → G(h′, c)

If <for all x, if x is noble, then x has been rich for a long time>, then old wealth is a necessary condition for nobility.

__________________________________

G(h′, c)

Therefore, old wealth is a necessary condition for nobility.

G(t, h′)

Time is a necessary condition for old wealth.

________________

G(t, h′) ∧ G(h′, c)

Therefore, time is a necessary condition for old wealth and old wealth is a necessary condition for nobility.

∀x, y, z [[G(x, y) ∧ G(y, z)] → G(x, z)]                    (4)

For all x, y, and z, if <x is a necessary condition for y and y is a necessary condition for z>, then x is a necessary condition for z.

__________________________________

G(t, c)

Therefore, time is a necessary condition for nobility.

 

____________

¬G(t, c) ∧ G(t, c)

Therefore, time is not a necessary condition for nobility and time is a necessary condition for nobility (contradiction).

____________________________________________________________________

[[∀x[[B(x) ∧ ¬C(x)] → ¬◊C′′(x)] ∧ ∀x[[(B(x) ∧ R(x, y) ∧ ¬C(y)] → ¬◊C(x)]] ∧ [I′(i₁) ∨ I′(i₂)]] → ⊥

Therefore, <for all x, if x is a man and x is not noble, then it is not possible that x becomes noble, and for all x, if x is a man and x descends from father y and y is not noble, then it is not possible that x is noble> and <i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is a correct definition of nobility or i₂ (the definition of nobility that says that nobility is old wealth) is a correct definition of nobility> implies a contradiction.

______________________________________________________________________

¬[[∀x[[B(x) ∧ ¬C(x)] → ¬◊C′′(x)] ∧ ∀x[[(B(x) ∧ R(x, y) ∧ ¬C(y)] → ¬◊C(x)]] ∧ [I′(i₁) ∨ I′(i₂)]]

Therefore, it is not the case that both <for all x, if x is a man and x is not noble, then it is not possible that x becomes noble, and for all x, if x is a man and x descends from father y and y is not noble, then it is not possible that x is noble> and <i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is a correct definition of nobility or i₂ (the definition of nobility that says that nobility is old wealth) is a correct definition of nobility>.

¬[I′(i₁) ∨ I′(i₂)] ≡ [¬I′(i₁) ∧¬I′(i₂)]

It is not the case that <i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is a correct definition of nobility or i₂ (the definition of nobility that says that nobility is old wealth) is a correct definition of nobility>.

is logically equivalent to

i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is not a correct definition of nobility and i₂ (the definition of nobility that says that nobility is old wealth) is not a correct definition of nobility.

__________________________________________________________________

¬[∀x[[B(x) ∧ ¬C(x)] → ¬◊C′′(x)] ∧ ∀x[[(B(x) ∧ R(x, y) ∧ ¬C(y)] → ¬◊C(x)]] ∨ [¬I′(i₁) ∧ ¬I′(i₂)]

Therefore, it is not the case that <for all x, if x is a man and x is not noble, then it is not possible that x becomes noble, and for all x, if x is a man and x descends from father y and y is not noble, then it is not possible that x is noble> or <i₁ (the definition of nobility that says that nobility is old wealth combined with pleasing manners) is not a correct definition of nobility and i₂ (the definition of nobility that says that nobility is old wealth) is not a correct definition of nobility>.

Q.E.D. (#3)

Né voglion che vil uom gentil divegna
né di vil padre scenda
nazion che per gentil già mai s’intenda:
quest’è da lor confesso…
Ancor segue di ciò che ’nnanzi ho messo
che siàn tutti gentili o ver villani
o che non fosse ad uom cominciamento;
ma ciò io non consento,
néd eglino altressì, se son cristiani.
Per ch’a ’ntelletti sani
è manifesto i lor diri esser vani,
ed io così per falsi li ripruovo (61-64, 69-76).

Now my opponents maintain that a base man can never himself become noble, and that the offspring of a base father can never be reckoned noble: this is what they say… In addition, it follows from the foregoing that either we are all noble or all base, or else that mankind didn’t have one beginning; but this alternative I do not admit, and neither do they if they are Christians. Consequently it is clear to every healthy mind that their statements are groundless; and so, I refute them as false.

Proof #4

∀x [[B(x) ∧ ¬C(x)] → ¬◊C′′(x)] ∧ ∀x [[B(x) ∧ R(x, y) ∧ ¬C(y)] → ¬◊C(x)]

For all x, if <x is a man and x is not noble>, then it is not possible that x becomes noble, and for all x, if <x is a man and x descends from father y and y is not noble>, then it is not possible that x is noble.

[∀x [[B(x) ∧ ¬C(x)] → ¬◊C′′(x)] ∧ ∀x [[B(x) ∧ R(x, y) ∧ ¬C(y)] → ¬◊C(x)]] →

[[∀x [B(x) → C(x)] ⊻ ∀x [B(x) → ¬C(x)]] ⊻ [∃x, y [B(x) ∧ B(y) ∧ S(x, b′) ∧ S(y, b′) ∧ ¬(x ≈ y)]]]

If it is the case that <for all x, if <x is a man and x is not noble>, then it is not possible that x becomes noble, and for all x, if <x is a man and x descends from father y and y is not noble>, then it is not possible that x is noble>, then <either for all x, if x is a man, then x is noble or (exclusive) for all x, if x is a man, then x is not noble, or (exclusive) else there exist an x and a y such that x is a man and y is a man and x was a beginning to mankind and y was a beginning to mankind and it is not the case that x is equivalent to y>.

___________________________________________________________________

[∀x [B(x) → C(x)] ⊻ ∀x [B(x) → ¬C(x)]] ⊻ [∃x, y [B(x) ∧ B(y) ∧ S(x, b′) ∧ S(y, b′) ∧ ¬(x ≈ y)]]

Therefore, either for all x, if x is a man, then x is noble or (exclusive) for all x, if x is a man, then x is not noble, or (exclusive) else there exist an x and a y such that x is a man and y is a man and x was a beginning to mankind and y was a beginning to mankind and it is not the case that x is equivalent to y.

∃x [B(x) ∧ C(x)] ∧ ∃x [B(x) ∧ ¬C(x)]

There exists an x such that x is a man and x is noble, and there exists an x such that x is a man and x is not noble.

__________________________________________________________________

¬[∀x [B(x) → C(x)] ⊻ ∀x [B(x) → ¬C(x)]]

Therefore, it is not the case that <either for all x, if x is a man, then x is noble or (exclusive) for all x, if x is a man, then x is not noble>.

∃!x [B(x) ∧ S(x, b′)]

There exists exactly one x such that x is a man and x was a beginning to mankind.

____________________________________

¬[∃x, y [B(x) ∧ B(y) ∧ S(x, b′) ∧ S(y, b′) ∧ ¬(x ≈ y)]]

Therefore, it is not the case that <there exist an x and a y such that x is a man and y is a man and x was a beginning to mankind and y was a beginning to mankind and it is not the case that x is equivalent to y>.

__________________________________________________________________

¬[∀x [B(x) → C(x)] ⊻ ∀x [B(x) → ¬C(x)]] ∧ ¬[∃x, y [B(x) ∧ B(y) ∧ S(x, b′) ∧ S(y, b′) ∧ ¬(x ≈ y)]]

Therefore, it is not the case that <either for all x, if x is a man, then x is noble or (exclusive) for all x, if x is a man, then x is not noble> and it is not the case that <there exist an x and a y such that x is a man and y is a man and x was a beginning to mankind and y was a beginning to mankind and it is not the case that x is equivalent to y>.

___________________________________________________________________

¬[[∀x [B(x) → C(x)] ⊻ ∀x [B(x) → ¬C(x)]] ⊻ [∃x, y [B(x) ∧ B(y) ∧ S(x, b′) ∧ S(y, b′) ∧ ¬(x ≈ y)]]]

Therefore, it is not the case that <either for all x, if x is a man, then x is noble or (exclusive) for all x, if x is a man, then x is not noble, or (exclusive) else there exist an x and a y such that x is a man and y is a man and x was a beginning to mankind and y was a beginning to mankind and it is not the case that x is equivalent to y>.

___________________________________________________________________

[[∀x [B(x) → C(x)] ⊻ ∀x [B(x) → ¬C(x)]] ⊻ [∃x, y [B(x) ∧ B(y) ∧ S(x, b′) ∧ S(y, b′) ∧ ¬(x ≈ y)]]] ∧

¬[[∀x [B(x) → C(x)] ⊻ ∀x [B(x) → ¬C(x)]] ⊻ [∃x, y [B(x) ∧ B(y) ∧ S(x, b′) ∧ S(y, b′) ∧ ¬(x ≈ y)]]]

Therefore, <either for all x, if x is a man, then x is noble or (exclusive) for all x, if x is a man, then x is not noble, or (exclusive) else there exist an x and a y such that x is a man and y is a man and x was a beginning to mankind and y was a beginning to mankind and it is not the case that x is equivalent to y> and it is not the case that <either for all x, if x is a man, then x is noble or (exclusive) for all x, if x is a man, then x is not noble, or (exclusive) else there exist an x and a y such that x is a man and y is a man and x was a beginning to mankind and y was a beginning to mankind and it is not the case that x is equivalent to y> (contradiction).

___________________________________________________________________

[∀x [[B(x) ∧ ¬C(x)] → ¬◊C′′(x)] ∧ ∀x [[(B(x) ∧ R(x, y) ∧ ¬C(y)] → ¬◊C(x)]] → ⊥

Therefore, <for all x, if x is a man and x is not noble, then it is not possible that x becomes noble, and for all x, if x is a man and x descends from father y and y is not noble, then it is not possible that x is noble> implies a contradiction.

_________________________________________________________

¬[∀x [[B(x) ∧ ¬C(x)] → ¬◊C′′(x)] ∧ ∀x [[(B(x) ∧ R(x, y) ∧ ¬C(y)] → ¬◊C(x)]]

Therefore, it is not the case that <for all x, if x is a man and x is not noble, then it is not possible that x becomes noble, and for all x, if x is a man and x descends from father y and y is not noble, then it is not possible that x is noble>.

Q.E.D. (#4)

Stanzas 5 and 6

Dico ch’ogni vertù principalmente
vien da una radice,
vertute, dico, che fa l’uom felice
in sua operazione.
Quest’è secondo che l’Etica dice,                                85
un abito eligente
lo qual dimora in mezzo solamente,
e tai parole pone.
Dico che nobiltate in sua ragione
importa sempre ben del suo subietto                        90
come viltate importa sempre male;
e vertute cotale
dà sempre altrui di sé buono intelletto;
per che in medesmo detto
convegnono amendue, ch’èn d’uno effetto.              95
Dunque convien che d’altra vegna l’una
o d’un terzo ciascuna;
ma se l’una val ciò che l’altra vale
ed ancor più, da lei verrà più tosto.
Ciò ch’i’ ho detto qui, sia per supposto.                  100

È gentilezza dovunqu’è vertute,
ma non vertute ov’ella,
sì com’è ’l cielo dovunqu’è la stella,
ma ciò non e converso.
E noi in donna ed in età novella                                105
vedem questa salute
in quanto vergognose son tenute,
ch’è da vertù diverso.
Dunque verrà come dal nero il perso
ciascheduna vertute da costei,                                   110
o vero il gener lor, ch’io misi avanti;
però nessun si vanti
dicendo ‘Per ischiatta i’ son con lei’,
ched e’ son quasi dei
que’ c’han tal grazia fuor di tutti rei;                         115
ché solo Iddio all’anima la dona
che vede in sua persona
perfettamente star, sì ch’ad alquanti
ch’è ’l seme di felicità s’accosta
messo da Dio nell’anima ben posta.                         120

A Few Notes on the Logic

In Proof #5:

As explained above, the fifth and sixth stanzas in Italian are cited one after the other since the logical argument of Proof #5 spans both stanzas. Textual citations in these two stanzas are kept in the order in which they appear in the poem. However, for clarity of logical argumentation, the corresponding logical translations of such citations are put in a different order. This choice does not affect the validity of the argument. Thus, principles (6) and (5) are numbered as such to retain the fact that principle (6) is used at a later verse in the poem than is principle (5).

---

I represent the statements “per che in medesmo detto / convegnono amendue, ch’èn d’uno effetto. / Dunque convien che d’altra vegna l’una / o d’un terzo ciascuna” (94-97) as follows:

V(c, v) → [T′(c, v) ⊻ T′(v, c) ⊻ ∃z [¬(c ≈ z) ∧ ¬(v ≈ z) ∧ T′(c, z) ∧ T′(v, z)]]

This translates to “If <nobility and virtue concur in one definition>, then <either nobility derives from virtue or (exclusive) virtue derives from nobility, or (exclusive) else there exists a z such that it is not the case that nobility is equivalent to z and it is not the case that virtue is equivalent to z and nobility derives from z and virtue derives from z>.” This formulation is consistent with Foster and Boyde’s comment on verses 89-95.

---

I use Dante’s statement that “ma se l’una val ciò che l’altra vale / ed ancor più, da lei verrà più tosto” (98-99) as a basis for my formulation of Principle (5):

∀x, y [X(x, y) → T′(y, x)]

This translates to “For all x and y, if x comprehends y and something else as well, then y derives from x.” For support of this formalization— i.e. T′(y, x) as opposed to T′(x, y), meaning y derives from x as opposed to x derives from y— see De Robertis’ comment on verse 99. I note, however, that this principle may not actually be true. That is, it is not necessarily the case that the less comprehensive quality y derives from the more comprehensive quality x. Thus, while Dante’s argument in which he concludes that virtue (the less comprehensive quality) derives from nobility (the more comprehensive quality) is valid, the argument, as expressed in the poem, may not be sound.

---

Dante’s use of the analogy of the sky and the star in verses 101-105 (i.e. with respect to the comprehensive relationship between nobility and virtue) supports my formalization of Principle (6) as follows:

∀x, y [[∀z [Y(x, z) → Y(y, z)] ∧ ¬∀z [Y(y, z) → Y(x, z)]] → X(y, x)]

This translates to “For all x and y, if it is the case that <for all z, if x is in place z, then y is in place z, and it is not the case that for all z, if y is in place z, then x is in place z>, then y comprehends x and something else as well.”

---

Dante’s claim that “ogni vertù principalmente / vien da una radice” (every virtue stems ultimately from one root [81-82]) is distinct from his claim that virtue derives from nobility. That is, the radice does not refer to nobility. Rather, it refers to the “abito eligente / lo qual dimora in mezzo solamente” (86-87). Accordingly, I formulate two distinct predicates based on Dante’s use of the verb venire: T(x, y) to mean “x principalmente vien da y” and T′(x, y) to mean “x vien da y.” The former is used to apply to virtue and the abito eligente and the latter is used to apply to virtue and nobility. For support of my use of two distinct predicates in this regard, see Foster and Boyde’s comment on verses 81-84 and De Robertis’ comment on verse 111.

---

In the Remaining Logic of the Sixth Stanza:

Dante’s conclusion expressed in Proof #5, namely, that virtue, and, more precisely, the abito eligente lo qual dimora in mezzo solamente, derives from nobility is itself used as a premise for further argumentation. Accordingly, I repeat “Dunque verrà come dal nero il perso / chiascheduna vertute da costei, / o vero il gener lor, ch’io misi avanti ” (Hence each virtue— or rather the above-mentioned common factor in virtue— derives from nobility as perse from black [109-111]) prior to presenting the logic of verses 109-120. As a reminder, Dante’s argument here does have missing premises. For this reason, I do not refer to the summation of my formulations into logic as a proof.

---

As noted above in the first stanza, for Dante, only man can be gentile. Therefore, I do not specify that x is a man in each case in which x is gentile— i.e. that B(x) when C(x).

Dico ch’ogni vertù principalmente
vien da una radice,
vertute, dico, che fa l’uom felice
in sua operazione.
Quest’è secondo che l’Etica dice,
un abito eligente
lo qual dimora in mezzo solamente,
e tai parole pone.
Dico che nobiltate in sua ragione
importa sempre ben del suo subietto
come viltate importa sempre male;
e vertute cotale
dà sempre altrui di sé buono intelletto;
per che in medesmo detto
convegnono amendue, ch’èn d’uno effetto.
Dunque convien che d’altra vegna l’una
o d’un terzo ciascuna;
ma se l’una val ciò che l’altra vale
ed ancor più, da lei verrà più tosto.
Ciò ch’i’ ho detto qui, sia per supposto (81-100).

È gentilezza dovunqu’è vertute,
ma non vertute ov’ella,
sì com’è ’l cielo dovunque’è la stella,
ma ciò non e converso…
Dunque verrà come dal nero il perso
chiascheduna vertute da costei,
o vero il gener lor, ch’io misi avanti (101-104, 109-111).

I affirm that every virtue stems ultimately from one root, meaning by virtue that which makes a man happy in his actions. This is, as the Ethics states, a ‘habit of choosing which keeps steadily to the mean’—those are the very words. And I say that nobility by definition always connotes good in him who has it, as baseness always connotes bad. Similarly virtue, as defined above, always connotes good. Hence, since both have the same effect, both concur in one definition. Hence it must be that either one is derived from the other, or each is derived from a third thing. But if one comprehends the other and something else as well, then this is the one more likely to be the origin. And let all this be presupposed in what follows.

Nobility is wherever virtue is, but virtue is not wherever nobility is; just as the sky is wherever a star is, but not e converso… Hence each virtue—or rather the above-mentioned common factor in virtue—derives from nobility as perse from black.

Proof #5

∀z U(c, e, z)

For all z, nobility connotes good in subject z.

∀z U(f, m, z)

For all z, baseness connotes bad in subject z.

∀z U(v, e, z)

For all z, virtue connotes good in subject z.

[∀z U(c, e, z) ∧ ∀z U(v, e, z)] → W(c, v)

If <for all z, nobility connotes good in subject z, and for all z, virtue connotes good in subject z>, then nobility has the same effect as virtue.

__________________________________

W(c, v)

Therefore, nobility has the same effect as virtue.

W(c, v) → V(c, v)

If nobility has the same effect as virtue, then nobility and virtue concur in one definition.

__________________

V(c, v)

Therefore, nobility and virtue concur in one definition.

V(c, v) → [T′(c, v) ⊻ T′(v, c) ⊻ ∃z [¬(c ≈ z) ∧ ¬(v ≈ z) ∧ T′(c, z) ∧ T′(v, z)]]

If <nobility and virtue concur in one definition>, then <either nobility derives from virtue or (exclusive) virtue derives from nobility, or (exclusive) else there exists a z such that it is not the case that nobility is equivalent to z and it is not the case that virtue is equivalent to z and nobility derives from z and virtue derives from z>.

____________________________________________________________________

T′(c, v) ⊻ T′(v, c) ⊻ ∃z [¬(c ≈ z) ∧ ¬(v ≈ z) ∧ T′(c, z) ∧ T′(v, z)]

Therefore, either nobility derives from virtue or (exclusive) virtue derives from nobility, or (exclusive) else there exists a z such that it is not the case that nobility is equivalent to z and it is not the case that virtue is equivalent to z and nobility derives from z and virtue derives from z.

∀x, y [[∀z [Y(x, z) → Y(y, z)] ∧ ¬∀z [Y(y, z) → Y(x, z)]] → X(y, x)]                    (6)

For all x and y, if it is the case that <for all z, if x is in place z, then y is in place z, and it is not the case that for all z, if y is in place z, then x is in place z>, then y comprehends x and something else as well.

∀z [Y(v, z) → Y(c, z)] ∧ ¬∀z [Y(c, z) → Y(v, z)]

For all z, if virtue is in place z, then nobility is in place z, and it is not the case that for all z, if nobility is in place z, then virtue is in place z.

___________________________________________________________

X(c, v)

Therefore, nobility comprehends virtue and something else as well.

∀x, y [X(x, y) → T′(y, x)]                    (5)

For all x and y, if x comprehends y and something else as well, then y derives from x.

_______________________

T′(v, c)

Therefore, virtue derives from nobility.

T(v, v′) ∧¬[∃x [T(v, x) ∧ ¬(x ≈ v′)]]

Virtue stems ultimately from the habit of choosing which keeps steadily to the mean, and there does not exist an x such that <virtue stems ultimately from x and it is not the case that x is equivalent to the habit of choosing which keeps steadily to the mean>.

[T(v, v′) ∧¬[∃x [T(v, x) ∧ ¬(x ≈ v′)]]] → [T′(v, c) → T′(v′, c)]

If <virtue stems ultimately from the habit of choosing which keeps steadily to the mean, and there does not exist an x such that virtue stems ultimately from x and it is not the case that x is equivalent to the habit of choosing which keeps steadily to the mean>, then <if virtue derives from nobility, then the habit of choosing which keeps steadily to the mean derives from nobility>.

_______________________________________________________

T′(v, c) → T′(v′, c)

Therefore, if virtue derives from nobility, then the habit of choosing which keeps steadily to the mean derives from nobility.

__________________

T′(v′, c)

Therefore, the habit of choosing which keeps steadily to the mean derives from nobility.

_________________

T′(v, c) ∧ T′(v′, c)

Therefore, virtue derives from nobility and the habit of choosing which keeps steadily to the mean derives from nobility.

Q.E.D. (#5)

Dunque verrà come dal nero il perso
ciascheduna vertute da costei,
o vero il gener lor, ch’io misi avanti;
però nessun si vanti
dicendo ‘Per ischiatta i’ son con lei’,
ched e’ son quasi dei
que’ c’han tal grazia fuor di tutti rei;
ché solo Iddio all’anima la dona
che vede in sua persona
perfettamente star, sì ch’ad alquanti
ch’è ’l seme di felicità s’accosta
messo da Dio nell’anima ben posta (109-120).

Hence each virtue— or rather the above-mentioned common factor in virtue— derives from nobility as perse from black. And therefore let no one boast saying: ‘I am noble because of my birth’; for those who have this grace without any flaw are almost godlike; for it is God alone who gifts it to a soul which He sees is perfectly in her body. Hence it is clear to some that nobility is the seed of happiness placed by God in a soul that is well positioned.

T′(v, c) ∧ T′(v′, c)

Virtue derives from nobility and the habit of choosing which keeps steadily to the mean derives from nobility.

(Missing Premises)

________________________________________________

∀x [C(x) ↔ ∀z [β(z, x) → [θ(z) ∧ ε(g, z) ∧ δ(g, c, z)]]]

Therefore, for all x, <x is noble if and only if <for all z, if z is the soul of x, then <z is perfectly in its body and god sees that z is perfectly in its body and god gifts nobility to z>>>.

∀x [C(x) ↔ ∀z [β(z, x) → [θ(z) ∧ ε(g, z) ∧ δ(g, c, z)]]] → ∀x [C(x) → α(x)]

If for all x, <x is noble if and only if <for all z, if z is the soul of x, then <z is perfectly in its body and god sees that z is perfectly in its body and god gifts nobility to z>>>, then <for all x, if x is noble, then x is godlike>.

____________________________________________________________________

∀x [C(x) → α(x)]

Therefore, for all x, if x is noble, then x is godlike.

∀x [C(x) → α(x)] → ∀x [C(x) → ¬Z(x)]

If <for all x, if x is noble, then x is godlike>, then <for all x, if x is noble, then it is not by birth that x is noble>.

____________________________________

∀x [C(x) → ¬Z(x)]

Therefore, for all x, if x is noble, then it is not by birth that x is noble.

∀z [β(z, x) → [θ(z) → ε(g, z)]]

For all z, <if z is the soul of x, then <if z is perfectly in its body, then god sees that z is perfectly in its body>>.

∀x [C(x) ↔ ∀z [β(z, x) → [θ(z) ∧ ε(g, z) ∧ δ(g, c, z)]]]

For all x, <x is noble if and only if <for all z, if z is the soul of x, then <z is perfectly in its body and god sees that z is perfectly in its body and god gifts nobility to z>>>.

_________________________________________________

∀x [C(x) ↔ ∀z [β(z, x) → [θ(z) ∧ δ(g, c, z)]]]

Therefore, for all x, <x is noble if and only if <for all z, if z is the soul of x, then <z is perfectly in its body and god gifts nobility to z>>>.

∀z [β(z, x) → [θ(z) → π(z)]]

For all z, <if z is the soul of x, then <if z is perfectly in its body, then z is well positioned>>.

∀z [β(z, x) → [δ(g, c, z) → λ(c, g, z)]]

For all z, <if z is the soul of x, then <if god gifts nobility to z, then nobility is placed by god in z>>.

________________________________________

∀x [C(x) ↔ ∀z [β(z, x) → [π(z) ∧ λ(c, g, z)]]]

Therefore, for all x, <x is noble if and only if <for all z, if z is the soul of x, then <z is well positioned and nobility is placed by god in z>>>.

(Missing Premises)

________________________________________

∀x [C(x) ↔ ∀z [β(z, x) → [π(z) ∧ λ(s, g, z)]]]

Therefore, for all x, <x is noble if and only if <for all z, if z is the soul of x, then <z is well positioned and the seed of happiness is placed by god in z>>>.

Stanza 7

L’anima cui adorna esta bontate
no·lla si tiene ascosa,
ché dal principio ch’al corpo si sposa
la mostra infin la morte.
Ubidente, soave e vergognosa                                    125
è nella prima etate,
e sua persona acconcia di beltate
colle sue parti accorte;
in giovanezza temperata e forte,
piena d’amore e di cortesi lode,                                 130
e solo in lealtà far si diletta;
è nella sua senetta
prudente e giusta, e larghezza se n’ode,
e ’n sé medesma gode
d’udire e ragionar dell’ altrui prode;                         135
poi nella quarta parte della vita
a Dio si rimarita
contemplando la fine ch’ell’aspetta,
e benedice li tempi passati.
Vedete omai quanti son gl’ingannati.                       140

No logic to present.

Congedo

Contra-li-erranti mia, tu te n’andrai,
e quando tu sarai
in parte dove sia la donna nostra,
non le tenere il tuo mestiere coverto:
tu le puoi dir per certo:                                                145
«I’ vo parlando dell’amica vostra».

No logic to present.

---

Acknowledgments:

Special thanks to Professor Teodolinda Barolini for her unparalleled wisdom, guidance, and support as this project developed. The depth of her expertise in Dante and her willingness to share it helped me to formulate the logic in a way that best reflects what Dante expressed in the poem. The genesis of this work was the final paper I submitted for her “Studies in Dante” seminar at Columbia University in Spring 2017. I thank Professor Barolini for her suggestion that I look at the arguments of Le dolci rime for a paper topic. My decision to formalize the arguments of the poem into logic and assess their validity grew out of that suggestion. Thank you also to Professor Achille Varzi for his thorough and attentive review of this work in the final editing stages. I was fortunate to study logic at Columbia University with Professor Varzi as an undergraduate, and I am deeply appreciative of the time he spent reviewing the logic, particularly given the valuable combination of his expertise in the field and his fluency as a native Italian speaker. Additionally, thank you to the Editorial Board of Digital Dante for their helpful comments, particularly with a view to making this piece accessible to readers without a background in logic. I especially thank Grace Delmolino for her suggestion that I include an interlinear translation of the logic, and Meredith Levin for the incredible care she took when inputting the logic onto this site. Lastly, thank you to Christina McGrath for listening on various occasions as I thought through portions of the logic aloud.

---

Recommended Citation: Schiff, Jenny Clark. “Dante’s Canzone Le dolci rime Translated into Formal Logic: With Interlinear Translation.” Digital Dante. New York: NY, Columbia University Libraries. 2020.