| 1. |
- Awodey, Steve, et al.
(author)
-
Ultrasheaves and double negation
- 2004
-
In: Notre Dame Journal of Formal Logic. - : Duke University Press. - 0029-4527 .- 1939-0726. ; 45:4, s. 235-245
-
Journal article (peer-reviewed)abstract
- Moerdijk has introduced a topos of sheaves on a category of filters. Following his suggestion, we prove that its double negation subtopos is the topos of sheaves on the subcategory of ultrafilters—the ultrasheaves. We then use this result to establish a double negation translation of results between the topos of ultrasheaves and the topos on filters.
|
|
| 2. |
- Bonnay, Dennis, et al.
(author)
-
Invariance and Definability, with and without Equality
- 2018
-
In: Notre Dame Journal of Formal Logic. - : Duke University Press. - 0029-4527. ; 59:1, s. 109-133
-
Journal article (peer-reviewed)abstract
- The dual character of invariance under transformations and definability by some operations has been used in classical works by, for example, Galois and Klein. Following Tarski, philosophers of logic have claimed that logical notions themselves could be characterized in terms of invariance. In this article, we generalize a correspondence due to Krasner between invariance under groups of permutations and definability in L∞∞L∞∞ so as to cover the cases (quantifiers, logics without equality) that are of interest in the logicality debates, getting McGee’s theorem about quantifiers invariant under all permutations and definability in pure L∞∞L∞∞ as a particular case. We also prove some optimality results along the way, regarding the kinds of relations which are needed so that every subgroup of the full permutation group is characterizable as a group of automorphisms.
|
|
| 3. |
- Cantwell, John
(author)
-
The Logic of Conditional Negation
- 2008
-
In: Notre Dame Journal of Formal Logic. - : Duke University Press. - 0029-4527 .- 1939-0726. ; 49:3, s. 245-260
-
Journal article (peer-reviewed)abstract
- It is argued that the "inner" negation ∼ familiar from 3-valued logic can be interpreted as a form of "conditional" negation: ∼A is read 'A is false if it has a truth value'. It is argued that this reading squares well with a particular 3-valued interpretation of a conditional that in the literature has been seen as a serious candidate for capturing the truth conditions of the natural language indicative conditional (e.g., "If Jim went to the party he had a good time"). It is shown that the logic induced by the semantics shares many familiar properties with classical negation, but is orthogonal to both intuitionistic and classical negation: it differs from both in validating the inference from A→∼B to ∼(A→B).
|
|
| 4. |
- Engström, Fredrik, 1977, et al.
(author)
-
Transplendent models : Expansions omitting a type
- 2012
-
In: Notre Dame Journal of Formal Logic. - : Duke University Press. - 0029-4527. ; 53:3, s. 413-428
-
Journal article (peer-reviewed)abstract
- We expand the notion of resplendency to theories of the kind T + p, where T is a first-order theory and p expresses that the type p is omitted. We investigate two different formulations and prove necessary and suffcient conditions for countable recursively saturated models of PA. Some of the results in this paper can be found in one of the author's doctoral thesis.
|
|
| 5. |
- Espíndola, Christian, 1984-
(author)
-
Semantic Completeness of First-Order Theories in Constructive Reverse Mathematics
- 2016
-
In: Notre Dame Journal of Formal Logic. - : Duke University Press. - 0029-4527 .- 1939-0726. ; 57:2, s. 281-286
-
Journal article (peer-reviewed)abstract
- We introduce a general notion of semantic structure for first-order theories, covering a variety of constructions such as Tarski and Kripke semantics, and prove that, over Zermelo–Fraenkel set theory (ZF), the completeness of such semantics is equivalent to the Boolean prime ideal theorem (BPI). Using a result of McCarty (2008), we conclude that the completeness of Kripke semantics is equivalent, over intuitionistic Zermelo–Fraenkel set theory (IZF), to the Law of Excluded Middle plus BPI. Along the way, we also prove the equivalence, over ZF, between BPI and the completeness theorem for Kripke semantics for both first-order and propositional theories.
|
|
| 6. |
- Fjellstad, Andreas, et al.
(author)
-
IKT! and Łukasiewicz-models
- 2021
-
In: Notre Dame Journal of Formal Logic. - : Duke University Press. - 0029-4527. ; 62:2, s. 247-256
-
Journal article (peer-reviewed)abstract
- In this note, we show that the first-order logic IK! is sound with regard to the models obtained from continuum-valued Łukasiewicz-models for first-order languages by treating the quantifiers as infinitary strong disjunction/conjunction rather than infinitary weak disjunction/conjunction. Moreover, we show that these models cannot be used to provide a new consistency proof for the theory of truth IKT! obtained by expanding IK! with transparent truth, because the models are inconsistent with transparent truth. Finally, we show that whether or not this inconsistency can be reproduced in the sequent calculus for IKT! depends on how vacuous quantification is treated.
|
|
| 7. |
- Hansson, Sven Ove
(author)
-
Past Probabilities
- 2010
-
In: Notre Dame Journal of Formal Logic. - : Duke University Press. - 0029-4527 .- 1939-0726. ; 51:2, s. 207-223
-
Journal article (peer-reviewed)abstract
- The probability that a fair coin tossed yesterday landed heads is either 0 or 1, but the probability that it would land heads was 0.5. In order to account for the latter type of probabilities, past probabilities, a temporal restriction operator is introduced and axiomatically characterized. It is used to construct a representation of conditional past probabilities. The logic of past probabilities turns out to be strictly weaker than the logic of standard probabilities.
|
|
