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, Pailos, Szmuc): 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. Cobreros, Égré, Ripley and van Rooij proved that Γ ⊨_ST φ holds exactly when Γ ⊨_CL φ, so ST and classical logic have the same valid sequents; the matrices are finite and three-valued, so ST is decidable and its validity problem is coNP-complete, as the classical one is. Reflexivity, weakening and contraction all survive, and cut is the sole structural casualty, failing only where a formula on the cut can take the value ½. Adding a transparent truth predicate then yields a non-trivial theory that conservatively extends classical logic in the truth-free vocabulary (Ripley 2012), which is precisely what the loss of cut secures: the Liar takes ½ and explosion, which required cut, does not follow. The identity with classical logic breaks one level up — Dicher and Paoli (2019) argue that the metainferences ST validates are exactly the inferences valid in LP — and iterating the construction produces the hierarchy STω of Barrio, Pailos and Szmuc. The same semantics validates the tolerance principle, so the sorites is blocked at the point where the Liar is.

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. Cut is not admissible in ST's sequent calculus, so derivations do not compose: a lemma derived from premises cannot be freely used as a premise in a later derivation, and the ordinary practice of proving lemmas and chaining them is not underwritten by the consequence relation. The identity with classical logic is therefore level-relative — every classical inference is ST-valid, not every classical metainference is — so "ST is classical logic" is a claim indexed to a level rather than an unqualified theorem, and the STω hierarchy exists because the qualification cannot be removed at any finite stage. TS, the dual reading, validates nothing at all, which shows how much the choice of standards is doing. And the semantics assigns the Liar ½ without interpreting ½: the third value carries the entire diagnosis and receives no independent explanation, so the account is model-theoretic rather than explanatory.

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