What's a Badass?

A badass is someone who's courageous, confident, tough, strong, fearless, and who doesn't back down. Someone who doesn’t hesitate to express themselves or show who they are. No tiptoeing around the bush. No walking on eggshells. A badass knows who they are, and they aren’t afraid to stand up for themselves. 

      A badass won't let themselves be bullied. They can tolerate minor harassment, but they don't let themselves be pushed around. When they are compelled to act, they have fighting and survival skills no one would have expected. 

      A badass is ready to speak truth to power. A badass will stand up to others in the face of threats and intimidation. A badass is ready to kick ass if necessary. A badass is also someone who’s loyal and who will be there for their friends and rock with them all the way out. 

      There’s no telling a badass they’re not big enough or pretty enough or good enough. A badass knows their own abilities and doesn't have to try to impress anybody, but they can also show some swagger when doubters are around. They don't brag about themselves, and that way they catch people off guard.

      A badass will do something that nobody else thought of doing and be so bold and bodacious about it that they catch their enemies by surprise. They have street smarts and know how to get out of a jam.

      “Badass” is a term of praise, so it doesn’t apply to someone who’s a thug, victimizer, sociopath, narcissist, misogynist, or creep. 

      Shaft is a badass. John Wick is a badass. Lisbeth Salander (The Girl with the Dragon Tattoo) is a badass. Lorraine Broughton (in Atomic Blonde) is a badass. Sandor Clegane (The Hound, in Game of Thrones) is a badass. 

      

Immanence and Transcendence

The following is an excerpt from a brief reflection I shared at the "Faith at Eight" service at church on Sunday, May 3, 2026, when I read Evelyn Underhill's poem "Immanence" to the other members of our group during our reflection period.

In today's reading from the Gospel of John, Jesus says, "Do you not believe that I am in the Father and the Father is in me? The words that I say to you I do not speak on my own; but the Father who dwells in me does his works. Believe me that I am in the Father and the Father is in me; but if you do not, then believe me because of the works themselves" (John 14:10-12).
      I think that one of the themes of today's reading from John is God's immanence. Jesus is in God, and God is in him. God is in us, and we are in him.
      Evelyn Underhill (1875-1941), a British poet and novelist who wrote several books about religious mysticism, wrote a poem called "Immanence" (1916), which consists of three stanzas, each of which begins with the lines "I come in the little things,/Saith the Lord." The poem is rather formal in style and language, and it's an example of Edwardian poetry, which may use an archaic vocabulary and may strictly adhere to conventional meter and rhyme schemes. But one reason I found the poem to be quite resonant and meaningful was that it says God comes "in the little things."
      We can feel God's presence among us. We can know God's immanence. But at the same time, we can't be aware of God's immanence without also being aware of God's transcendence. God's immanence and transcendence are inseparable. The mystery of God's presence is God's simultaneous immanence and transcendence. And I think that is also the mystery of the Holy Eucharist. When we consume the bread and wine of the Holy Eucharist, God's presence is within us, and we are in God.The Holy Spirit can find a dwelling place within us. We can transcend ourselves if the Holy Spirit is within us. But God also transcends human reality.
      Jesus is called Emmanuel because the name Emmanuel means "God with us." The name Emmanuel signifies God's presence. And the mystery of God's presence, immanently and transcendently, can be found everywhere we look for God.

Chronology of the Life of Sister Thea Bowman (1937-1990)

Sister Thea Bowman was a Black Catholic religious sister, teacher, preacher, singer, evangelist, and social justice advocate.

1937 - On December 29, 1937, she was born Bertha Elizabeth Bowman, to Mary Esther Bowman, a teacher, and Theon Edward Bowman, a physician, in Yazoo City, Mississippi. She later described herself as an "old folks' child," because her parents were middle-aged when she was born. She was their only child.

    

1947 - Her parents were Methodists, but Bertha attended a Catholic school, the Holy Child Jesus School in Canton, Mississippi, which had been founded for Black children by the Missionary Servants of the Most Holy Trinity and the Franciscan Sisters of Perpetual Adoration, a religious order from La Crosse, Wisconsin. At the age of nine, with her parents' permission, she converted to Catholicism and was baptized into the Catholic Church.

1953 - At the age of fifteen, she decided to become a religious sister, and despite her parents' initial concerns regarding how she would be received by other members of the convent, she became the first African American member of the Franciscan Sisters of Perpetual Adoration, in La Crosse, Wisconsin.

1955 - She was diagnosed with tuberculosis and was treated for ten months at River Pines Sanatorium, in Stevens Point, Wisconsin.

1956 - She took the name Sister Mary Thea, in honor of the Blessed Virgin Mary and her father, Theon. The name Thea also meant "of God."

1958 - She taught at Blessed Sacrament School, in La Crosse, Wisconsin.

1961 - She taught at Holy Child Jesus School, in Canton, Mississippi. She directed the school choir, and they recorded an album to raise money to build a new wing at the school.¹

1965 - She earned a B.A. in English from Viterbo College (now Viterbo University), a Catholic college in La Crosse, Wisconsin.

1968 - She became s founding member of the National Black Sisters' Conference (NBSC), an organization of Black Catholic sisters and nuns. 

     

1969 - She earned an M.A. in English at The Catholic University of America, in Washington, D.C. As a graduate student, she created the first course in Black literature taught at the university.

1972 - She earned a Ph.D. in English at The Catholic University of America. Her doctoral dissertation was on the relationship of pathos and style in Thomas More's A Dialogue of Comfort Against Tribulation. She traveled in Europe and studied during the summer at Oxford University.

1972 - She started teaching at Viterbo College, eventually becoming chair of the English department.

1978 - She became a consultant (and later the director) of the Office of Intercultural Awareness for the Diocese of Jackson, Mississippi.² She had speaking engagements across the country, and she became a nationally recognized advocate for social justice. 

      Following the modernizing changes made by the Second Vatican Council (1962-1965), and because her order, the Franciscan Sisters of Perpetual Adoration, allowed members to forgo the traditional requirement of wearing a nun's habit, she started wearing African dress in order to celebrate her ancestry and more closely reflect the people whom she served.³

      She advocated for the full inclusion of African American religious expression within the Catholic Church, and she urged the Church to combat racism.

1980 - She was a founding faculty member of the Institute of Black Catholic Studies at Xavier University of Louisiana, in New Orleans.

       

1984 - Both of her parents died, and she was diagnosed with breast cancer, which had already spread to her lymph nodes and bones. She underwent surgery, radiation, and chemotherapy.

1985 - She received The Harriet Tubman Award from the National Black Sisters' Conference. She traveled to Nairobi, Kenya to participate in the International Eucharistic Congress, and she also traveled to Zimbabwe and Nigeria.⁴

1987 - She was interviewed by Mike Wallace for a segment of 60 Minutes, which was broadcast on May 3rd. After seeing the 60 Minutes program, Harry Belafonte began to make plans for a film about her life, starring Whoopi Goldberg. She met both of them during a trip to California in 1988. Harry Belafonte also visited her home in Canton, Mississippi, and he visited her at Xavier University in New Orleans, where he spoke with her students.

      At the Conference of The National Congress of the Religious Formation, in New Orleans, she delivered an address entitled "Cosmic Spirituality: Formation in a New Age," in which she said,

      "We invite men and women from the cultures of the world to come into our congregations...So often in formation and in community, their spiritual gifts and spiritual journeys are ignored. Talk with the people of color in your congregations, in your formation programs. Ask them to what extent they believe that you are serious about understanding them--their history, their experience, their culture, their heritage, their art, their music, their styles of prayer, their styles of meeting, their songs, their dances, their modalities of relationship. To what extent are you serious about sharing their spirituality, their styles of life and prayer and relationship?

      ...When Jesus is among us, to work among us miracles of transformation and miracles of love, there is no neutral ground. Neutral ground becomes loving ground, loving ground becomes holy ground, holy ground becomes Kingdom ground. We are the children of the cosmos, the children of the universe."⁵

      She was an inspirational figure in the creation of the first African American Catholic hymnal, which was published in 1987 and was entitled "Lead Me, Guide Me."

1988 - She recorded the albums "Songs of My People" and "Round the Glory Manger."

      Among the honorary degrees she received were doctoral degrees from Clarke College, Xavier University, Sacred Heart University, Viterbo College, Marygrove College, and Georgetown University. She was the first African-American woman to receive an honorary doctorate in religion from Boston College.

      

1989 - Wearing African dress, and sitting in a wheelchair due to her weakness from advanced cancer, she delivered a historic address at the general assembly of U.S. Catholic Bishops, at Seton Hall University, in South Orange, NJ. 

      She began by singing, in her beautiful, operatic voice, "Sometimes I feel like a motherless child," and she finished by having all the bishops stand, and join arms and sing "We Shall Overcome." In a very moving and charismatic speech, she addressed the question of "What does it mean to be Black and Catholic?" She explained that Black people are sometimes viewed by the Church with a patronizing and paternalistic attitude, and that some Catholics (and even some Black Catholics) may not fully approve of Black religious expression within the Catholic liturgy. Some members of the Church may feel that Black religious expression is not properly Catholic, and that it's not appropriately solemn or dignified. But she also explained that being Black and Catholic means that

"I bring myself, my Black self, all that I am, all that I have, all that I hope to become, I bring my whole history, my traditions, my experience, my culture, my African American song and dance, and gesture and movement, and teaching and preaching, and healing and responsibility as gifts to the Church."⁶

      Thus, she called for the Church to be a place in which the gifts of all people are welcome.⁷ 

She called for the Church to enable all Catholics to worship and pray, to feed the hungry and clothe the poor, to shelter the homeless and comfort the sick, to teach, and to do the work of the Church in the modern world.

1990 - Just a few weeks before she died, as she reflected on the meaning of Holy Week, she wrote, "Old folks used to say, "God is bread when you'e hungry. God is water when you're thirsty. God is a shelter from the storm. God is rest when you're weary. God's my doctor. God's my lawyer. God's my captain who never lost a battle. God is my lily of the valley."⁸

      On March 30th, she died of breast cancer, at the age of 52, in Canton, Mississippi. On April 4th, she was buried next to her parents at Elmwood Cemetery, in Memphis, Tennessee. 

      She was posthumously awarded the Laetare Medal, the oldest and most prestigious award given to American Catholics, from Notre Dame University. She was the first African-American to receive this award.

      An obituary in the NY Times on April 1st began by saying,

      "Sister Thea Bowman, a nationally active black educator who with song, prayer and persistent exhortation urged the Roman Catholic Church to embrace the culture of African Americans, died of cancer on Friday at her home in Canton, Miss. She was 52 years old."⁹

2018 - She was endorsed for sainthood by the U.S. Catholic Bishops at their fall Plenary Assembly in Baltimore.

FOOTNOTES

¹"Going Home Like a Shooting Star: Thea Bowman's Journey to Sainthood," film documentary, New Group Media and the Diocese of Jackson, Mississippi, written and produced by Sister Judith Ann Zielinski, 2022.

²Sister Thea Bowman, Shooting Star: Selected Writings and Speeches, edited by Celestine Cepress (Winona Minnesota: Saint Mary's Press, 1993), p. 14.

³René Ostberg, "Thea Bowman," in Encyclopedia Brittanica, 26 March 2026, online at https://www.britannica.com/biography/Thea-Bowman.

⁴Sister Thea Bowman, Shooting Star: Selected Writings and Speeches, edited by Celestine Cepress, p. 13.

⁵Ibid., pp. 106-107.

⁶Ibid., p. 32.

⁷Ibid. p. 29.

⁸Thea Bowman, In My Own Words, compiled and edited by Maurice J. Nutt (Liguori, Missouri: Liguori Publications, 2009), p. 3.

⁹Dennis Hevesi, "Sister Thea Bowman, 52, Worker for Catholic Sharing With Blacks," in The New York Times, April 1, 1990.

Artscape, 2026, Baltimore

Time in The Wilderness

Soundness and Completeness of Normal Modal Logics

In normal modal logic, system K is the minimal (or weakest) system. System K is characterized by the necessitation rule, ⊨ A → ⊨ □A ("if A is valid, then necessarily A is valid"), and the distribution axiom, □(A → B) → (□A → □B) (also called the K axiom, "if it is necessary that if A then B, then if it is necessary that A, then it is necessary that B"). All the other normal modal logics are extensions of system K.

      Normal modal logics are characterized by the necessitation rule, the distribution axiom, and the duality (or interdefinability) of the possibility and necessity operators (□p ↔ ~♢~p, and ♢p ↔ ~□~p). 

      In modal logic, a frame (W, R) is a structure of possible worlds W and accessibility relations R, defining the structure of a given system without assigning truth values, while a model adds a valuation function (M = (W, R, V)) to the frame.¹ A formula is valid in a frame if it is true in all possible models in that frame. 

      The accessibility relation (R) determines which worlds are accessible from others. If accessibility is serial (wRv), then every world has access to at least one other world. If accessibility is reflexive (wRw), then every world is accessible to itself. If accessibility is symmetric (wRv → vRw), then if world v is accessible to world w, then world w is accessible to world v. If accessibility is transitive ((wRv ∧ vRu) → wRu), then if world v is accessible to world w, and world u is accessible to world v, then world u is accessible to world w. If accessibility is Euclidean ((wRv ∧ wRu) → vRu), then if worlds v and u are accessible to world w, then u is accessible to v. 

       It should be noted that all Euclidean relations aren't necessarily reflexive, symmetric, or transitive. The standard formulation of Euclidean accessibility, (wRx ∧ wRy) → xRy, doesn't require symmetric Euclidean accessibility (wRx ∧ wRy) → (xRy ∧ yRx). On the other hand, if a Euclidean relation is combined with reflexivity, it will also be symmetric and transitive, transforming it into an equivalence relation. 

       Equivalence relations are reflexive, symmetric, and transitive. Universal relations are relations where every member of a set is connected to every other member of the set, including itself. Thus, a difference between equivalence relations and universal relations is that equivalence relations may partition a set into smaller, disjoint clusters or subsets, while universal relations connect every element in a set to every other element.²

      The K axiom (□(p → q) → (□p → □q)), doesn't correspond to any accessibility relation. System K doesn't have any requirement regarding the accessibility of possible worlds. 

      The D axiom (□φ → ♢φ) corresponds to serial accessibility, because seriality requires that for every world w, there must be at least one world v accessible to it (wRv), ensuring that if something is necessary, it is also possible. 

      The T axiom (□φ → φ), also called the M axiom, corresponds to reflexivity, because reflexivity requires every world to be accessible to itself (wRw), ensuring that if a proposition is necessarily true, then it must be true in our own world. 

      The B axiom (φ → □♢φ) corresponds to symmetry, because symmetry requires that if a world v is accessible from w, then w is also accessible from v (wRv → vRw), ensuring that if a proposition p is true, then it is necessary that p is possible. 

      The S4 axiom (□φ → □□φ) corresponds to transitivity, because transitivity requires that if a world u is accessible from w, and v is accessible from u, then v is directly accessible from w ((wRu ∧ uRv) → wRv) ensuring that if a proposition p is necessarily true, then it is necessary that p is necessarily true.

      The S5 axiom (♢φ → □♢φ) corresponds to reflexive and Euclidean accessibility, because it requires for all worlds w, x, and y, that if x is accessible to w (wRx), and y is accessible to w (wRy), then y is accessible to x (xRy). Thus, if a world w can see a world x where p is possible, then any world y accessible from w will be accessible to x, ensuring that if p is possible, then it is necessarily possible.

      In first-order logic, seriality can be expressed as ∀x∃yRxy (for every x, there is a y it's related to), reflexivity can be expressed as ∀xRxx (everything is related to itself), symmetry can be expressed as ∀x∀y(Rxy → Ryx) (for every x and every y, if x is related to y, then y is related to x), transitivity can be expressed as ∀x∀y∀z((Rxy ∧ Ryz) → Rxz) (for every x and every y and every z, if x is related to y, and y is related to z, then x is related to z), and Euclidean accessibility can be expressed as ∀x∀y∀z((Rxy ∧ Rxz) → Ryz) (for every x and every y and every z, if x is related to y, and x is related to z, then y is related to z).³      

      In normal modal logic, system K (named after philosopher and logician Saul Kripke) doesn't have any accessibility constraints. It includes the axioms of propositional logic, the distribution axiom, and the necessitation rule. However, it is too weak to prove that necessary truths are actually true (□φ → φ), because it doesn't require reflexive accessibility.      

      System D in normal modal logic is an extension of system K that includes the D axiom (□φ → ♢φ), corresponding to serial accessibility. System D is the foundational system for standard deontic logic (SDL). However, like system K, it's too weak to prove that necessary truths are actually true, because it doesn't require reflexive accessibility.      

      System T (or M) is an extension of K and D that includes the T (or M) axiom (□φ → φ), corresponding to reflexivity.      

      System B is an extension of K, D, and T that includes the B axiom (φ → □♢φ), corresponding to symmetry.      

      System S4 is an extension of K, D, and T that includes the S4 axiom (□φ → □□φ), corresponding to transitivity.      

      System S5 is an extension of B or S4 that includes the S5 axiom (♢φ → □♢φ), corresponding to Euclidean accessibility.      

      The frame conditions for system K are none, D serial, T reflexive, B reflexive and symmetric, S4 reflexive and transitive, and S5 reflexive, symmetric, and transitive. Both KTB4 and KTB5 are reflexive, symmetric, transitive, and Euclidean.      

      System K is sound and complete with respect to the class of all Kripke frames (which have no constraints on their accessibility relations).      

      System D is sound and complete with respect to the class of all serial frames.       

      System T is sound and complete with respect to the class of all reflexive frames.       

      System B is sound and complete with respect to the class of all frames that are reflexive and symmetric.       

      System S4 is sound and complete with respect to the class of all frames that are reflexive and transitive.       

      System S5 is sound and complete with respect to the class of all frames that are reflexive, symmetric, and transitive.

      Another way of saying this is that in all serial models, all formulas that are provable are valid, and all formulas that are valid are provable. In all reflexive models, all formulas that are provable are valid, and all formulas that are valid are provable. In all models that are both reflexive and symmetric, all formulas that are provable are valid, and all formulas that are valid are provable, and so on.     

      If a wff is K-valid, then it is also D-, T-, B-, S4-, and S-5 valid. If a wff is D-valid, then it is also T-, B-, S4-, and S-5 valid. If a wff is T-valid, then it is also B-, S4-, and S-5 valid. If a wff is B-valid, then it is also S-5 valid, and if a wff is S-4 valid, then it is also S-5 valid.

      The S-5 models are a subset of the B models and the S4 models, the B models and the S-4 models are subsets of the T models (but not of each other), the T models are a subset of the D models, and the D models are a subset of the K models.⁴

     K is the weakest system, because it has the fewest valid formulas and the most potentially falsifying models. S-5 is the strongest system, because it has the most valid formulas, and the fewest potentially falsifying models.⁵

      The soundness of normal modal logics can be established by showing that the axioms of a particular system (such as D, T, B, S4, or S5) are valid, and then by inference rules such as modus ponens and the necessitation rule, showing that any formulas derived from the axioms will also be valid. So, by induction, every formula derivable from the axioms will also be valid.

      If a wff is shown to be K-valid, then it will also be valid in all the stronger systems (D, T, B, S4, and S5). If a wff is D-valid, then it will also be valid in all the stronger systems (B, S4, and S5). If a wff is shown to be T-valid, then it will also be valid in all the stronger systems (B, S4, and S5).

      However, not every formula that is B-valid is S4-valid, and not every formula that is S4-valid is B-valid. The B-axiom is B-valid but not S4-valid (because it is not valid in frames that are reflexive and transitive but not symmetric), and the formula □♢A → ♢□A is S4-valid but not B-valid (because it is not valid in frames that are reflexive and symmetric but not transitive). However, every formula that is B-valid or S4-valid is S5-valid.

      The completeness of normal modal logics can be established by the canonical model method. A canonical model of a logic, M = (W, R, V), consists of W (a set of all maximal consistent sets of formulas), R (an accessibility relation), and V (a valuation), and it may serve as a universal counter-model by showing that every formula not provable in the logic is false in the model. (A set of formulas is maximally consistent when no additional formula can be added to it without making it inconsistent.)

      However, while many normal modal logics (such as K, D, T, B, S4 and S5) are both sound and complete with respect to their frame conditions, some normal modal logics (such as van Bentham's logic, denoted vB), are incomplete. Incomplete normal modal logics are systems that cannot be characterized by any class of Kripke frames, meaning that they cannot prove all the formulas that are valid in their own frame semantics.⁶

      Non-normal modal logics differ from normal modal logics, because they lack the necessitation rule or the distribution axiom. While normal modal logics are interpreted using Kripke semantics, non-normal modal logics are typically interpreted using neighborhood semantics.

      Neighborhood semantics differ from Kripke semantics, because instead of using a relational frame (W, R) consisting of a set W of worlds and an accessibility relation R that indicates which worlds are accessible from others, they use a neighborhood frame (W, N) consisting of a set W of worlds and a neighborhood function N that assigns to each element of W a set of subsets (or neighborhoods) of W.⁷

      Non-normal modal logics are generally sound and complete with respect to their neighborhood semantics.⁸

      Logic E is the minimal or weakest system of non-normal modal logic. It serves as a basis for constructing stronger systems, and it includes the congruence rule (if ⊢ φ ↔ ψ, then ⊢ □φ ↔ □ψ). Additional axioms, such as T, C, or N, can be added to Logic E to form stronger modal systems.  

      Axiom C (□A ∧ □ B) → □(A ∧ B) signifies that if two propositions are individually necessary, then their conjunction is also necessary.

      Axiom N (□T) signifies that all tautologies are necessarily true.

      Logic E is sound and complete with respect to general neighborhood frames.⁹ Adding axiom T to Logic E yields the system ET, which is characterized by reflexive accessibility in its neighborhood semantics. 

      System ET includes the congruence rule and the T axiom, and it is sound and complete with respect to all reflexive neighborhood models. All formulas provable in system ET are valid in all reflexive neighborhood models, and all valid formulas in all reflexive neighborhood models are provable in system ET.

      If Axioms, T, C, and N are all added to E, then the resulting system becomes a normal modal logic equivalent to K, which is also sound and complete in the corresponding semantic models.

      Strong completeness may be distinguished from weak completeness. Blackburn et al. (2001) explain that a logic Λ is strongly complete with respect to a class of frames S, if for any set of formulas 𝛤, where φ ∈ 𝛤, if 𝛤 ⊨_S 𝜙, then 𝛤 ⊢_S 𝜙. A logic Λ is weakly complete with respect to a class of frames S if for any formula φ, if S ⊨ 𝜙, then ⊢_Λ 𝜙. Weak completeness is therefore a special case of strong completeness in which 𝛤 is empty. Strong completeness with respect to a class of frames also implies weak completeness with respect to that same class of frames.¹⁰ 

      Most normal logic systems (like K, D, T, B, S4, and S5) are strongly complete with respect to their corresponding classes of Kripke frames. 

      Normal modal logic systems that are weakly complete but not strongly complete are typically characterized by having Kripke semantics that are not compact, meaning that an infinite set of formulas 𝛤 may be unsatisfiable in the given system, even though every finite subset of 𝛤 is satisfiable.      

      The Gödel-Löb (GL) logic and the Grz (Grzegorczyk) logic are two such logical systems. Their respective semantics are not compact, and thus they are weakly complete but not strongly complete.       

      The GL logic results from adding the Gödel-Löb axiom, □(□p → p) → □p, to system K. The GL logic is sound and weakly complete with respect to the class of all finite, transitive, converse well-founded frames. 

       A well-founded frame is a frame in which the accessibility relation has no infinite descending sequences of worlds. A converse well-founded frame is a frame in which the accessibility relation has no infinite ascending sequences of worlds (if you start at any world w, and you follow the accessibility relation upward, you can't go on forever). An ascending sequence of worlds (w₀Rw₁Rw₂R...) is a sequence in which each world has access to the next one in the sequence. A descending sequence of worlds (...Rw₂Rw₁Rw₀) is a sequence in which each world is accessible from the previous world. 

      Both well-founded and converse well-founded frames are irreflexive, because the absence of infinite descending or ascending sequences means that no world is accessible to itself (wRw), insofar as an infinite cycle (wRwRwR...) could be created, diverging from the well-foundedness condition.¹¹ 

     The Grz logic results from adding the Grz axiom, □(□p → □p)→ p) → p, to system K. It is sound and weakly complete with respect to the class of all finite, reflexive, transitive, converse well-founded frames. (Reflexivity is allowed by its accessibility relation only to the extent that there are no loops or infinite ascending sequences of worlds).  

      For compact logics (like K, D, T, B, S4, and S5), strong completeness follows from weak completeness. However, in non-compact logics, (like GL and Grz), a system can be weakly complete without being strongly complete.

FOOTNOTES

¹Jordan Hebert, "Completeness in Modal Logic," 2020, p. 3, online at

https://math.uchicago.edu/~may/REU2020/REUPapers/Hebert.pdf

²Open Logic Project Builds, "Equivalence Relations and S5," online at
https://builds.openlogicproject.org/content/normal-modal-logic/frame-definability/equivalence-S5.pdf

³Mark Jago, "Systems of Modal Logic," 2021, online at

https://www.youtube.com/watch?v=fl4JWORXOLY&list=PLwSlKSRwxX0qXTZKnIT7l4_YAIWpJcZJ9&index=4

⁴Theodore Sider, Logic for Philosophy (Oxford: Oxford University Press, 2010), p. 186.

⁵Ibid, p. 186.

⁶J.F.A.K. van Benthem, "Two Simple Incomplete Modal Logics," in Theoria, Vol. 44 (1), April 1978, pp. 25-37.

⁷Eric Pacuit, "Neighborhood Semantics for Modal Logic: An Introduction," July 3, 2007, p. 11, online at https://www-cs.stanford.edu/~epacuit/classes/esslli/nbhdesslli.pdf

⁸Atefeh Rohani and Thomas Studer, "Explicit Non-Normal Modal Logic," in Journal of Logic and Computation, Volume 35, Issue 6, September 2025, online at

https://academic.oup.com/logcom/article/35/6/exae052/7930603

⁹Brian Chellas, Modal Logic: An Introduction (Cambridge: Cambridge University Press, 1980), p. 257.

¹⁰Patrick Blackburn, Maarten de Rijke, and Yde Venema, Modal Logic (Cambridge: Cambridge University Press, 2001), p. 194.

¹¹Rineke Verbrugge, "Provability Logic," Stanford Encyclopedia of Philosophy, 2010, online at https://plato.stanford.edu/archives/fall2012/entries/logic-provability/