Strict-Tolerant Logic (ST)

Origin. Pablo Cobreros, Paul Égré, David Ripley, and Robert van Rooij, "Tolerant, classical, strict" (Journal of Philosophical Logic, 2012), introduced for vagueness; Ripley, "Paradoxes and failures of cut" (2013) and "Conservatively extending classical logic with transparent truth" (2012), turned it on the semantic paradoxes. The claim that made it notorious: ST is classical logic, and it has a transparent truth predicate.

Models. Strong Kleene's three values, read twice. A premise is asserted strictly — it must take value 1. A conclusion is granted tolerantly — it need only take value ≥ ½. Validity is preservation from strict premises to tolerant conclusions. Every classical sequent is ST-valid, so nothing in the object language is lost; what fails is chaining two valid sequents together.

Formalism.

The three values and two standards: Values {0, ½, 1}, Strong Kleene tables. Strict truth: v(φ) = 1. Tolerant truth: v(φ) ≥ ½.

The four consequence relations: SS = strict premises, strict conclusion = K3 TT = tolerant premises, tolerant conclusion = LP ST = strict premises, tolerant conclusion = classical logic TS = tolerant premises, strict conclusion = empty (no valid sequents)

The result (Cobreros–Égré–Ripley–van Rooij): Γ ⊨_ST φ iff Γ ⊨_CL φ ST and classical logic have exactly the same valid sequents. So ST is not a weakening of classical logic at the object level. It is classical logic.

Where the difference is: Cut fails. ⊨_ST λ ∨ ¬λ (classical, so ST-valid) λ ∧ ¬λ ⊨_ST ⊥ (classical, so ST-valid) but the composition is blocked. A sequent may be ST-valid with a ½ formula on both sides, and two such sequents do not compose.

Transparent truth: Add T with T⌜φ⌝ and φ intersubstitutable everywhere. ST + transparent T is a conservative extension of classical logic (Ripley 2012). The Liar takes ½ and nothing explodes — because the explosion needed cut.

The metainferential hierarchy (Barrio, Rosenblatt, Pailos): ST validates every classical inference and not every classical metainference (cut). ST/ST validates the classical metainferences and not the meta-metainferences. Iterating gives a hierarchy STω whose fixed point is classical at every level. Whether "being classical logic" means the inferences or the whole hierarchy is the live dispute.

Symbols.

Metatheory. ST's identity with classical logic at the inference level, and its divergence at the metainference level, is the result the whole literature turns on: it locates the cost of a transparent truth predicate precisely, in cut, and nowhere else. The metainferential hierarchy is the counterattack — Barrio, Rosenblatt, and Pailos argue that a logic is its metainferences too, in which case ST is not classical after all, and Ripley's reply is that the hierarchy's fixed point is. The same apparatus handles vagueness: tolerance principles are ST-valid, and the sorites is blocked at the same joint the Liar is, which is the strongest evidence that the diagnosis is not ad hoc.

Applies to. Semantic paradox with a transparent truth predicate. Vagueness and the sorites, where the tolerance principle is recovered. The metainferential turn in philosophical logic. Substructural approaches to paradox generally, alongside the noncontractive treatments of Curry.

Limitations. Whether a nontransitive consequence relation deserves the name is the standing objection: chaining inferences is what consequence is for, and a logician who cannot compose two valid arguments has been sold classical logic and given something else. The metainferential hierarchy makes "ST is classical" a claim indexed by level, which drains it. And ST's treatment of the Liar assigns it ½ without saying what ½ means — the three values do the work and the semantics does not interpret them, which is Kripke's problem inherited rather than solved.

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