This blog post formalizes my conception of Stoicism using propositional modal logic.
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| This post examines how Stoicism interacts with the assumptions of Psychoanalysis and Behaviorism. Pictured in the illustration are the representative figures of these ideas: respectively, Seneca the Younger (center), Sigmund Freud (left), and B.F. Skinner (right). Of course, the post examines these ideas in a rather abstract sense that might not be attributable to these authors. |
- Syntax and semantics
- Axioms for modal logic and Stoicism, and their philosophical justification
- Theorems and proofs (P+R+K+T fragment)
- Psychological methodology axioms and their philosophical justification
- Why a Stoic can endorse Psychoanalysis (Ps)
- Why a Stoic can endorse Behaviorism (Bh)
- How to read the choice
- Theorems and proofs (Ps/Bh fragment)
Syntax and semantics
The language is first-order propositional modal logic. The definition ◇φ ≝ ¬□¬φ is used freely.
The language is interpreted under a novel way, which once explained, will justify the use of axioms K and T. Although in other contexts I may endorse the use of S5 and the kitchen sink, other axioms were omitted for economy and because I’m not always sure how to interpret the theorems which they would prove.
The interpretation is that all propositions are the attribution of a predicate to a given observed action. The propositions of interest are “the action is Emotional” (E) and “the action is morally Bad” (B). Stoicism asserts a relationship between these attributions which will be explained, justified, and worked out in detail. To aid this explanation, the modal operators are introduced, interpreted in a unique way. Given an action-attribution A, when A was attributed to an action in an observable uncontroversial way, we say □A: for instance, crying is necessarily emotional (□E), and murder is necessarily bad (□B). To say A ∧ ¬□A is interpreted as saying that the action was inwardly emotional although it did not show it (E ∧ ¬□E), or that the action was morally moved by bad motives although it was not observably a bad action (B ∧ ¬□B). The assertions of Stoicism are then clearly stated as the axioms P and R, suggestively named Passion and Reason, which are explained and justified below.
Axioms for modal logic and Stoicism, and their philosophical justification
Classical propositional logic is assumed, and appeals to its tautologies (contraposition, explosion, ∧-intro, etc.) are called PC in proofs.
Modal axiom schemes
K: □(p→q)→(□p→□q)
T: □p→p
T should clearly be true under my interpretation of the modal operator: if something is manifest, then it is (actually) the case. K means manifestness respects implication, which also seems plausible enough.
Rules of inference
US, Uniform Substitution: ⊢ φ ⇒ ⊢ σ(φ)
MP, Modus Ponens: α, α → β ⊢ β
N, Necessitation: ⊢ φ ⇒ ⊢ □φ
The use of Necessitation may seem too strong for an observability operator, but as I see it, whoever believes the axioms will believe them to be analytic, part simply of the meaning of emotionality and badness attributions.
Stoic axioms
Stoicism consists in the axioms P and R:
P, Passion: □E → □B
R, Reason: B → E
These axioms are meant to formalize the informal notion of Stoicism, which is characterized as the belief that actions are morally bad if, and only if, they are moved by Passion rather than Reason. The axioms are justified below.
Justification of Stoic axioms
The justification will employ some propositions which will not ultimately be kept as axioms or mentioned in theorems or proofs. For instance, to justify the formalization, it seems helpful to mention a proposition meaning “the action was moved by Reason”, which, since the letter R is already taken by our axiom R which we intend to justify, I will gloss as “the action was Wise” and abbreviate to W. Then Stoicism clearly supports the Dichotomy:
D, Dichotomy: E ∨ W
The Dichotomy says that every action is moved either by emotion, or by reason, or both (inclusive disjunction). The Dichotomy is not yet a full statement of Stoicism, it just asserts one of its doctrines: besides emotion and reason, there is no third type of motive that can be attributed to an action. (For instance, if we attribute the action to demonic possession, then the true agent was the demon, and we again have the question whether the demon acted rationally or emotionally, or both.) Passion is, if not equivalent to emotion, at least a kind of emotion; if someone believes in “rational emotions”, they need not object yet, since E ∧ W is still possible under this axiom. A fuller statement of Stoicism is added by TR:
TR, Thick Rationality: W→¬B
As the gloss that called a reason-moved action Wise had already hinted, if a Stoic has attributed an action to Reason, then he does not believe that it was morally Bad. The Stoic believes in “thick rationality”: if an action correctly chose the means to pursue the agent’s ends, but the agent’s ends were not themselves good ends, then the agent was irrational in his choice of ends, and his action was irrational as a result.
But from D and TR we have our Axiom R.
Proof: From TR (W→¬B) take the contrapositive to get B→¬W; assume B, then ¬W. From D (E∨W) and ¬W, Disjunctive Syllogism yields E. Thus from B we derived E, so by →-introduction B→E, i.e., R. ∎
Axiom R represents that if a Stoic has called an action morally Bad, then he must attribute it to Emotion, since Reason is simply not available for Bad actions. A similar informal consideration supports P: if a Stoic can conclusively call an action Emotional just from observing it, this must be because he has already ruled out Reason as a motive for it, which means that he is looking at a manifestly morally Bad action. After all, to suppose that we ever have □E ∧ □W is to suppose that there exist observably overdetermined actions, where both motives are observably attributed to an action even though either motive would be, under Stoicism, sufficient to explain it. So now we have R and P both justified by our informal beliefs about Stoicism, as desired.
A Stoic who believes further that all emotions are Passions might also endorse the converse of R, i.e., E→B. This collapses all emotion-attributions into badness-attributions. To keep our assumptions light, we are not assuming this here. In the section on psychological methodology, it will follow as a theorem (part of T10) when Behaviorism, my preferred psychological theory, is assumed.
Theorems and proofs (P+R+K+T fragment)
Some lemmas are used to speed up proofs.
Lemmas (P+R+K+T fragment)
L1 (Converse Passion). ⊢ □B → □E
Proof:
- ⊢ B → E (R)
- ⊢ □(B → E) (1, N)
- ⊢ □B → □E (2, K) ■
L2 (Box Equivalence). ⊢ □E ↔ □B
Proof: From P: □E→□B and L1: □B→□E by PC obtain the biconditional. ■
L3 (Reason—Contraposition). ⊢ ¬E → ¬B
Proof: From R: B→E by PC (contraposition). ■
L4 (Necessitated Contraposition). ⊢ □¬E → □¬B
Proof:
- ⊢ ¬E → ¬B (L3)
- ⊢ □(¬E → ¬B) (1, N)
- ⊢ □¬E → □¬B (2, K) ■
Theorems (P+R+K+T fragment)
T1 (Possible Badness Requires Possible Emotion). ⊢ ◇B → ◇E.
Proof: ◇B → ◇E ≡ (¬□¬B → ¬□¬E) which is the contrapositive of L4 (□¬E→□¬B). PC gives the contrapositive, then rewrite with ◇. ■
Remark: The converse, ◇E → ◇B, is not derivable. It has a countermodel: M = (W,R,V) with W = {w0,w1}, R = W × W (so ∀x∀y xRy), V(E) = {w1} and V(B) = ∅ – hence M, w0 ⊨ ◇E ∧ ¬◇B.
T2 (Possibly Dispassionate Just When Possibly Innocent). ⊢ ◇¬E ↔ ◇¬B.
Proof: By L2, ⊢ □E ↔ □B, hence by PC ⊢ ¬□E ↔ ¬□B, i.e. ⊢ ◇¬E ↔ ◇¬B. ■
T3 (Manifest Passion Condemns). ⊢ □E → B.
Proof: P gives □E→□B; T (with p:=B) gives □B→B; compose by PC: □E→B. ■
T4 (Manifest Vice Reveals Passion). ⊢ □B → E.
Proof: L1 gives □B→□E; T (with p:=E) gives □E→E; compose by PC: □B→E. ■
T5 (Emotion May Require Public Badness). ⊢ ◇(E → □B).
Proof: Let A := □(E ∧ ¬□B).
- Tautology t1: (E∧¬□B)→E. By N and K: A → □E. With P: □E→□B, so A → □B.
- Tautology t2: (E∧¬□B)→¬□B. By N and K: A → □¬□B. By T (p:=□B): □¬□B→¬□B, so A → ¬□B.
- From (A→□B) and (A→¬□B), PC yields ¬A. Thus ¬□(E∧¬□B). Now, ¬□(E∧¬□B) ≡ ◇¬(E∧¬□B) and by PC/US, ¬(E∧¬□B) ≡ (E→□B). Hence ◇(E→□B). ■
T6 (Badness May Require Public Emotion). ⊢ ◇(B → □E).
Proof: Mirror of T5 using L1 instead of P. Let A′ := □(B ∧ ¬□E).
- (B∧¬□E)→B ⇒ A′→□B; with L1: □B→□E, so A′→□E.
- (B∧¬□E)→¬□E ⇒ A′→□¬□E ⇒ (with T, p:=□E) A′→¬□E.
- PC: (A′→□E) & (A′→¬□E) ⇒ ¬A′. Thus ¬□(B∧¬□E) ≡ ◇¬(B∧¬□E) ≡ ◇(B→□E). ■
T7 (Cold Vice Impossible). ⊢ ¬◇(B ∧ ¬E).
Proof: By N on R: ⊢ □(B→E). Using PC/US, (B→E) ≡ ¬(B∧¬E); by N+K this yields ⊢ □¬(B∧¬E), i.e. ⊢ ¬◇(B∧¬E). ■
T8 (Covert Vice Requires Covert Passion). ⊢ (B ∧ ¬□B) → (E ∧ ¬□E).
Proof: From R we have B→E, so from B infer E. From P we have □E→□B; by PC (contraposition) obtain ¬□B→¬□E, so from ¬□B infer ¬□E. Conjoin by PC to get (E ∧ ¬□E). ■
Psychological methodology axioms and their philosophical justification
The Stoic core (P and R) is deliberately neutral about how inner attributions (E, B) are evidenced. The two “methodology” schemata say what you must think about the relation between inner states and their public marks if you want to use AMS as a model of inquiry into character:
Ps, Psychoanalysis: ◇¬(E→□E) ∧ ◇¬(B→□B)
It is possible for Emotion and Badness to be present without being manifest.
Bh, Behaviorism: □(E→□E) ∧ □(B→□B)
Necessarily, if Emotion or Badness are present, they are manifest (in principle) as such.
These are contraries rather than contradictories because our evidential situation could, in principle, be mixed or indeterminate.
Why a Stoic can endorse Psychoanalysis (Ps)
On the present reading, □ is an observability operator: □A means there are uncontroversial, public criteria that settle A for the action at hand. Ps says only that sometimes those criteria underdetermine the truth about motives and moral quality. This is a modest and very Stoic-friendly thesis.
Opacity of assent. For Stoics, the moral core of action lies in assent—the inner commitment that makes an impulse ours. Assent can be swift, conflicted, or suppressed without leaving unmistakable behavioral residue. One can act with a Stoic face while inwardly submitting to a turbulent evaluation; conversely, one can perform an outwardly admirable deed from a vicious maxim. That is exactly the content of ◇(E ∧ ¬□E) and ◇(B ∧ ¬□B): sometimes the motivational reality outruns what observers (or even the agent) can certify.
Self-deception and social masking. Agents can misconstrue or conceal their motives—for status, harmony, or fear. The philanthropist moved chiefly by spite toward a rival, the polite insult delivered with a smile, the calm saboteur who harms by omission: each illustrates how inner Emotion or Badness can be present though the public checklists do not light up.
Methodological humility. Ps is not a metaphysical posit about hidden homunculi; it is a research policy: do not assume that the public marks exhaust the moral or emotional facts. That policy fits R (Bad → Emotional) and P (□Emotional → □Bad): the Stoic linkage between passion and vice can hold even when the passion or the vice do not surface in an uncontroversial way. In the Ps regime you therefore expect, and methodologically allow for, cases of inward passion without manifest signs and inward vice behind respectable conduct. (This is precisely what our theorems T11 and T12 will record at the modal level.)
Why a Stoic can endorse Behaviorism (Bh)
Bh takes a different tack: it operationalizes the vocabulary. It says, in effect, “we will only count as ‘Emotional’ or ‘Bad’ those attributions that admit public criteria strong enough to settle them.” Under this stance, E and B are not hidden properties to which behavior gives fallible clues; they are roles defined by their outward criteria. Hence □(E→□E) and □(B→□B) are analytic of how we are choosing to use the words in inquiry.
Publicness and accountability. If moral and emotional classifications are to guide law, pedagogy, or communal correction, they must rest on intersubjectively checkable marks. “Hidden badness” that leaves no behavioral trace becomes, on this policy, a misuse of Bad rather than a genuine counterexample. This secures clean evidential pipelines: attributing E or B already licenses—and indeed collapses into—their manifestability (hence our theorem T9: E↔□E and B↔□B, even necessarily so).
Guardrails against mind-reading. By building □ into the very use of E and B, Bh avoids speculative diagnostics about motives. That is a virtue if you worry that the Stoic critique of passion otherwise tempts us to over-interpret ordinary actions. Combined with P and R, this yields a sharp, behaviorally anchored ethics: necessarily, what counts as Emotional counts as Bad, and conversely (our theorem T10). The theory becomes a calibration norm for observers rather than a depth psychology.
Not metaphysics, but operational choice. Bh does not deny that people have inner lives; it declares that for the purposes of this theory’s attributions, only what can, in principle, be publicly settled will count. The box thus records a convention about evidence, not a claim that private states do not exist.
How to read the choice
Both axioms are live options because they address method, not the Stoic core:
- Ps prioritizes explanatory depth: it leaves room for inner assent and vice to outrun public marks, aligning with the Stoic focus on the hegemonikon (the ruling faculty). Expect hidden passion/vice to be possible, and treat attributions as defeasible by deeper interpretation.
- Bh prioritizes intersubjective reliability: it identifies moral and emotional predicates with their public criteria, making the theory action-guiding, teachable, and legally tractable, at the cost of collapsing the inner/outer distinction for E and B.
Either stance is consistent with P and R; they are incompatible with each other because one builds in underdetermination while the other bans it. In that sense AMS functions as a core Stoic logic that can be coupled with a hermeneutic (Ps) or an operational (Bh) methodology, depending on what you want your theory to do.
Theorems and proofs (Ps/Bh fragment)
Lemmas (Ps/Bh fragment)
L5 (Box–Conjunction). ⊢ (□A ∧ □B) → □(A ∧ B)
Proof: Tautology t: A→(B→A∧B). By N: □t.
By K twice: from □A get □(B→A∧B), then with □B get □(A∧B). ■
L6 (◇–Monotonicity). ⊢ □(A→B) → (◇A → ◇B)
Proof: From PC: (A→B)→(¬B→¬A). By N and K: □(A→B)→□(¬B→¬A)→(□¬B→□¬A).
PC (contraposition): □(A→B)→(¬□¬A→¬□¬B), i.e. □(A→B)→(◇A→◇B). ■
(Recall L2: ⊢ □E ↔ □B, hence by PC ⊢ ¬□E ↔ ¬□B.)
Theorems (Ps/Bh fragment)
T9 (Behaviorist Transparency). Bh ⊢ □(E↔□E) and Bh ⊢ □(B↔□B); hence Bh ⊢ (E↔□E) and (B↔□B).
Proof: E-part:
- From T: ⊢ □E→E, so by N: ⊢ □(□E→E).
- Bh gives □(E→□E).
- By L5: □[(E→□E) ∧ (□E→E)].
- PC tautology: [(E→□E)∧(□E→E)]→(E↔□E); by N+K and MP: □(E↔□E).
B-part is identical with E/B swapped.
By T on each: □(… )→(… ), so Bh ⊢ (E↔□E) and (B↔□B). ■
T10 (Behaviorist Collapse). Bh ⊢ □(E↔B) (hence Bh ⊢ E↔B).
Proof:
From L2 and N: ⊢ □(□E↔□B).
From T9: Bh ⊢ □(E↔□E) and □(B↔□B).
PC tautology (transitivity): (X↔Y)∧(Y↔Z)→(X↔Z).
- With X:=E, Y:=□E, Z:=□B, by N+K and L5: □(E↔□B).
- With X:=E, Y:=□B, Z:=B, again N+K and L5: □(E↔B). Finally T yields the unboxed E↔B. ■
T11 (Psychoanalytic Opacity). Ps ⊢ ◇E ∧ ◇¬□E and Ps ⊢ ◇B ∧ ◇¬□B.
Proof: E-part:
- PC: ¬(E→□E) ↔ (E∧¬□E) gives ⊢ ¬(E→□E)→(E∧¬□E); by N and L6: ◇¬(E→□E)→◇(E∧¬□E).
- PC: (E∧¬□E)→E and (E∧¬□E)→¬□E; by N and L6: ◇(E∧¬□E)→◇E and ◇(E∧¬□E)→◇¬□E.
- From Ps: ◇¬(E→□E); chain the arrows to get ◇E ∧ ◇¬□E.
T12 (Psychoanalytic Cross-Opacity). Ps ⊢ ◇(E ∧ ¬□B) and Ps ⊢ ◇(B ∧ ¬□E).
Proof: First show the E-claim.
- From L2 and PC: ⊢ ¬□E→¬□B. Hence ⊢ (E∧¬□E)→(E∧¬□B) (PC).
- By N and L6: ◇(E∧¬□E)→◇(E∧¬□B).
- From T11 (under Ps): ◇(E∧¬□E). So Ps ⊢ ◇(E∧¬□B).
The B-claim is symmetric using ⊢ ¬□B→¬□E. ■
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