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 smallest number of valid formulas and the greatest number of potentially falsifying models. S-5 is the strongest system, because it has the greatest number of valid formulas, and the smallest number of 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 (2001) explains 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). Such a class of frames is called irreflexive, because the absence of infinite ascending sequences means that no world can relate to itself, since a self-relation could create an infinite cycle and diverge 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.       

      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/  

John MacFarlane, on the Normativity of Logic for Thought

John McFarlane (2004) asks, "In what sense (if any) is logic normative for thought?" He explains that it is surprisingly difficult to say how facts about the validity of logical arguments relate to norms of reasoning, but he offers some bridge principles in order to show how claims about logical validity may be connected to norms of belief.

      The bridge principles each take the form of a conditional statement, "If A,B ⊨ C, then..." 

      If the deontic operator specifies strict obligation, and it is embedded in the consequent, then we have "If you believe A and B, then you ought to believe C" or "If you believe A and B, then you ought not to disbelieve C."

      If the deontic operator specifies permission, and it is embedded in the consequent, then we have "If you believe A and B, then you may believe C" or "If you believe A and B, then you are permitted not to disbelieve C."

      On the other hand, if the deontic operator specifies strict obligation, and it is embedded in both the antecedent and the consequent, then we have "If you ought to believe A and B, then you ought to believe C" or "If you ought to believe A and B, then you ought not to disbelieve C."

      If the deontic operator specifies permission, and it is embedded in both the antecedent and the consequent, then we have "If you may believe A and B, then you may believe C" or "If you may believe A and B, then you are permitted not to disbelieve C."

      If the deontic operator specifies strict obligation, and it scopes over the whole conditional, then we have "You ought to see to it that if you believe A and B, you believe C" or "You ought to see to it that if you believe A and B, you do not disbelieve C."

      In these various bridge principles, the strength of the deontic operator may vary, according to whether obligation, permission, or having a defeasible reason to have a particular belief is being proposed.

      Each of these bridge principles may also be governed by three parameters: (1) the type of deontic operator (according to whether an obligation, permission, or reason to have a particular belief is being proposed), (2) the polarity of the directive to conform to a norm of belief (according to whether an obligation, permission, or reason to believe or merely not to disbelieve is being proposed), and (3) the scope of the deontic operator (according to whether it governs only the consequent of the conditional, both the antecedent and the consequent, or the whole conditional).

      In MacFarlane's system, class C bridge principles have the deontic operator embedded in the consequent, class B have the deontic operator embedded in both the antecedent and the consequent, and class W have the deontic operator scoping over the whole conditional. Each of these three classes of bridge principles include obligations, permissions, or defeasible reasons to believe or not to disbelieve a proposition C, based on believing propositions A and B.

      McFarlane criticizes the class C bridge principles by saying that beliefs are not self-justifying, in the sense that if you believe A, then you ought to or may believe A. There is in fact no reason to say that you are entitled to believe A simply because you believe A.

      He criticizes class B bridge principles by saying that they don't account for cases in which we wrongly believe the premises of an argument, and thus they are only normative when our beliefs are already in order, i.e. when we already believe what we ought to believe or may believe or have reason to believe.

      He critically evaluates the class W bridge principles by examining whether they are hindered by (1) excessive demands, (2) the paradox of the preface, (3) the strictness test, (4) the priority question, or (5) logical obtuseness.

      Excessive demands require that if we believe a set of axioms, then we must believe all the consequences of those axioms, no matter how infinite they may be.

      The paradox of the preface requires us not to believe in a conjunction of claims if we think one of them might be false, even when we believe in all the rest of them.

      The strictness test requires us to acknowledge that if the relation between believing p and believing q is strict, then this strictness should be reflected in the proposed bridge principle that links them.

      The priority question (regarding the normativity of logic for thought) is whether we are subject to logical norms only insofar as we have logical knowledge. It may be argued that if we are ignorant of the logical consequences of our beliefs, then we may be more free to believe whatever we please. But MacFarlane rejects this argument, saying that one of the reasons we seek logical knowledge is to learn how to revise our beliefs, even if we are in a state of ignorance.

      Logical obtuseness may be exemplified by our believing A and also believing B, but nevertheless refusing to take a stand on believing their conjunction, A and B. Thus, the bridge principle "You ought to see to it that if you believe A and you believe B, you do not disbelieve C" seems too weak in its deontic modality.

      Conflicting obligations may pose a problem for wide-scope (class W) bridge principles if there are conflicting evidential justifications for believing  or disbelieving a given proposition. For example, if premises A and B are reasons for believing C, but premises D and E are reasons for disbelieving C, then we can't simultaneously see to it that if we believe A and B, we believe C, while also seeing to it that if we believe D and E, we disbelieve C.

      The applications of the bridge principles (as explanations for the connection between logical validity and norms of belief) may also depend on the relevance, necessity, and formality of the inferences that are derived from various sets of premises.

      Relevance requires that the premises of an argument be relevant to the conclusion. Thus, the contradictory premises that Frank is six feet tall and that Frank is not six feet tall do not logically imply, from a relevantist standpoint, that Lisa has a pet lizard.

      Necessity requires that an argument be necessarily truth preserving (that a false conclusion cannot be derived from true premises if the argument is valid).

     Formality requires that an argument be truth preserving by virtue of form. If an argument is valid, then its form ensures that if the premises are true, the conclusion is also true.

     As an example of the connection between formality and transparency, McFarlane offers the following syllogism: (1) Cicero talked the talk, (2) Tully walked the walk, (3) Someone walked the walk and talked the talk. 

      If we didn't know that Cicero and Tully were the same person, then we wouldn't be able to discern the necessary connection between (1), (2), and (3). If the syllogism were: (1) Cicero talked the talk, (2) Cicero walked the walk, and (3) Someone walked the walk and talked the talk, then its logical validity would be much more transparent. The importance of transparency in such cases, according to MacFarlane, is due to the normative implications of logical validity.¹ The formal structure of an argument must be easily recognizable if its logical validity, and therefore its normative implications, are to be transparent.

      In order to avoid or remedy the transparency problem, McFarlane provides another bridge principle, based on the presupposition that the normativity of logic has it source not in the formal validity of inferences, but in the formal validity of inference schemata:

Bridge Principle: "If [you know that] the schema S is formally valid and you apprehend the inference A, B / C as an instance of S, then you ought to see to it that if you believe A and B, you believe C."²

      He concludes that by systematically exploring such bridge principles we may come to a better understanding of the relation between logical validity and norms of reasoning.

      In another unpublished paper, entitled "Is Logic a Normative Discipline?" (2017), MacFarlane distinguishes between the weak and strong senses of normativity. He says, 

"A discipline is normative in the weak sense iff one can derive normative claims about the disciplines's subject matter from the principle of the discipline plus some true normative claims that are not part of the discipline."³
"A discipline is normative in the strong sense iff some of its fundamental principles are explicitly normative or evaluative, or are reducible to explicitly normative or evaluative terms."⁴

He admits that it may be difficult to maintain that logic is normative in the strong sense, but he says that if we analyze intertheoretic validity in normative terms by saying, for example, that "An inference form is valid just in case, for every instance with premises P₁... P_n and conclusion Q, one ought not to believe P₁... P_n without believing Q," then such an analysis may vindicate the claim that logic is normative in the strong sense and demonstrate that claims about validity may be reduced to normative claims.⁵

      Philosophers who have described logic as a normative discipline include Kant, Frege, Peirce, MacFarlane, Florian Steinberger, and Hartry Field.

      Kant says in the Logic (1800) that logic is a science of the necessary laws of thinking, and that in logic the question is not how we think, but how we ought to think.⁶ 

       Kant also says, in the Critique of Pure Reason (1781), that logic may be twofold, insofar as it may involve the general or particular understanding. Logic of the general understanding contains the necessary laws of thought, while logic of the particular understanding contains laws of correct thinking upon a particular class of objects. Pure general logic is concerned with pure a priori principles, while applied general logic is concerned with the laws of the use of the understanding.⁷ 

      Frege (1893) says that logic, like ethics, is a normative science, and that laws of logic prescribe the way in which we ought to think.⁸

      Peirce (1903) divides the normative sciences into logic, ethics, and aesthetics. Logic is the science of the laws of signs, while ethics is the science of right and wrong, and aesthetics is the science of ideals. Logic may be divided into (1) speculative grammar (the theory of the nature and meaning of signs), (2) critical logic (the theory of arguments and their validity), and (3) methodeutic (the study of the methods that should be pursued in the investigation, exposition, and application of truth). Logic is normative, because it is a theory of the kind of reasoning that should be employed in order to discover truth.⁹

      Florian Steinberger (2019) describes three normative functions that logic may have and that may be conflated: (1) it may provide directives, (2) it may offer evaluations, and (3) it may present appraisals. Directives guide us as to what to do, choose, or believe. Evaluations provide standards against which to assess acts or states as good or bad, correct, or incorrect, etc. Appraisals provide the basis for attributions of praise or blame to agents.¹⁰ Bridge principles that connect claims about logical validity to norms of belief must therefore be assessed according to the particular normative role they are intended to perform.

      Hartry Field (2009), in addressing the question "What is the normative role of logic?", also asks "What is the connection between (deductive) logic and rationality?" He explains that, rather than assuming a position of normative realism in which logic imposes objective normative constraints upon our reasoning, we might instead look at what it means for us to employ a logic in terms of the norms that we follow--norms that govern our beliefs by directing that those beliefs be in accord with the inference rules provided by logic.¹¹

      Steinberger (2022) explains that logical pluralism is a theory that there is more than one correct logic, while logical monism is a theory that there is only one correct logic. According to logical pluralists, there is no absolute sense, but only system-relative senses, in which logic can normatively bind us. "Pluralism about logic thus seems to give rise to pluralism about logical normativity: If there are several equally legitimate consequence relations, there are also several equally legitimate sets of logical norms. Consequently, it is hard to see prima facie how substantive conflicts can arise."¹²

      However, in what Steinberger has called "Harman's skeptical challenge" to the normativity of logic, Gilbert Harman (1984) has argued that logical principles are not directly rules of belief revision.¹³ Logical principles hold universally and without exception, and thus they differ from principles of belief revision, which are at best prima facie principles.¹⁴ Furthermore, a rational person may reasonably believe that at least one of his or her beliefs is false, and thus they may be prepared to admit that there is some inconsistency in some of their beliefs.¹⁵ Thus, logical principles are different from rules of practical reasoning.

      Gillian Russell (2020) argues that logic isn't normative, and that if it were, then there would be a question about how logical pluralism could be maintained, since different notions of logical validity could lead to different (and possibly conflicting) notions of its normative consequences. According to Russell, logic is the study of patterns of truth-preservation on truth-bearers.¹⁶ Logic's apparent normative consequences are merely the result of background norms about the relations between belief, reasoning, and truth, and they do not demonstrate that logic has its own normativity. Logical pluralists therefore needn't worry about objections to pluralism that are based on the supposed normativity of logic, because logic isn't normative.¹⁷ 

FOOTNOTES

¹John McFarlane, "In What Sense (If Any) Is Logic Normative for Thought?", 2004, p. 20, online at https://johnmacfarlane.net/normativity_of_logic.pdf.

²Ibid., p. 22.

³John MacFarlane, "Is Logic A Normative Discipline?" (2017), p. 2, online at 

https://johnmacfarlane.net/normative.pdf

⁴Ibid., p. 3.

⁵Ibid., p. 6.

⁶Immanuel Kant, Logic: A Manual for Lectures [Logik: ein Handbuch zu Vorlesungen, 1800], translated by Robert S. Hartman and Wolfgang Schwarz (Indianapolis: Bobbs-Merrill Company, 1974) p. 16.

⁷Immanuel Kant, Critique of Pure Reason [Kritik der reinen Vernunft, 1781], translated by J.M.D. Meiklejohn (Amherst, NY: Prometheus Books, 1990), p. 45.

⁸Gottlob Frege, Basic Laws of Arithmetic [Grundgesetze der Arithmetik, 1893], translated by Montgomery Furth (Berkeley: University of California Press, 1967), p. 12.

⁹Charles S. Peirce, Collected Papers of Charles Sanders Peirce, Volume I, paragraph 191, 1903 (Cambridge: Harvard University Press, 1960), p. 79.

¹⁰Florian Steinberger, "Three Ways In Which Logic Might Be Normative," in The Journal of Philosophy, Vol 116, No 1, January 2019), p. 16.

¹¹Hartry Field, "What is the Normative Role of Logic?", in Proceedings of the Aristotelian Society, Supplementary Volume 83 (1), 2009, p. 262.

¹²Florian Steinberger, "The Normative Status of Logic," in Stanford Encyclopedia of Philosophy, 2022, online at 

https://plato.stanford.edu/archives/spr2025/entries/logic-normative/

¹³Gilbert Harman, "Logic and Reasoning," in Synthese, Vol. 60, No 1, July 1984, p. 107.

¹⁴Ibid., pp. 107-108.

¹⁵Ibid., p. 109.

¹⁶ Gillian Russell, "Logic isn't normative," in Inquiry, Volume 63, Issue 3-4, 2020, p 382.

¹⁷Ibid., p. 387.

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.