Graham Priest's Logic of Paradox

Classical logic is characterized by such rules as (1) the principle of bivalence (that a proposition can only be assigned one of two truth values, true or false), (2) the law of excluded middle, (p v ~p, that every proposition is either true or false), (3) the law of non-contradiction (~(p ∧ ~p), that a proposition and its negation cannot both be true at the same time), (4) the law of self-implication (p → p, that if a proposition is true, then it is true), and (5) double negation elimination (~~p → p, that if a proposition is not false, then it is true).  
      The two main branches of classical logic are propositional logic and predicate logic. Propositional logic employs variables to represent propositions, and it connects them with logical operators like "and", "or", and "not". Predicate logic extends propositional logic by using predicates and quantifiers to describe properties of objects and the relations between them. 

       ~(L & H) → ~G (I will not graduate if I don't pass both logic and history) is a sentence of propositional logic, while ∀xFx → ∀xSx (if anyone fails the exam, then everyone will be sad) is a sentence of predicate logic. 

       First-order predicate logic is a logic in which predicates take only individuals as arguments, and quantifiers only bind individual variables. Higher-order predicate logic is a logic in which predicates take other predicates as arguments, and quantifiers bind predicate variables.¹

       Non-classical logics include three-valued logics (such as Łukasiewicz's, Kleene's, and Priest's), 4-valued logics (such as Belnap's), many-valued logics (such as infinitely-valued fuzzy logic), paraconsistent logic, intuitionistic logic, modal logics (including alethic modal logic, deontic logic, temporal logic, epistemic logic, and doxastic logic), free logics, and substructural logics (such as relevance logic and linear logic). 

       Some differences between classical and non-classical logics include: (1) classical logic uses only two truth values, but non-classical logics may allow multiple truth values, (2) classical logic accepts the law of excluded middle, but intuitionistic logic rejects it (the sentence p v ~p is not valid in intuitionistic logic, so ⊬ p v ~p), (3) indirect proof (or proving a sentence by assuming its negation and then showing a contradiction) is a valid inference rule in classical logic, but not in intuitionistic logic (because in intuitionistic logic, proving that the negation of a statement yields a contradiction doesn't prove the statement is true), (4) classical logic accepts the law of non-contradiction, but paraconsistent logic rejects it (so some contradictions may still be assigned a truth value), (5) in classical logic, a contradiction implies anything (p ∧ ~ p → q), according to the principle of explosion (ex falso quodlibet, "from falsehood anything follows"), but paraconsistent logic rejects this, and (6) in classical logic, double negation elimination is a valid inference rule, but in intuitionistic logic, it is not (because in intuitionistic logic, ruling out the falsity of a statement doesn't prove that the statement is true, ~~p ⊬ p). 

       Motivations for three-valued logic include: (1) vagueness, semantic underdetermination, and borderline cases where statements do not seem to be clearly true or false, (2) future contingents, where statements about the future do not seem to be clearly true or false, (3) presupposition failure, where statements with false presuppositions do not seem to be clearly true or false, and (4) empty names, where statements about nonexistent or fictional entities do not seem to be clearly true or false.²

       Graham Priest's Logic of Paradox (LP) is motivated by the need to respond to logical paradoxes--both semantic ones, such as the Liar Paradox, and set-theoretic ones, such as Russell's Paradox.³ Priest explains that all paradoxes of self-reference (including both semantic and set-theoretic paradoxes) fit the "inclosure schema," where a set of all things with a property exists, but the property itself contradicts the set's membership.⁴ Beyond such paradoxes, LP is also designed to deal with problems of vagueness (e.g. the Sorites paradox) by allowing for "truth-value gluts" (where a statement may be both true and false) in borderline cases. 

       Theodore Sider (2010) explains that in Priest's Logic of Paradox, a set of wffs Γ semantically implies wff φ iff φ is either 1 or # in every trivalent interpretation in which every member of Γ is either 1 or #. (Priest uses the symbols t for true and only true, f for false and only false, and p for "paradoxical" or both true and false, so in Sider's notation # would correspond to p in Priest's notation). Thus, the definitions of validity and semantic consequence in Priest's Logic of Paradox are:

• φ is LP-valid (“⊨_LP φ”) iff KV_I (φ) ≠ 0 for each trivalent interpretation I
• φ is an LP-semantic-consequence of Γ (“Γ ⊨_LP φ”) iff for every trivalent interpretation, I, if KV_I (γ ) ≠ 0 for each γ ∈ Γ, then KV_I (φ) ≠ 0 

(where KV_I is a Kleene valuation function).⁵

      Sider also explains that for Priest, p represents the state of being both true and false (a truth-value "glut"), rather than the state of being neither true nor false (a truth-value “gap”). Correspondingly, Priest takes 1 to represent true and only true, and 0 to represent false and only false. The position that natural language sentences can be both true and false is known as dialetheism. LP represents logical consequence as the preservation of either 1 or p; and in LP, a formula is thought of as being true iff it is either 1 or p (in the latter case the formula is false as well).⁶ LP is a paraconsistent logic, because P, ~P ⊭_LP Q. 

       Priest (2024) distinguishes between paraconsistency and dialetheism by saying that paraconsistent logicians need not be dialetheists, and that they may reject the principle of explosion for other reasons.⁷ He explains that J.C. Beall and Greg Restall (2006) distinguish between four grades of paraconsistency: (1) gentle-strength, which is simply the rejection of explosion with respect to logical consequence, (2) full-strength, which holds that there are interesting or important theories that are inconsistent but not trivial, (3) industrial-strength, which holds that some inconsistent but non-trivial theories are possibly true, and (4) dialetheic, which holds that some inconsistent but non-trivial theories are indeed true.⁸

       Sider notes that ex falso quodlibet is not the only classical inference rule that fails in LP. Modus ponens, modus tollens, and reductio ad absurdum also fail in LP. Thus, the relation of logical consequence is treated much differently in LP than in classical logic.⁹ 

       Why modus ponens fails in LP can be demonstrated as follows: P, P → Q ⊭_LP Q. In other words, Q is not a semantic consequence of the premises P and P → Q, because if we assign P a truth value of #, and Q a truth value of 0, then #, # → 0 ⊭_LP 0, because the truth value for # → 0 is #.

FOOTNOTES

¹Peter Suber, "Glossary of First-Order Logic," 2002, online.

²Lisa Cassell, "Beyond Standard Propositional Logic," Philosophy of Logic, lecture, 2026.

³Graham Priest, "The Logic of Paradox," Journal of Philosophical Logic, Vol. 8, No. 1, Jan. 1979, p. 219.

⁴Graham Priest, "Inclosures, Vagueness, and Self-Reference," Notre Dame Journal of Formal Logic, Vol. 51, No. 1, 2010, p. 70.

⁵Theodore Sider, Logic for Philosophy (Oxford: Oxford University Press, 2010), p. 102.

⁶Ibid., p. 103.

⁷Graham Priest, "Dialetheism," Stanford Encyclopedia of Philosophy, 2024, online at 

https://plato.stanford.edu/entries/dialetheism/.

⁸J.C. Beall and Greg Restall, Logical Pluralism (Oxford: Oxford University Press), 2006.

⁹Sider, Logic for Philosophy, p. 103.

Sequent Proofs for DeMorgan's Laws

Theodore Sider (2009) presents a sequent proof for a "DeMorgan" sequent as follows:

∼(P ∨ Q) ⇒ (∼P ∧ ∼Q):

1. ∼(P ∨ Q) ⇒ ∼(P ∨ Q)                            RA
2. P ⇒ P                                                  RA
3. P ⇒ P ∨ Q                                           2,∨I
4. ∼(P ∨ Q), P ⇒ (P ∨ Q) ∧ ∼(P ∨ Q)       1,3,∧I
5. ∼(P ∨ Q) ⇒ ∼P                                     4,RAA
6. Q ⇒ Q                                                  RA
7. Q ⇒ P ∨ Q                                           6,∨I
8. ∼(P ∨ Q), Q ⇒ (P ∨ Q) ∧ ∼(P ∨ Q)       1,7,∧I
9. ∼(P ∨ Q) ⇒ ∼Q                                     8,RAA
10. ∼(P ∨ Q) ⇒ ∼P ∧ ∼Q                          5,9,∧I 

(Each line is numbered, and to the right of each line is written the sequent line and the inference rule that justify it: ∨I stands for "v introduction," ∧I stands for "∧ introduction," DN stands for "double negation," RA stands for "rule of assumption," and RAA stands for "reductio ad absurdum.")¹

      DeMorgan's Laws are two rules of inference that define the relation between negation, disjunction, and conjunction. They are: ~(p v q) ↔ (~p ∧ ~q), which can be read as "the negation of the disjunction of two statements is logically equivalent to the conjunction of their negations," and ~(p ∧ q) ↔ (~p v ~q), which can be read as "the negation of the conjunction of two statements is logically equivalent to the disjunction of their negations."

      Below are sequent proofs for the other side of the first law and both sides of the second. These proofs are formulated by Fergus Duniho in his "Logic Lesson 15: Proving DeMorgan's Theorem with Indirect Proof."²

SHOW: (∼P ∧ ∼Q) ⇒ ∼(P v Q)

1. (∼P ∧ ∼Q) ⇒ (∼P ∧ ∼Q)                         RA
2. (P ∨ Q) ⇒ (P ∨ Q)                                 RA
3. (∼P ∧ ∼Q) ⇒ ∼P                                    1,∧E
4. (∼P ∧ ∼Q) ⇒ ∼Q                                    1,∧E
5. (P ∨ Q), ∼P ⇒ Q                                   3,vE

6. (P v Q), (∼P ∧ ∼Q) ⇒ Q ∧ ∼Q                4,5,∧I
7. (∼P ∧ ∼Q) ⇒ ∼(P v Q)                            RAA

SHOW: ∼(P ∧ Q) ⇒ (∼P ∨ ∼Q)

1. ∼(P ∧ Q) ⇒ ∼(P ∧ Q)                               RA
2. ∼(∼P v ∼Q) ⇒ ∼(∼P v ∼Q)                        RA
3. ∼P ⇒ ∼P                                                  RA
4. ∼P ⇒ ∼P v ∼Q                                         3,vI
5. ∼P, ∼(∼P v ∼Q) ⇒ (∼P v ∼Q) ∧ ∼(∼P v ∼Q)
                                                                   2,4,∧I
6. ∼(∼P v ∼Q) ⇒ P                                       RAA
7. ∼Q ⇒ ∼Q                                                 RA
8. ∼Q ⇒ ∼P v ∼Q                                         8,vI
9. ∼Q,∼(∼P v ∼Q) ⇒ (∼P v ∼Q) ∧ ∼(∼P v ∼Q)
                                                                   2,8,∧I
10. ∼(∼P v ∼Q) ⇒ Q                                    RAA
11. ∼(∼P v ∼Q) ⇒ P ∧ Q                              6,10,∧I
12. ∼(P ∧ Q), ∼(∼P v ∼Q) ⇒ (P ∧ Q) ∧ ∼(P ∧ Q)
                                                                   1,11,∧I
13. ∼(P ∧ Q) ⇒ (∼P v ∼Q)                            RAA

SHOW: (∼P ∨ ∼Q) ⇒ ∼(P ∧ Q)

1. (∼P ∨ ∼Q) ⇒ (∼P ∨ ∼Q)                           RA
2. P ⇒ P                                                     RA
3. (P ∧ Q) ⇒ (P ∧ Q)                                   RA
4. (P ∧ Q) ⇒ P                                             3,∧E
5. (P ∧ Q) ⇒ Q                                            3,∧E
6. P ⇒ ∼∼P                                                  DN
7. (∼P ∨ ∼Q), ∼∼P ⇒ ∼Q                              1,vE
8. (P ∧ Q), (∼P v ∼Q) ⇒ Q ∧ ∼Q                5,7,∧E
9. (∼P v ∼Q) ⇒ ∼(P ∧ Q)                              RAA


FOOTNOTES

¹Theodore Sider, Logic for Philosophy (Oxford: Oxford University Press, 2010), p. 55.

²Fergus Duniho, "Logic Lesson 15: Proving DeMorgan's Theorem with Indirect Proof" (2015), online on YouTube, https://www.youtube.com/watch?v=HqJIoz1lCXE

Chandrakirti's Prasannapada

Chandrakirti (c. 600 - c. 650) was an Indian Madhyamaka Buddhist monk whose writings included commentaries on the teachings of Nagarjuna (c. 150 - c. 250). His Prasannapada ("Clear Words") and Madhyamakavatara ("Introduction to the Middle Way") examined and interpreted Nagarjuna's Mulamadhyamakakarika ("Fundamental Verses on the Middle Way"). 

      His Sunyatasaptativritti ("Commentary on the Seventy Stanzas on Emptiness") and Yuktisastikavritti ("Commentary on the Seventy Stanzas on Reasoning") also examined and interpreted Nagarjuna's teachings.
      In the Prasannapada, Chandrakirti says that Nagarjuna teaches that the true nature of things is that they are neither arising nor perishing, neither coming nor going, neither temporary nor eternal, neither differentiable nor non-differentiable.¹ The true nature of things is that they are without self-existence. 

      Things do not arise spontaneously or independently; rather, they are caused to exist, and they depend on causes and conditions of existence. Thus, they arise through a process of dependent origination (and not spontaneous origination).
      The true nature of things is marked by eight characteristics: not arising independently, not perishing, not coming, not going, not terminating, not enduring eternally, not being differentiable, and not being non-differentiable.
      Nothing is self-caused or arises of itself. Everything is interdependent. The concept of a divine being as someone or something that is self-existent and not caused by anything other than itself is therefore unintelligible.²
      The self-existence of things is illusory. There is no self, and there is no other, so it doesn't make sense to say that things arise from themselves or from what is other than themselves.
      Nothing truly arises at all, insofar as if something exists, it cannot be said to have been brought into existence, because it must have already existed (at least in some respect).
      Everything is unreal, in the sense that all our perceptions of things as existing in their own right are illusory.
      Motion is illusory, in the sense that if something were in motion, then it would have to be conceived of as having already traversed a path of motion or as not yet having traversed a path of motion or as traversing some path that is distinct from what has already or has not yet been traversed. But there is no motion in what has already been traversed or in what has not yet been traversed or in what is somehow distinct from these two alternatives.
      Rest is illusory, in the sense that a mover does not come to rest, nor does a non-mover. Rest cannot be said to be the cessation of motion, if there is no such thing as motion. 
      The sensory faculties of vision, hearing, smell, taste, and touch do not exist, insofar as if they do not see, hear, smell, taste, or touch themselves, then they do not see, hear, smell, taste, or touch other things (because the self and the other don't exist).
      The self as a subject of perception is also illusory, because it neither exists nor does not exist in its own right. All subjects (and all objects) are unreal, because they are not what they seem to be. All things are without self-nature, insofar as they are not self-caused, and insofar as their essential nature is impermanent and changeable. If the essential nature of things were invariable or unchangeable, then their transformation would be impossible, because the alteration of things that continue to exist as they did in their previous state is impossible.³ 
      Material objects don't exist, because matter can't be understood as their cause, and because if they existed apart from matter as their cause, then they would be uncaused, which is impossible (because nothing is ever without a cause). 
      Space doesn't exist, because if it did, then it would have to be a subject of characterization or a characteristic itself, but since neither subjects of characterization nor characteristics exist, space doesn't either.⁴ A subject of characterization is unintelligible without definable characteristics, and since we can't establish the existence of any subject of characterization (because nothing inherently exists or is self-existent), we can't establish the existence of any characteristics either. Thus, space neither exists nor does not exist, because there is nothing of which we can say that it inherently exists or does not exist.
      Time is unreal, insofar as it depends on the self-existence of things. While the present and the future may not be able to be established independently of the past, the past may also not be able to be established independently of the present and the future, so the nature of things that apparently exist independently of each other in the past, present, or future is illusory.
      Emptiness of self-existence is therefore a characteristic of all things (although it neither exists nor does not exist). There are no non-empty things, and there is no state of non-emptiness. All things are empty, and there are no self-existent things; and just as there is no self-existence, there is no other-existence.⁵ 
      Eternalism holds that things exist inherently, and that they never do not exist. Nihilism, on the other hand, holds that things that previously existed can cease to exist.⁶ However, to be entangled in either of these two dogmas is also to be entangled in the realm of samsara (the endless cycle of birth, death, and rebirth).⁷ The Madhyamaka view (the Middle Way) is a path between the two dogmas, and it holds that the existence or non-existence of things is only appearance and not true reality.
      Chandrakirti argues that Madhyamaka is not a form of nihilism, although it holds that nothing is real in itself or has any inherent existence, because it accepts the conventional reality of things for the purposes of the everyday world.
      But even though Madhyamikas (adherents of the Madhyamaka school) accept the conventional reality of things in the everyday world, they recognize the difference between conventional reality and ultimate reality. The ultimate reality of things is that they do not exist in their own right, and that their apparently self-existing nature is illusory. The ultimate reality of things is also that they are neither real nor unreal, insofar as reality is seen (by naive realists) as belonging to things in the everyday world.
      Thus, there are two truths (or distinct kinds of truth): the truth of the everyday world, and the truth of ultimate reality.
      Basic afflictions, such as desire, aversion, and illusion, are causes of suffering, and their eradication leads to release from the realm of samsara. When we extirpate these basic afflictions and understand the true nature of things, we no longer mistake ignorance for true knowledge, or conventional reality for ultimate reality.
      Chandrakirti describes four misbeliefs that lead to illusion: (1) the belief that there is something imperishable in the five perishable factors of personal existence (form, feelings, perceptions, mental formations, and consciousness), (2) the belief that whatever is perishable is subjected to suffering, so happiness rather than suffering can be found in the five factors of personal existence, (3) the belief that the body is pure, and (4) the belief that there is an enduring self among the five perishable factors of personal existence.⁸
      The Four Noble Truths are (1) the truth of suffering, (2) the truth of the origin of suffering, (3) the truth of the cessation of suffering, and (4) the truth of the path to the cessation of suffering. Through the Four Noble Truths, clear knowledge of the nature of afflicted existence is possible, as is understanding of the way to overcome the source of affliction, the acceptance of the way leading to cessation of affliction, and the final realization of liberation.⁹ If the Four Noble Truths did not hold, then none of these stages would be possible.
      Nirvana or release from suffering is attained by the extinction of the afflictions, and by the cessation of the perishable factors of personal existence. It is "neither something that can be extirpated, like desire, nor something that can be realized through action, like the fruit of moral striving...It is only by the dissipation of all named things that it is attained."¹⁰ 
      Jay L. Garfield and Sonam Thakchoe (2025) criticize the view that Chandrakirti is a radical nihilist who denies the possibility of any knowledge, and they instead characterize his epistemological position as a moderate realism about the conventional world. However, they say that Chandrakirti also synthesizes this position with panfictionalism and illusionism. They argue that he believes that ordinary people may be warranted in their beliefs, because even if people are deluded with regard to the mode of existence of phenomena, this position does not entail that they are also deluded with respect to the conventional properties of phenomena, so it is possible for them to obtain valid knowledge of those properties.¹¹ 
      Garfield and Thakchoe also describe Chandrakirti's epistemological position as a pragmatic coherentism, insofar as he sees epistemic practices as recursively self-correcting on the basis of perception, judgment, and reasoning. They deny that his position is a form of global error theory, because he believes that we can distinguish between conventional truth and falsehood.¹² This position also provides a middle way between foundationalism (the view that knowledge is based on foundational or self-evident truths) and relativism (the view that all truths are relative to a person's viewpoint), since conventional truths and ultimate truths aren't taken to be foundational to each other, and since they don't depend on anyone's particular viewpoint.¹³


FOOTNOTES

¹Chandrakirti, Lucid Exposition of the Middle Way: The Essential Chapters from the Prasannapada of Candrakirti, translated from the Sanskrit by Mervyn Sprung (Boulder: Prajna Press, 1979), p. 32.

²Ibid., p. 43.

³Ibid., p. 147.

⁴Ibid., p. 106.

⁵Ibid., p. 157.

⁶Ibid., p. 161.

⁷Ibid., p. 163.

⁸Ibid., pp. 214-215.

⁹Ibid., p. 225.

¹⁰Ibid., p. 249.
¹¹Jay L. Garfield and Sonam Thakchoe, By the Light of the Moon: Candrakirti's Prasangika Madhyamaka (Oxford: Oxford University Press, 2025), p. 50.

¹²Ibid., p. 53.

¹³Ibid., p. 59.

Marcus Steinweg, on The Aporias of Love

Marcus Steinweg, in his book Aporien der Liebe (Aporias of Love, 2010), says that the aporias of love are conditions of its possibility, insofar as love means resistance to narcissistic self-mirroring and openness to otherness. Love is aporetic, because it is both singular and universal, and because it exposes the lovers to the conflict not only between singularity and universality, but also between uniqueness and structuring convention.
      According to Steinweg, there are at least five aporias (insoluble problems) belonging to the experience of love: (1) the aporia of emptiness, (2) the aporia of happiness, (3) the aporia of hyperbolism, (4) the aporia of intensity, and (5) the aporia of the impossible.
      The aporia of emptiness is that the lover feels an emptiness in the absence of the beloved, and yet this absence is also the ground of possibility for another encounter between the lover and the beloved.
      The aporia of happiness is that, as difficult as it is not to expect happiness from love, those who free themselves from this expectation may also be those most likely to be happy.
      The aporia of hyperbolism is that love may become a kind of fever that drives us to reach beyond ourselves, opening us to the presence of the Other. Rather than promoting greater stability, love may take us through peaks and valleys of experience.
      The aporia of intensity is that love may be an experience of something that eludes us and is beyond us and yet is capable of being experienced.
      The aporia of the impossible is that for love, the possible and the impossible (like reality and ideality) are pseudo-alternatives, because every love creates its own aporias. Love is situated in the aporetic compossibility of finitude and infinity, and the aporia is also revealed by the unity in which the finite singularity of lovers encounters the infinite universality of love.¹ 
      Still another aporia is that of the "who" and the "what" of love. Do we love someone because of who they are as a whole person or because of the particular qualities (such as attractiveness and intelligence) that they possess? The blindness of love may consist in our being willing to lose sight of the "what" in order to affirm the "who."² 
      Still another aporia is that to affirm the Other as constitutive of oneself is also to affirm the incompleteness of oneself without the Other, and the incommensurability of oneself with the Other as Other.
      Other aporias of love include those of reason and unreason, presence and absence, and immanence and transcendence.

FOOTNOTES

¹Marcus, Steinweg, Aporien der Liebe (Berlin: Merve Verlag, 2010), p. 15.

²Ibid., p. 38.

Exam Question in Sentential Logic

This semester, I'm taking a class in deductive logic at The University of Maryland, Baltimore County (UMBC). Below is my answer to a question on our second exam (construct a derivation of the conclusion in line 2 from the premise in line 1). The exam covered derivations in sentential logic, based on Gary M. Hardegree's Symbolic Logic: A First Course, second edition (New York: McGraw-Hill, 1999). The annotation to the right of each line shows the justification for its presence in the derivation, with "Pr" standing for premise, "CD" standing for conditional derivation, "As" standing for assumption, "ID" standing for indirect derivation, "DD" standing for direct derivation, and "X" standing for contradiction.

Below is my answer to another question.