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."

      If the deontic operator specifies "having a defeasible reason for," and it is embedded in the consequent, then we have "If you believe A and B, then you have reason to believe C" or "If you believe A and B, then you have reason 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.

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.

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.