## Revamping the Restriction Strategy by (2007)

Citations: | 2 - 1 self |

### BibTeX

@MISC{Tennant07revampingthe,

author = {Neil Tennant},

title = {Revamping the Restriction Strategy by},

year = {2007}

}

### OpenURL

### Abstract

This study continues the anti-realist’s quest for a principled way to avoid Fitch’s paradox. It is proposed that the Cartesian restriction on the anti-realist’s knowability principle ‘ϕ, therefore ✸Kϕ ’ should be formulated as a consistency requirement not on the premise ϕ of an application of the rule, but rather on the set of assumptions on which the relevant occurrence of ϕ depends. It is stressed, by reference to illustrative proofs, how important it is to have proofs in normal form before applying the proposed restriction. A similar restriction is proposed for the converse inference, the so-called Rule of Factiveness ‘✸Kϕ therefore ϕ’. The proposed restriction appears to block another Fitch-style derivation that uses the KK-thesis in order to get around the Cartesian restriction on applications of the knowability principle. ∗ To appear in Joseph Salerno, ed., All Truths are Known: New Essays on the Knowability Paradox, Oxford University Press. This paper would not have been written without the stimulation, encouragement and criticism that I have enjoyed from Joseph Salerno, Salvatore Florio, Christina Moisa, Nicholaos Jones, and Patrick Reeder.

### Citations

142 |
Elements of Intuitionism
- Dummett
- 1977
(Show Context)
Citation Context ...ence. (See [7] and [8].) Reduction procedures also form the centerpiece of Michael Dummett’s inferential theory of meaning, and his arguments in favor of intuitionistic logic as the right logic. (See =-=[3]-=-, the two famous essays ‘The Philosophical Basis of Intuitionistic Logic’ and ‘The Justification of Deduction’ in [4], and [5].) Normalization lies at the heart also of the present author’s characteri... |

96 |
The Logical Basis of Metaphysics
- Dummett
- 1991
(Show Context)
Citation Context ...d his arguments in favor of intuitionistic logic as the right logic. (See [3], the two famous essays ‘The Philosophical Basis of Intuitionistic Logic’ and ‘The Justification of Deduction’ in [4], and =-=[5]-=-.) Normalization lies at the heart also of the present author’s characterization of relevance in deduction. (See [12] for an exposition and for relevant sources.) 4 Conclusion We have undertaken here ... |

64 |
Truth and Other Enigmas
- Dummett
- 1978
(Show Context)
Citation Context ...aning, and his arguments in favor of intuitionistic logic as the right logic. (See [3], the two famous essays ‘The Philosophical Basis of Intuitionistic Logic’ and ‘The Justification of Deduction’ in =-=[4]-=-, and [5].) Normalization lies at the heart also of the present author’s characterization of relevance in deduction. (See [12] for an exposition and for relevant sources.) 4 Conclusion We have underta... |

24 |
The Taming of The True
- Tennant
- 1997
(Show Context)
Citation Context ... that contains no maximal sentence occurrences. 4s2 The new restriction on Knowability 2.1 How the new restriction works on Fitch’s original proof We recall the proof of the Fitch paradox as given in =-=[9]-=-, at pp. 260–1: 3 Σ (∧I) ϕ ¬Kϕ K(ϕ ∧ ¬Kϕ) (✸K) ϕ ∧ ¬Kϕ Ξ (✸⊥) ✸K(ϕ ∧ ¬Kϕ) ⊥ (1) ⊥ where the embedded proof Ξ is (1) Ξ ϕ ∧ ¬Kϕ (K∧) K(ϕ ∧ ¬Kϕ) ¬Kϕ Kϕ (1) (I) K(ϕ ∧ ¬Kϕ) ⊥ (1) ⊥ The reader will easily v... |

9 |
On the idea of a general proof theory
- Prawitz
- 1974
(Show Context)
Citation Context ...for which the technique of converting proofs into normal form has afforded useful insights. Dag Prawitz used normalization techniques to frame a fertile conception of intuitionistic consequence. (See =-=[7]-=- and [8].) Reduction procedures also form the centerpiece of Michael Dummett’s inferential theory of meaning, and his arguments in favor of intuitionistic logic as the right logic. (See [3], the two f... |

9 |
The Language of Science
- Došen, Logic
(Show Context)
Citation Context ...cal Basis of Intuitionistic Logic’ and ‘The Justification of Deduction’ in [4], and [5].) Normalization lies at the heart also of the present author’s characterization of relevance in deduction. (See =-=[12]-=- for an exposition and for relevant sources.) 4 Conclusion We have undertaken here only the most preliminary explorations of prooftheoretic measures designed to stave off certain threats of paradox in... |

8 |
Meaning and proofs: On the conflict between classical and intuitionistic logic
- Prawitz
- 1977
(Show Context)
Citation Context ...h the technique of converting proofs into normal form has afforded useful insights. Dag Prawitz used normalization techniques to frame a fertile conception of intuitionistic consequence. (See [7] and =-=[8]-=-.) Reduction procedures also form the centerpiece of Michael Dummett’s inferential theory of meaning, and his arguments in favor of intuitionistic logic as the right logic. (See [3], the two famous es... |

5 | Clues to the Paradoxes of Knowability: Reply to Dummett and Tennant
- Brogaard, Salerno
- 2009
(Show Context)
Citation Context ...ill ever actually be known (i.e., Kψ)? Of course it does not!—for the envisaged possibility might never be actualized. There is 5 Compare Brogaard and Salerno’s proof of the KK-knowability paradox in =-=[1]-=-. 6 This was observed by Brogaard and Salerno in [2]. 10sa degree of serendipity in empirical (and even mathematical) inquiry, which even the anti-realist must recognize. So we cannot treat ✸K as reli... |

1 |
Knowability and the closure principle
- Brogaard, Salerno
(Show Context)
Citation Context ...charged by applying the rule in question. A discharged assumption no longer counts among the assumptions on which the conclusion of the newly created proof depends. 4 Personal communication. See also =-=[2]-=-. 5sfollowing proof Θ of ψ from the set of assumptions {p, ¬Kp}. Note that the application of (✸K) uses the old, ‘local’, restriction. Θ : (2) Kp ¬Kp ⊥ (1) Kψ (2) K(p ∧ (Kp→Kψ)) (1) p Kp→Kψ p ∧ (Kp→Kψ... |

1 |
Knowability and Cartesian Propositions
- Florio
- 2005
(Show Context)
Citation Context ...s undischarged assumptions on which the premise ϕ (for an application of the Knowability Principle) depends. An example illustrating this point would be the following, which I owe to Salvatore Florio =-=[6]-=-. It exploits the rule called (λ) in [10], at p. 837: (i) K, Kϕ (λ) ✸Kϕ ⊥ (i) ⊥ The proof using (λ) is as follows. . (2) (1) p∧¬Kp Kp ¬Kp (3) (2) ⊥ (3) K(p∧(Kp→¬Kq)) p∧¬Kp ¬Kq K(p∧(Kp→¬Kq)) p∧(Kp→¬Kq)... |

1 |
Anti-Realist Aporias
- Tennant
(Show Context)
Citation Context ...x, which exploits the KK -thesis but does not fall foul of the global restriction on its application of the Knowability Principle. 1.1 A retraction, for the record The present author’s claim, made in =-=[10]-=-, at p. 829 and repeated in [11], at p. 140, to the effect that ✸Kϕ is factive, was incautious. While it is valid so long as the sentence ϕ in question concerns only non-epistemic facts, the inference... |