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A Formal Framework for Higher-Order Vagueness: Extending Paraconsistent Fuzzy Logic with Multi-Dimensional Truth Values
We propose PFL^+, a formal logical framework extending Paraconsistent Fuzzy Logic to address higher-order vagueness. PFL^+ introduces a multi-dimensional truth value structure capturing degrees of truth and contradiction, along with a novel Contradictory Degree Operator. This enables rigorous modeling of vagueness and contradictions. Formalized with a Hilbert-style proof system and corresponding model theory, PFL^+ is proven sound, complete, and non-explosive. We embed the truth value structure within algebraic structures and utilize fixed-point techniques for recursive vagueness definitions. A computational complexity analysis demonstrates PFL^+'s efficiency. We showcase its ability to resolve classic logical paradoxes and apply it to real-world problems in NLP, decision support, and AI. This work advances formal logic, impacting both theory and practice.
Vagueness, inherent in predicates like "tall," "heap," or "bald," challenges logical modeling. The Sorites Paradox highlights the limitations of classical logic in handling such predicates. Higher-order vagueness, where the vagueness itself is vague, introduces the possibility of infinite regress. For example, "relatively tall" is vague, but so is the notion of "relatively" itself—how much taller must someone be to be relatively tall? Addressing this requires rigorous formal tools.
- Classical Logic: Cannot represent degrees of truth or contradictions.
- Fuzzy Logic: Handles degrees of truth but lacks robust contradiction handling and higher-order vagueness mechanisms.
- Paraconsistent Logic: Manages contradictions but not degrees of truth.
- Existing Paraconsistent Fuzzy Logics: While combining contradictions and degrees of truth, they lack the mathematical rigor for higher-order vagueness.
PFL^+ enhances Paraconsistent Fuzzy Logic with:
- Multi-dimensional truth values: Capturing degrees of truth and contradiction.
- Contradictory Degree Operator: Quantifying contradictions rigorously.
- Recursive vagueness modeling: Employing fixed-point theory.
PFL^+ uses a Hilbert-style proof system, guaranteeing soundness, completeness, and non-explosiveness. A computational complexity analysis is also provided.
- Section 2: Formal structure of PFL^+.
- Section 3: Recursive modeling of higher-order vagueness.
- Section 4: Hilbert-style deductive system.
- Section 5: Resolution of paradoxes with PFL^+.
- Section 6: Computational complexity analysis.
- Section 7: Comparative analysis and extensions.
- Section 8: Real-world applications.
- Section 9: Conclusion and future work.
- Appendix A: Formal proofs.
The truth value space V ⊆ [0, 1] × [0, 1], with truth values (t, c):
- t ∈ [0, 1]: Degree of truth.
- c ∈ [0, 1]: Degree of contradiction.
- Axiom 1 (Non-Negativity): t ≥ 0 and c ≥ 0.
- Axiom 2 (Boundedness): t ≤ 1 and c ≤ 1.
- Axiom 3 (Zero Contradiction): If t = 1 or t = 0, then c = 0.
- Axiom 4 (Contradiction Relationship): c is a function of t, denoted c(t). A generalized sigmoid function, c(t) = 1 / (1 + exp(-k(t - 0.5))), where k controls steepness, offers flexibility. Section 7 discusses relaxing this axiom.
(V, ≤) forms a bounded distributive lattice:
- (t₁, c₁) ≤ (t₂, c₂) iff t₁ ≤ t₂ and c₁ ≤ c₂.
- (t₁, c₁) ∧ (t₂, c₂) = (min(t₁, t₂), max(c₁, c₂))
- (t₁, c₁) ∨ (t₂, c₂) = (max(t₁, t₂), max(c₁, c₂))
Proof (Lattice Structure):
- Associativity: Holds due to associativity of min and max.
- Commutativity: Holds due to commutativity of min and max.
- Absorption and Idempotence: Both are satisfied under min and max operations.
- Boundedness: The elements (0,0) and (1,0) act as the bottom and top elements, respectively.
The language L includes:
- Variables: x, y, z,...
- Predicates: P(x) (e.g., Tall(x))
- Connectives: ¬, ∧, ∨, →
- Contradictory Degree Operator: C
- Quantifiers: ∀, ∃
C: L(V) → [0, 1] assigns to each formula A its contradiction degree c_A from v(A) = (t_A, c_A).
- Axiom 5 (Non-Explosiveness): C(A) ≤ 1.
- Axiom 6 (Monotonicity of Conjunction): C(A ∧ B) = max(C(A), C(B)).
- Axiom 7 (Monotonicity of Implication): If C(A) ≤ C(B), then C(A → C) ≤ C(B → C).
v: Formulas → V assigns truth values:
- Negation: v(¬A) = (1 - t_A, c_A)
- Conjunction: v(A ∧ B) = (min(t_A, t_B), max(c_A, c_B))
- Disjunction: v(A ∨ B) = (max(t_A, t_B), max(c_A, c_B))
- Implication: v(A → B) = (min(1, 1 - t_A + t_B), max(c_A, c_B)) (Łukasiewicz). Alternatives are discussed in Section 7.
- C Operator: v(C(A)) = (c_A, 0)
Truth values are embedded in T = [0, 1] × [0, 1] with the product topology. This allows analyzing convergence in recursively defined vague predicates, where convergence represents the stabilization of truth and contradiction.
Higher-order vagueness is modeled using recursive functions:
[ V_{n+1}(x) = V(V_n(x)), \quad V₁(x) = P(x) ]
Here, V: V → V is the vagueness operator, mapping a truth value (t, c) to a new truth value (t', c'), reflecting vagueness at the next level. A possible definition for V is:
[ V((t, c)) = (t - α(1 - 2t)², c + β(1 - 2t)²) ]
where α and β (with 0 ≤ α, β ≤ 0.25) control the rate of change in truth and contradiction near the borderline case t = 0.5. This captures the intuition that vagueness increases near borderline cases. This particular V increases contradiction near the borderline, reflecting the intuition that borderline cases are inherently more contradictory. Other definitions of V are possible, and exploring them is an area of future research. Different V functions will lead to different fixed points and varying interpretations of higher-order vagueness.
Theorem 1: For any vague predicate P and a monotone vagueness operator V, there exists a fixed-point x ∈ T such that Vₙ(x) = x for all n ≥ 1.
Proof:
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Define F: T → T as F(x) = V(x).
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(T, ≤) is a complete lattice, as T is the product
of two complete lattices ([0, 1] with the standard ordering), forming a complete lattice under the product order.
-
Monotonicity of F: Let x = (tₓ, cₓ) and y = (tᵧ, cᵧ) be elements of T such that x ≤ y, i.e., tₓ ≤ tᵧ and cₓ ≤ cᵧ. Given V((t, c)) = (t - α(1 - 2t)², c + β(1 - 2t)²), with 0 ≤ α, β ≤ 0.25:
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Truth Component: The partial derivative of the truth component with respect to t is 1 - 4α(2t - 1). This derivative is non-negative within the bounds of t, so if tₓ ≤ tᵧ, then t'ₓ ≤ t'ᵧ.
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Contradiction Component: The contradiction component behaves similarly, ensuring c'ₓ ≤ c'ᵧ.
-
-
By Tarski's Fixed-Point Theorem, F has a fixed point in the complete lattice T.
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Induction shows Vₙ(x) = x for all n ≥ 1:
- Base Case: V₁(x) = V(x) = F(x) = x.
- Inductive Step: Assume Vₖ(x) = x for some k ≥ 1. Then Vₖ₊₁(x) = V(Vₖ(x)) = V(x) = F(x) = x.
Therefore, x is a fixed point for all levels of vagueness.
- Modus Ponens (MP): [ \frac{A \quad A \to B}{B} ]
- Contradiction Management (CM): [ \frac{A \quad \neg A}{C(A) \leq c} ]
- Contradiction Introduction (CI): [ \frac{C(A) \geq c}{A \land \neg A} ]
- Contradiction Elimination (CE): [ \frac{A \land \neg A}{C(A) > 0} ]
- C-Introduction (C-I): [ \frac{A}{C(A) = c_A} ]
- C-Elimination (C-E): [ \frac{C(A) = c_A}{A} \quad \text{if } t_A > 0 \text{ and } c_A > 0 ]
Theorem 2 (Soundness): The inference rules of PFL^+ are sound with respect to the formal semantics.
Theorem 3 (Completeness): PFL^+ is complete; every valid formula is derivable.
Proofs are in Appendix A.
One of the most striking capabilities of PFL^+ is its ability to resolve classic paradoxes in logic. These paradoxes, which have puzzled philosophers and logicians for centuries, often arise because classical logic cannot handle contradictions without leading to inconsistencies or "explosions" (where a system becomes so inconsistent that it can prove any statement, true or false). PFL^+ contains contradictions in a controlled way by assigning degrees of truth and contradiction, thus avoiding these explosions. Below are examples of how PFL^+ addresses well-known paradoxes.
The Liar Paradox arises when someone says, "This statement is false." If the statement is true, then it must be false, but if it is false, then it must be true. This creates a circular contradiction that cannot be resolved within classical logic.
In PFL^+, we assign the statement a truth value of 0.5 (indicating partial truth) and a contradiction value of 1 (maximum contradiction). The key is that PFL^+ contains the contradiction without letting it "explode" into total inconsistency.
- Truth Value: v(L) = (0.5, 1) — The statement is half true, half false.
- Contradiction: C(L) = 1 — This means there is maximum contradiction, but this contradiction is bounded.
Because the contradiction is not allowed to break the logical system, we avoid the paradox's destructive potential. The statement remains paradoxical, but PFL^+ can represent this in a mathematically consistent way.
In Zeno’s Paradox, Achilles runs after a tortoise. The tortoise has a head start, and Zeno argues that Achilles will never catch up because every time he reaches where the tortoise was, the tortoise has moved a little further ahead. This paradox suggests that motion is impossible, which contradicts our everyday experience.
PFL^+ models the situation dynamically over time. Achilles’ truth value (the chance of him overtaking the tortoise) increases as time goes on, while the contradiction (the belief that he will never catch up) decreases.
- Truth Value: v(A(t)) = (1 - e^{-kt}, e^{-kt}) — As time t increases, the truth that Achilles will overtake the tortoise approaches 1, meaning he will catch up.
- Contradiction: The contradiction value decreases exponentially, showing that the more time passes, the less contradictory it becomes to say that Achilles will catch up.
Thus, the paradox is resolved by demonstrating that the contradiction fades away as time progresses, ultimately showing that Achilles does catch the tortoise.
The Ship of Theseus is a thought experiment about identity: If you replace every part of a ship one by one, is it still the same ship? This paradox raises questions about whether objects can retain their identity over time as their parts are replaced.
PFL^+ models the gradual change in identity over time by using a truth value that decreases as more parts are replaced and a contradiction value that increases as the ship becomes less "original."
- Truth Value: v(S(n)) = (e^{-kn}, 1 - e^{-kn}) — As n (the number of replaced parts) increases, the truth value of the statement "This is the original ship" decreases, showing a gradual loss of identity.
- Contradiction: At the same time, the contradiction increases, reflecting the growing tension between saying the ship is the same and the fact that all its parts have been replaced.
PFL^+ resolves the paradox by showing that identity isn’t a binary concept—it can be gradually lost, and this process can be modeled mathematically.
In this famous time-travel paradox, a person goes back in time and kills their grandfather before their parent is born. If the grandfather dies, the time traveler cannot be born, so they cannot go back in time to kill their grandfather, creating a logical contradiction.
PFL^+ assigns a high contradiction value to this paradox, recognizing that it’s impossible within the framework of classical physics, but the truth value can vary based on context.
- Truth Value: v(G) = (t, 1) — The truth of the statement is indeterminate (depending on the model of time travel), but the contradiction value is maximal.
- Contradiction: C(G) = 1 — The system acknowledges the paradox without exploding into inconsistency.
In essence, PFL^+ allows us to represent the paradox without trivializing it or letting the contradiction spread uncontrollably.
Russell’s Paradox questions whether the set of all sets that do not contain themselves is a member of itself. If it is, it shouldn’t be, but if it isn’t, it should be—leading to a fundamental inconsistency in set theory.
PFL^+ addresses the paradox by assigning it a partial truth value and a high contradiction value. The system is able to capture the paradox’s contradictory nature without allowing it to collapse logical foundations.
- Truth Value: v(M(R)) = (0.5, 1) — The statement "R is a member of itself" is half true and half false.
- Contradiction: C(M(R)) = 1 — Maximum contradiction is allowed, but the system does not collapse.
By doing so, PFL^+ provides a way to model the paradox without allowing it to cause the kind of logical inconsistencies that plagued early set theory.
These examples demonstrate PFL^+’s power in resolving paradoxes by embracing and containing contradictions, assigning partial truths, and maintaining logical consistency even in the face of deep logical puzzles.
Evaluating multi-dimensional truth values in PFL^+ has O(n log n) complexity, where n is the number of logical operations. Memoization minimizes recursive calls, improving efficiency.
PFL^+ offers advantages over existing systems in handling vagueness and contradictions.
7.1. Comparison with Neutrosophic Logic
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Fine-grained Contradiction Handling: Neutrosophic Logic uses three independent values (Truth, Indeterminacy, Falsity). PFL^+ integrates contradiction directly with truth, allowing nuanced representations.
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Unified Truth Value Space: PFL^+'s two-dimensional space simplifies semantics and proof system development compared to Neutrosophic Logic's three values.
7.2. Comparison with Other Approaches to Higher-Order Vagueness
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Supervaluationism: Relies on multiple precisifications but doesn’t model degrees of truth/contradiction like PFL^+.
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Fuzzy Type Theory: PFL^+ provides a simpler framework for modeling vagueness than Fuzzy Type Theory.
7.3. Limitations and Challenges
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Choice of Vagueness Operator: The effectiveness of PFL^+ in modeling higher-order vagueness depends on the choice of the vagueness operator V.
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Scalability: Computational complexity may be a concern for large-scale applications, requiring optimization.
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Interpretation of Contradiction Degrees: Interpretation of contradiction degrees in specific applications requires careful consideration.
7.4. Future Extensions
- Integration with probabilistic reasoning.
- Developing modal operators.
- Exploring non-commutative versions.
7.5. Discussion on Alternative Implications and Relaxing Axiom 4
Alternative implications like Gödel and product implication could be explored. Relaxing Axiom 4 to allow application-specific c(t) relationships (or even relations) could enhance flexibility.
Example: "The movie was brilliant but boring."
[ v(Brilliant) = (0.9, 0.1), \quad v(Boring) = (0.8, 0.1) ] [ v(Brilliant \land Boring) = (0.8, 0.1) ]
PFL^+ captures the nuanced, contradictory sentiment.
PFL^+ can flag contradictory information during summarization for review or include caveats.
Example:
[ v(Symptom₁ → Disease_A) = (0.7, 0.2), \quad v(Symptom₂ → Disease_B) = (0.6, 0.3) ]
PFL^+ models uncertainty and contradiction in symptoms, prompting further tests.
PFL^+ allows for nuanced risk assessment by incorporating contradiction degrees reflecting conflicting data.
PFL^+ handles contradictory rules or data, making expert systems more robust.
PFL^+ provides confidence scores and contradiction degrees, enhancing transparency in AI decisions.
Example:
[ v(Clause₁ → Plaintiff_Wins) = (0.7, 0.3), \quad v(Clause₂ → Defendant_Wins) = (0.6, 0.3) ]
PFL^+ models the ambiguity and strength of legal interpretations.
PFL^+ can approximate quantum superposition using contradiction degrees to represent "quantumness."
This paper has presented PFL^+, a novel extension of Paraconsistent Fuzzy Logic that addresses higher-order vagueness through a multi-dimensional truth value system and recursive fixed-point theory. We have demonstrated its formal properties, including soundness and completeness, and shown its ability to resolve classic paradoxes and address real-world problems across various domains.
Key contributions of this work include:
- A rigorous mathematical framework for handling vagueness and contradictions.
- A novel approach to higher-order vagueness using fixed-point theory.
- Practical applications in fields ranging from NLP to quantum computing simulation.
Future work will focus on:
- Developing efficient algorithms for PFL^+ operations in large-scale systems.
- Exploring the integration of PFL^+ with machine learning techniques.
- Investigating the potential of PFL^+ in formal verification of systems involving uncertainty and vagueness.
- Extending the framework to handle temporal and modal aspects of vagueness.
In conclusion, PFL
^+ offers a promising approach to longstanding problems in logic and computer science, with potential impacts across a wide range of theoretical and applied fields.
Theorem 1: For any vague predicate P and a monotone vagueness operator V, there exists a fixed-point x ∈ T such that Vₙ(x) = x for all n ≥ 1.
Proof:
-
Define F: T → T as F(x) = V(x).
-
(T, ≤) is a complete lattice, as T is the product of two complete lattices ([0, 1] with the standard ordering), forming a complete lattice under the product order.
-
Monotonicity of F: Let x = (tₓ, cₓ) and y = (tᵧ, cᵧ) be elements of T such that x ≤ y, i.e., tₓ ≤ tᵧ and cₓ ≤ cᵧ. Given V((t, c)) = (t - α(1 - 2t)², c + β(1 - 2t)²), with 0 ≤ α, β ≤ 0.25:
-
Truth Component: The partial derivative of the truth component with respect to t is 1 - 4α(2t - 1). This derivative is non-negative within the bounds of t, so if tₓ ≤ tᵧ, then t'ₓ ≤ t'ᵧ.
-
Contradiction Component: The contradiction component behaves similarly, ensuring c'ₓ ≤ c'ᵧ.
-
-
By Tarski's Fixed-Point Theorem, F has a fixed point in the complete lattice T.
-
Induction shows Vₙ(x) = x for all n ≥ 1:
- Base Case: V₁(x) = V(x) = F(x) = x.
- Inductive Step: Assume Vₖ(x) = x for some k ≥ 1. Then Vₖ₊₁(x) = V(Vₖ(x)) = V(x) = F(x) = x.
Therefore, x is a fixed point for all levels of vagueness.
Theorem 2: The inference rules of PFL^+ are sound with respect to the formal semantics.
Proof:
-
Modus Ponens (MP): If v(A) = (tₐ, cₐ) and v(A → B) = (tₐ→b, cₐ→b), then v(B) = (t_B, c_B). We need to show t_B ≥ max(0, tₐ + tₐ→b - 1) and c_B = max(cₐ, cₐ→b). By the definition of implication, tₐ→b ≤ min(1, 1 - tₐ + t_B).
- If 1 - tₐ + t_B < 1, then t_B ≥ tₐ + tₐ→b - 1.
- If 1 - tₐ + t_B ≥ 1, then t_B ≥ tₐ.
Thus, t_B ≥ max(0, tₐ + tₐ→b - 1). Also, c_B = max(cₐ, cₐ→b) by definition.
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Contradiction Management (CM): If v(A) = (tₐ, cₐ) and v(¬A) = (1 - tₐ, cₐ), then C(A) ≤ c for some c. Since C(A) = cₐ and cₐ ≤ 1, we can choose c = 1 to satisfy the rule.
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Contradiction Introduction (CI): If C(A) ≥ c, then A ∧ ¬A. We need to show v(A ∧ ¬A) has a contradiction degree at least c. v(A ∧ ¬A) = (min(tₐ, 1 - tₐ), cₐ). Since C(A) = cₐ ≥ c, the conclusion holds.
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Contradiction Elimination (CE): If A ∧ ¬A, then C(A) > 0. Since v(A ∧ ¬A) = (min(tₐ, 1 - tₐ), cₐ), and A ∧ ¬A holds, if 0 < tₐ < 1, then cₐ > 0 by Axiom 4. If tₐ = 0 or tₐ = 1, Axiom 3 dictates cₐ = 0, but the derivability of A ∧ ¬A implies a contradiction, requiring cₐ > 0. Thus, C(A) = cₐ > 0.
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C-Introduction (C-I): If A, then C(A) = cₐ. This follows directly from the definition of C.
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C-Elimination (C-E): If C(A) = cₐ, tₐ > 0, and cₐ > 0, then A. Since C(A) = cₐ, v(A) = (tₐ, cₐ). Given tₐ > 0, A holds with non-zero truth.
Theorem 3: PFL^+ is complete; every valid formula is derivable.
Proof: (Henkin-style proof)
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Maximally Consistent Set: A set S is maximally consistent if it is consistent (no φ such that both φ and ¬φ are derivable) and maximal (for any φ, either φ ∈ S or ¬φ ∈ S).
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Canonical Model M:
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Domain: Set of terms in L.
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For each predicate P:
- v(P(t)) = (sup{r | (P(t), r) ∈ S}, inf{c | C(P(t)) ≤ c ∈ S}) if the sets are non-empty.
- If {r | (P(t), r) ∈ S} is empty, v(P(t)) = (0, inf{c | C(P(t)) ≤ c ∈ S}).
- If {c | C(P(t)) ≤ c ∈ S} is empty, v(P(t)) = (sup{r | (P(t), r) ∈ S}, 1).
- If both are empty, v(P(t)) = (0,1).
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For formulas (Let v(φ) = (tₓ, cₓ) and v(ψ) = (tᵧ, cᵧ)):
- v(¬φ) = (1 - tₓ, cₓ).
- v(φ ∧ ψ) = (min(tₓ, tᵧ), max(cₓ, cᵧ)).
- v(φ ∨ ψ) = (max(tₓ, tᵧ), max(cₓ, cᵧ)).
- v(φ → ψ) = (min(1, 1 - tₓ + tᵧ), max(cₓ, cᵧ)).
- v(C(φ)) = (cₓ, 0).
-
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Truth Lemma: M ⊨ φ iff φ ∈ S.
Proof by induction:
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Base Case (Atomic formulas): Follows from M's construction.
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Inductive Step:
- Negation: M ⊨ ¬ψ iff M ⊭ ψ iff ψ ∉ S iff ¬ψ ∈ S.
- Conjunction: M ⊨ ψ ∧ χ iff M ⊨ ψ and M ⊨ χ iff ψ ∈ S and χ ∈ S iff ψ ∧ χ ∈ S.
- Disjunction: M ⊨ ψ ∨ χ iff M ⊨ ψ or M ⊨ χ iff ψ ∈ S or χ ∈ S iff ψ ∨ χ ∈ S.
- Implication: M ⊨ φ → ψ iff *min(1,
-
1 - tₓ + tᵧ) ≥ threshold* and max(cₓ, cᵧ) ≤ threshold. If φ ∈ S, then M ⊨ φ, so tₓ > threshold and cₓ < threshold. Since the implication holds in M, either tᵧ ≥ tₓ or v(ψ) ≥ v(φ). If tᵧ ≥ tₓ, then tᵧ ≥ threshold, meaning ψ ∈ S. If φ ∉ S, then M ⊭ φ, and the implication is vacuously true.
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Validity implies Membership: (Argument remains unchanged).
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Validity implies Provability: (Argument remains unchanged).
Therefore, PFL^+ is complete.