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# Pertinence Construed Modally

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### Pertinence Construed Modally

1. 1. Pertinence Construed Modally Arina Britz1,2 Johannes Heidema2 Ivan Varzinczak1 1 Meraka Institute, CSIR 2 University of South Africa Pretoria, South Africa Pretoria, South Africa Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 1 / 29
2. 2. A Simple Example (attributed to Russell) Let p: “Mars orbits the Sun” q: “a red teapot is orbiting Mars” In Classical Logic ¬p ∧ q |= q ¬p |= ¬p ∨ q ¬p |= ⊥ |= ¬p |= p → (q → p) Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 2 / 29
3. 3. A Simple Example (attributed to Russell) Let p: “Mars orbits the Sun” q: “a red teapot is orbiting Mars” In Classical Logic ¬p ∧ q |= q ¬p |= ¬p ∨ q ¬p |= ⊥ |= ¬p |= p → (q → p) Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 2 / 29
4. 4. A Simple Example (attributed to Russell) Let p: “Mars orbits the Sun” q: “a red teapot is orbiting Mars” In Classical Logic ¬p ∧ q |= q (disjunctive syllogism: ¬p ∧ (p ∨ q) |= q) ¬p |= ¬p ∨ q ¬p |= ⊥ |= ¬p (ex contradictione quodlibet) |= p → (q → p) (positive paradox) Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 2 / 29
5. 5. A Simple Example (attributed to Russell) Let p: “Mars orbits the Sun” q: “a red teapot is orbiting Mars” In Classical Logic ¬p ∧ q |= q (disjunctive syllogism: ¬p ∧ (p ∨ q) |= q) ¬p |= ¬p ∨ q ¬p |= ⊥ |= ¬p (ex contradictione quodlibet) |= p → (q → p) (positive paradox) Do we want this? Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 2 / 29
6. 6. Classical Logic: the Logic of ‘Complete Ignorance’ α |= β Fact W Every α-world is a β-world β β ∧ ¬α-worlds completely free and arbitrary α Nothing to do with α or any of the α-worlds Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 3 / 29
7. 7. Classical Logic: the Logic of ‘Complete Ignorance’ α |= β But W β One intuitive connotation of entailment is that more, some notion of relevance or pertinence, should hold between α and β α Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 3 / 29
8. 8. Classical Logic: the Logic of ‘Complete Ignorance’ α |= β Usually W Extra information expressed either as β Syntactic rules, or as Semantic constraints α Binary relation on sets of sentences Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 3 / 29
9. 9. Less Attractive Features of Traditional Relevance Logics Following Avron [1992]: Conﬂation of |= with → [Anderson and Belnap, 1975, 1992] Start with proof theory, then ﬁnd a proper semantics Moreover Sometimes metaphysical ideas get admixed into the relevance endeavour Relevance logics traditionally pay scant attention to contexts Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 4 / 29
10. 10. Less Attractive Features of Traditional Relevance Logics Following Avron [1992]: Conﬂation of |= with → [Anderson and Belnap, 1975, 1992] Start with proof theory, then ﬁnd a proper semantics Moreover Sometimes metaphysical ideas get admixed into the relevance endeavour Relevance logics traditionally pay scant attention to contexts Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 4 / 29
11. 11. Less Attractive Features of Traditional Relevance Logics Following Avron [1992]: Conﬂation of |= with → [Anderson and Belnap, 1975, 1992] Start with proof theory, then ﬁnd a proper semantics Moreover Sometimes metaphysical ideas get admixed into the relevance endeavour Relevance logics traditionally pay scant attention to contexts Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 4 / 29
12. 12. Less Attractive Features of Traditional Relevance Logics Following Avron [1992]: Conﬂation of |= with → [Anderson and Belnap, 1975, 1992] Start with proof theory, then ﬁnd a proper semantics Moreover Sometimes metaphysical ideas get admixed into the relevance endeavour Relevance logics traditionally pay scant attention to contexts Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 4 / 29
13. 13. Less Attractive Features of Traditional Relevance Logics Following Avron [1992]: Conﬂation of |= with → [Anderson and Belnap, 1975, 1992] Start with proof theory, then ﬁnd a proper semantics Moreover Sometimes metaphysical ideas get admixed into the relevance endeavour Relevance logics traditionally pay scant attention to contexts Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 4 / 29
14. 14. Outline 1 Logical Preliminaries Modal Logic 2 Pertinent Entailment Infra-Modal Entailment Properties Examples 3 Conclusion Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 5 / 29
15. 15. Outline 1 Logical Preliminaries Modal Logic 2 Pertinent Entailment Infra-Modal Entailment Properties Examples 3 Conclusion Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 5 / 29
16. 16. Outline 1 Logical Preliminaries Modal Logic 2 Pertinent Entailment Infra-Modal Entailment Properties Examples 3 Conclusion Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 5 / 29
17. 17. Outline 1 Logical Preliminaries Modal Logic 2 Pertinent Entailment Infra-Modal Entailment Properties Examples 3 Conclusion Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 6 / 29
18. 18. Modal Logic Propositional modal language Atoms: p, q, . . . and Normal modal operator 2 Formulas: α, β, . . . α ::= p | | ¬α | α ∧ α | 2α Other connectives deﬁned as usual 3α ≡def ¬2¬α Given 2 and 3, we can speak of their converses: 2 and 3 ˘ ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 7 / 29
19. 19. Modal Logic Propositional modal language Atoms: p, q, . . . and Normal modal operator 2 Formulas: α, β, . . . α ::= p | | ¬α | α ∧ α | 2α Other connectives deﬁned as usual 3α ≡def ¬2¬α Given 2 and 3, we can speak of their converses: 2 and 3 ˘ ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 7 / 29
20. 20. Modal Logic Propositional modal language Atoms: p, q, . . . and Normal modal operator 2 Formulas: α, β, . . . α ::= p | | ¬α | α ∧ α | 2α Other connectives deﬁned as usual 3α ≡def ¬2¬α Given 2 and 3, we can speak of their converses: 2 and 3 ˘ ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 7 / 29
21. 21. Modal Logic Standard semantics Deﬁnition A model is a tuple M = W, R, V , where W is a set of worlds R ⊆ W × W is an accessibility relation on W V : P × W −→ {0, 1} is a valuation w2 ¬p, q p, q w3 M : w1 ¬p, ¬q p, ¬q w4 Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 8 / 29
22. 22. Modal Logic Standard semantics Deﬁnition A model is a tuple M = W, R, V , where W is a set of worlds R ⊆ W × W is an accessibility relation on W V : P × W −→ {0, 1} is a valuation w2 ¬p, q p, q w3 M : w1 ¬p, ¬q p, ¬q w4 Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 8 / 29
23. 23. Modal Logic Standard semantics Deﬁnition Given a model M = W, R, V , M w p iﬀ V(p, w ) = 1 M w for every w ∈ W M M w ¬α iﬀ w α M M M w α ∧ β iﬀ w α and w β M M w 2α iﬀ w α for every w such that (w , w ) ∈ R truth conditions for the other connectives are as usual M M w 2 α iﬀ w ˘ α for every w such that (w , w ) ∈ R M M w 3 α iﬀ w ˘ α for some w such that (w , w ) ∈ R M M If w α for every w ∈ W, we say that α is valid in M , denoted |= α Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 9 / 29
24. 24. Modal Logic Standard semantics Deﬁnition Given a model M = W, R, V , M w p iﬀ V(p, w ) = 1 M w for every w ∈ W M M w ¬α iﬀ w α M M M w α ∧ β iﬀ w α and w β M M w 2α iﬀ w α for every w such that (w , w ) ∈ R truth conditions for the other connectives are as usual M M w 2 α iﬀ w ˘ α for every w such that (w , w ) ∈ R M M w 3 α iﬀ w ˘ α for some w such that (w , w ) ∈ R M M If w α for every w ∈ W, we say that α is valid in M , denoted |= α Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 9 / 29
25. 25. Modal Logic Standard semantics Deﬁnition Given a model M = W, R, V , M w p iﬀ V(p, w ) = 1 M w for every w ∈ W M M w ¬α iﬀ w α M M M w α ∧ β iﬀ w α and w β M M w 2α iﬀ w α for every w such that (w , w ) ∈ R truth conditions for the other connectives are as usual M M w 2 α iﬀ w ˘ α for every w such that (w , w ) ∈ R M M w 3 α iﬀ w ˘ α for some w such that (w , w ) ∈ R M M If w α for every w ∈ W, we say that α is valid in M , denoted |= α Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 9 / 29
26. 26. Modal Logic Classes of models Sets of models we work with Determined by additional constraints Axiom schemas (reﬂexivity, transitivity, etc.) Global axioms (see later) Here we are interested in the class of reﬂexive models Given M = W, R, V , idW ⊆ R Axiom schema 2α → α Modal logic KT Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 10 / 29
27. 27. Modal Logic Classes of models Sets of models we work with Determined by additional constraints Axiom schemas (reﬂexivity, transitivity, etc.) Global axioms (see later) Here we are interested in the class of reﬂexive models Given M = W, R, V , idW ⊆ R Axiom schema 2α → α Modal logic KT Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 10 / 29
28. 28. Modal Logic Classes of models Sets of models we work with Determined by additional constraints Axiom schemas (reﬂexivity, transitivity, etc.) Global axioms (see later) Here we are interested in the class of reﬂexive models Given M = W, R, V , idW ⊆ R Axiom schema 2α → α Modal logic KT Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 10 / 29
29. 29. Modal Logic Local consequence Deﬁnition M α entails β in M = W, R, V (denoted α |= β) iﬀ for every w ∈ W, if M M w α, then w β. Deﬁnition Let C be a class of models C M α entails β in C (denoted α |= β) iﬀ α |= β for every M ∈ C Validity and satisﬁability in C deﬁned as usual C When C is clear from the context, we write α |= β instead of α |= β Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 11 / 29
30. 30. Modal Logic Local consequence Deﬁnition M α entails β in M = W, R, V (denoted α |= β) iﬀ for every w ∈ W, if M M w α, then w β. Deﬁnition Let C be a class of models C M α entails β in C (denoted α |= β) iﬀ α |= β for every M ∈ C Validity and satisﬁability in C deﬁned as usual C When C is clear from the context, we write α |= β instead of α |= β Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 11 / 29
31. 31. Modal Logic Local consequence Deﬁnition M α entails β in M = W, R, V (denoted α |= β) iﬀ for every w ∈ W, if M M w α, then w β. Deﬁnition Let C be a class of models C M α entails β in C (denoted α |= β) iﬀ α |= β for every M ∈ C Validity and satisﬁability in C deﬁned as usual C When C is clear from the context, we write α |= β instead of α |= β Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 11 / 29
32. 32. Outline 1 Logical Preliminaries Modal Logic 2 Pertinent Entailment Infra-Modal Entailment Properties Examples 3 Conclusion Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 12 / 29
33. 33. The Flow of Entailment Asymmetric, directed Access from premiss to consequence Entailment as ‘access’: natural analogue in the accessibility relation However Relevance cannot be captured by standard modalities [Meyer, 1975] Relevance is a relation between sentences (sets of worlds), and not between worlds alone Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 13 / 29
34. 34. The Flow of Entailment Asymmetric, directed Access from premiss to consequence Entailment as ‘access’: natural analogue in the accessibility relation However Relevance cannot be captured by standard modalities [Meyer, 1975] Relevance is a relation between sentences (sets of worlds), and not between worlds alone Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 13 / 29
35. 35. The Flow of Entailment Asymmetric, directed Access from premiss to consequence Entailment as ‘access’: natural analogue in the accessibility relation However Relevance cannot be captured by standard modalities [Meyer, 1975] Relevance is a relation between sentences (sets of worlds), and not between worlds alone Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 13 / 29
36. 36. The Flow of Entailment Asymmetric, directed Access from premiss to consequence Entailment as ‘access’: natural analogue in the accessibility relation However Relevance cannot be captured by standard modalities [Meyer, 1975] Relevance is a relation between sentences (sets of worlds), and not between worlds alone Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 13 / 29
37. 37. Pertinence in the Meta-Level Notion of pertinence in the meta-level Pertinence of α and β to each other In our new entailment of β by α, the condition that we impose upon the (previously wild) β ∧ ¬α-worlds is that now each of them must be accessible from some α-world. W β • α • Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 14 / 29
38. 38. Pertinence in the Meta-Level Notion of pertinence in the meta-level Pertinence of α and β to each other In our new entailment of β by α, the condition that we impose upon the (previously wild) β ∧ ¬α-worlds is that now each of them must be accessible from some α-world. W β • α • Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 14 / 29
39. 39. Pertinence in the Meta-Level Deﬁnition M M M α pertinently entails β in M (denoted α |< β) iﬀ α |= β and β |= 3 α ˘ Deﬁnition C α pertinently entails β in the class C of models (denoted α |< β) iﬀ for M every M ∈ C , α |< β C When C is clear from the context, we write α |< β instead of α |< β Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 15 / 29
40. 40. Pertinence in the Meta-Level Deﬁnition M M M α pertinently entails β in M (denoted α |< β) iﬀ α |= β and β |= 3 α ˘ Deﬁnition C α pertinently entails β in the class C of models (denoted α |< β) iﬀ for M every M ∈ C , α |< β C When C is clear from the context, we write α |< β instead of α |< β Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 15 / 29
41. 41. Pertinence in the Meta-Level Deﬁnition M M M α pertinently entails β in M (denoted α |< β) iﬀ α |= β and β |= 3 α ˘ Deﬁnition C α pertinently entails β in the class C of models (denoted α |< β) iﬀ for M every M ∈ C , α |< β C When C is clear from the context, we write α |< β instead of α |< β Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 15 / 29
42. 42. Pertinence in the Meta-Level • 3α ˘  3α ˘      W    β  |< • ... • β •     α      • α Clearly, |< is infra-modal: if α |< β, then α |= β ‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’ Proposition α |< β iﬀ α ∨ β ≡ β ∧ 3 α ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 16 / 29
43. 43. Pertinence in the Meta-Level • 3α ˘  3α ˘      W    β  |< • ... • β •     α      • α Clearly, |< is infra-modal: if α |< β, then α |= β ‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’ Proposition α |< β iﬀ α ∨ β ≡ β ∧ 3 α ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 16 / 29
44. 44. Pertinence in the Meta-Level • 3α ˘  3α ˘      W    β  |< • ... • β •     α      • α Clearly, |< is infra-modal: if α |< β, then α |= β ‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’ Proposition α |< β iﬀ α ∨ β ≡ β ∧ 3 α ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 16 / 29
45. 45. Pertinence in the Meta-Level • 3α ˘  3α ˘      W    β  |< • ... • β •     α      • α Clearly, |< is infra-modal: if α |< β, then α |= β ‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’ Proposition α |< β iﬀ α ∨ β ≡ β ∧ 3 α ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 16 / 29
46. 46. Pertinence in the Meta-Level • 3α ˘  3α ˘      W    β  |< • ... • β •     α      • α Clearly, |< is infra-modal: if α |< β, then α |= β ‘|<’ vs. ‘|=’ like ‘<’ vs. ‘=’ Proposition α |< β iﬀ α ∨ β ≡ β ∧ 3 α ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 16 / 29
47. 47. A Spectrum of Entailment Relations Only restriction on R: idW ⊆ R ⊆ W × W (modal logic KT) The minimum (w.r.t. ⊆) case: R = idW maximum pertinence: |< = ≡ The maximum case: R = W × W (assume α ≡ ⊥, cf. later) minimum pertinence: |< = |= Theorem If the underlying modal logic is at least KT, then ≡ ⊆ |< ⊂ |= Note how, psychologically speaking, with increased pertinence between premiss and consequence ‘if’ tends to drift in the direction of ‘if and only if’ [Johnson-Laird & Savary, 1999] Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 17 / 29
48. 48. A Spectrum of Entailment Relations Only restriction on R: idW ⊆ R ⊆ W × W (modal logic KT) The minimum (w.r.t. ⊆) case: R = idW maximum pertinence: |< = ≡ The maximum case: R = W × W (assume α ≡ ⊥, cf. later) minimum pertinence: |< = |= Theorem If the underlying modal logic is at least KT, then ≡ ⊆ |< ⊂ |= Note how, psychologically speaking, with increased pertinence between premiss and consequence ‘if’ tends to drift in the direction of ‘if and only if’ [Johnson-Laird & Savary, 1999] Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 17 / 29
49. 49. A Spectrum of Entailment Relations Only restriction on R: idW ⊆ R ⊆ W × W (modal logic KT) The minimum (w.r.t. ⊆) case: R = idW maximum pertinence: |< = ≡ The maximum case: R = W × W (assume α ≡ ⊥, cf. later) minimum pertinence: |< = |= Theorem If the underlying modal logic is at least KT, then ≡ ⊆ |< ⊂ |= Note how, psychologically speaking, with increased pertinence between premiss and consequence ‘if’ tends to drift in the direction of ‘if and only if’ [Johnson-Laird & Savary, 1999] Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 17 / 29
50. 50. A Spectrum of Entailment Relations Only restriction on R: idW ⊆ R ⊆ W × W (modal logic KT) The minimum (w.r.t. ⊆) case: R = idW maximum pertinence: |< = ≡ The maximum case: R = W × W (assume α ≡ ⊥, cf. later) minimum pertinence: |< = |= Theorem If the underlying modal logic is at least KT, then ≡ ⊆ |< ⊂ |= Note how, psychologically speaking, with increased pertinence between premiss and consequence ‘if’ tends to drift in the direction of ‘if and only if’ [Johnson-Laird & Savary, 1999] Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 17 / 29
51. 51. Outline 1 Logical Preliminaries Modal Logic 2 Pertinent Entailment Infra-Modal Entailment Properties Examples 3 Conclusion Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 18 / 29
52. 52. Properties of |< Decidability Straightforward from deﬁnition Non-explosiveness falsum is not omnigenerating, in fact, only self-generating if ⊥ |< α, then α ≡ ⊥ More generally Theorem C C C Let α |< β. Then if |= α → ⊥, then |= β → ⊥ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 19 / 29
53. 53. Properties of |< Decidability Straightforward from deﬁnition Non-explosiveness falsum is not omnigenerating, in fact, only self-generating if ⊥ |< α, then α ≡ ⊥ More generally Theorem C C C Let α |< β. Then if |= α → ⊥, then |= β → ⊥ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 19 / 29
54. 54. Properties of |< Decidability Straightforward from deﬁnition Non-explosiveness falsum is not omnigenerating, in fact, only self-generating if ⊥ |< α, then α ≡ ⊥ More generally Theorem C C C Let α |< β. Then if |= α → ⊥, then |= β → ⊥ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 19 / 29
55. 55. Properties of |< |< is paratrivial verum is not omnigenerated α |< in general |< preserves valid modal formulas Theorem |< α iﬀ |= α Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 20 / 29
56. 56. Properties of |< |< is paratrivial verum is not omnigenerated α |< in general |< preserves valid modal formulas Theorem |< α iﬀ |= α Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 20 / 29
57. 57. Properties of |< |< rules out disjunctive syllogism (¬α ∨ β) ∧ α |= β (equivalent to β ∧ α |= β) β ∧ α |< β means that β ∧ α |= β and β |= 3 (β ∧ α) ˘ Every β-world, even if not an α-world, can be reached from some β ∧ α-world Does not hold in general Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 21 / 29
58. 58. Properties of |< |< rules out disjunctive syllogism (¬α ∨ β) ∧ α |= β (equivalent to β ∧ α |= β) β ∧ α |< β means that β ∧ α |= β and β |= 3 (β ∧ α) ˘ Every β-world, even if not an α-world, can be reached from some β ∧ α-world Does not hold in general Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 21 / 29
59. 59. Properties of |< |< rules out disjunctive syllogism (¬α ∨ β) ∧ α |= β (equivalent to β ∧ α |= β) β ∧ α |< β means that β ∧ α |= β and β |= 3 (β ∧ α) ˘ Every β-world, even if not an α-world, can be reached from some β ∧ α-world Does not hold in general Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 21 / 29
60. 60. Properties of |< |< does not satisfy contraposition Classically and modally we have contraposition: α |= β is equivalent to ¬β |= ¬α Not so for |<, and proof by contradiction does not hold in general ¬β |< ¬α says that α |= β and ¬α |= 3 ¬β: Every α-world is a ˘ β-world and every ¬α-world can be reached from some ¬β-world |< does not satisfy the deduction theorem α |< β iﬀ |< α → β (⇒) direction: OK (⇐) direction: Fails! Let |< α → β, i.e., |= α → β and α → β |= 3 . The latter is just the triviality α → β |= . We do ˘ not (in general) get the needed β |= 3 α. ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 22 / 29
61. 61. Properties of |< |< does not satisfy contraposition Classically and modally we have contraposition: α |= β is equivalent to ¬β |= ¬α Not so for |<, and proof by contradiction does not hold in general ¬β |< ¬α says that α |= β and ¬α |= 3 ¬β: Every α-world is a ˘ β-world and every ¬α-world can be reached from some ¬β-world |< does not satisfy the deduction theorem α |< β iﬀ |< α → β (⇒) direction: OK (⇐) direction: Fails! Let |< α → β, i.e., |= α → β and α → β |= 3 . The latter is just the triviality α → β |= . We do ˘ not (in general) get the needed β |= 3 α. ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 22 / 29
62. 62. Properties of |< |< does not satisfy contraposition Classically and modally we have contraposition: α |= β is equivalent to ¬β |= ¬α Not so for |<, and proof by contradiction does not hold in general ¬β |< ¬α says that α |= β and ¬α |= 3 ¬β: Every α-world is a ˘ β-world and every ¬α-world can be reached from some ¬β-world |< does not satisfy the deduction theorem α |< β iﬀ |< α → β (⇒) direction: OK (⇐) direction: Fails! Let |< α → β, i.e., |= α → β and α → β |= 3 . The latter is just the triviality α → β |= . We do ˘ not (in general) get the needed β |= 3 α. ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 22 / 29
63. 63. Properties of |< |< does not satisfy contraposition Classically and modally we have contraposition: α |= β is equivalent to ¬β |= ¬α Not so for |<, and proof by contradiction does not hold in general ¬β |< ¬α says that α |= β and ¬α |= 3 ¬β: Every α-world is a ˘ β-world and every ¬α-world can be reached from some ¬β-world |< does not satisfy the deduction theorem α |< β iﬀ |< α → β (⇒) direction: OK (⇐) direction: Fails! Let |< α → β, i.e., |= α → β and α → β |= 3 . The latter is just the triviality α → β |= . We do ˘ not (in general) get the needed β |= 3 α. ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 22 / 29
64. 64. Properties of |< |< does not satisfy contraposition Classically and modally we have contraposition: α |= β is equivalent to ¬β |= ¬α Not so for |<, and proof by contradiction does not hold in general ¬β |< ¬α says that α |= β and ¬α |= 3 ¬β: Every α-world is a ˘ β-world and every ¬α-world can be reached from some ¬β-world |< does not satisfy the deduction theorem α |< β iﬀ |< α → β (⇒) direction: OK (⇐) direction: Fails! Let |< α → β, i.e., |= α → β and α → β |= 3 . The latter is just the triviality α → β |= . We do ˘ not (in general) get the needed β |= 3 α. ˘ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 22 / 29
65. 65. Pertinent Conditional Deﬁnition α → β ≡def (α → β) ∧ (β → 3 α) ˘ Theorem α |< β iﬀ |< α → β Positive paradox: α → (β → α) Proposition |< α → (β → α) Corollary α |< β → α Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 23 / 29
66. 66. Pertinent Conditional Deﬁnition α → β ≡def (α → β) ∧ (β → 3 α) ˘ Theorem α |< β iﬀ |< α → β Positive paradox: α → (β → α) Proposition |< α → (β → α) Corollary α |< β → α Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 23 / 29
67. 67. Pertinent Conditional Deﬁnition α → β ≡def (α → β) ∧ (β → 3 α) ˘ Theorem α |< β iﬀ |< α → β Positive paradox: α → (β → α) Proposition |< α → (β → α) Corollary α |< β → α Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 23 / 29
68. 68. Pertinent Conditional Deﬁnition α → β ≡def (α → β) ∧ (β → 3 α) ˘ Theorem α |< β iﬀ |< α → β Positive paradox: α → (β → α) Proposition |< α → (β → α) Corollary α |< β → α Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 23 / 29
69. 69. Properties of |< Modus Ponens |< α, |< α → β |< β Non-Monotonicity: For |<, the monotonicity rule fails: α |< β, γ |= α γ |< β Substitution of Equivalents Transitivity: If the underlying logic is at least S4 α |< β, β |< γ α |< β, α |< β → γ α |< γ α |< γ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 24 / 29
70. 70. Properties of |< Modus Ponens |< α, |< α → β |< β Non-Monotonicity: For |<, the monotonicity rule fails: α |< β, γ |= α γ |< β Substitution of Equivalents Transitivity: If the underlying logic is at least S4 α |< β, β |< γ α |< β, α |< β → γ α |< γ α |< γ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 24 / 29
71. 71. Properties of |< Modus Ponens |< α, |< α → β |< β Non-Monotonicity: For |<, the monotonicity rule fails: α |< β, γ |= α γ |< β Substitution of Equivalents Transitivity: If the underlying logic is at least S4 α |< β, β |< γ α |< β, α |< β → γ α |< γ α |< γ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 24 / 29
72. 72. Properties of |< Modus Ponens |< α, |< α → β |< β Non-Monotonicity: For |<, the monotonicity rule fails: α |< β, γ |= α γ |< β Substitution of Equivalents Transitivity: If the underlying logic is at least S4 α |< β, β |< γ α |< β, α |< β → γ α |< γ α |< γ Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 24 / 29
73. 73. Outline 1 Logical Preliminaries Modal Logic 2 Pertinent Entailment Infra-Modal Entailment Properties Examples 3 Conclusion Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 25 / 29
74. 74. ‘Paraconsistent’ Character of |< Example p: “Mars orbits the Sun”; q: “a red teapot is orbiting Mars” w2 ¬p, q p, q w3 Background assumption: p-worlds are ‘preferred’ M : B = {¬p → 2¬p} w1 ¬p, ¬q p, ¬q w4 Premiss compatible with B: p ∧ q |< p; p |< p ∨ q; p |< Premiss incompatible with B: ¬p ∧ 3p |< ¬p; ⊥ |< p; 2p |< 2p ∨ ¬q Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 26 / 29
75. 75. ‘Paraconsistent’ Character of |< Example p: “Mars orbits the Sun”; q: “a red teapot is orbiting Mars” w2 ¬p, q p, q w3 Background assumption: p-worlds are ‘preferred’ M : B = {¬p → 2¬p} w1 ¬p, ¬q p, ¬q w4 Premiss compatible with B: p ∧ q |< p; p |< p ∨ q; p |< Premiss incompatible with B: ¬p ∧ 3p |< ¬p; ⊥ |< p; 2p |< 2p ∨ ¬q Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 26 / 29
76. 76. ‘Paraconsistent’ Character of |< Example p: “Mars orbits the Sun”; q: “a red teapot is orbiting Mars” w2 ¬p, q p, q w3 Background assumption: p-worlds are ‘preferred’ M : B = {¬p → 2¬p} w1 ¬p, ¬q p, ¬q w4 Premiss compatible with B: p ∧ q |< p; p |< p ∨ q; p |< Premiss incompatible with B: ¬p ∧ 3p |< ¬p; ⊥ |< p; 2p |< 2p ∨ ¬q Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 26 / 29
77. 77. ‘Paraconsistent’ Character of |< Example p: “Mars orbits the Sun”; q: “a red teapot is orbiting Mars” w2 ¬p, q p, q w3 Background assumption: p-worlds are ‘preferred’ M : B = {¬p → 2¬p} w1 ¬p, ¬q p, ¬q w4 Premiss compatible with B: p ∧ q |< p; p |< p ∨ q; p |< Premiss incompatible with B: ¬p ∧ 3p |< ¬p; ⊥ |< p; 2p |< 2p ∨ ¬q Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 26 / 29
78. 78. Pertinence and Causation Example s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking” Background assumption: B = {w → a, s → ¬a, 3s} Question: Is α the pertinent cause of β? M : ¬s, a, w w3 ¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w |< ¬w w2 ¬s, a, ¬w s, ¬a, ¬w w4 a ∧ 2¬s |< a ; a ∧ 23s |< a s |< ¬a ; s ∨ ¬a |< ¬a w1 ¬s, ¬a, ¬w Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
79. 79. Pertinence and Causation Example s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking” Background assumption: B = {w → a, s → ¬a, 3s} Question: Is α the pertinent cause of β? M : ¬s, a, w w3 ¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w |< ¬w w2 ¬s, a, ¬w s, ¬a, ¬w w4 a ∧ 2¬s |< a ; a ∧ 23s |< a s |< ¬a ; s ∨ ¬a |< ¬a w1 ¬s, ¬a, ¬w Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
80. 80. Pertinence and Causation Example s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking” Background assumption: B = {w → a, s → ¬a, 3s} Question: Is α the pertinent cause of β? M : ¬s, a, w w3 ¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w |< ¬w w2 ¬s, a, ¬w s, ¬a, ¬w w4 a ∧ 2¬s |< a ; a ∧ 23s |< a s |< ¬a ; s ∨ ¬a |< ¬a w1 ¬s, ¬a, ¬w Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
81. 81. Pertinence and Causation Example s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking” Background assumption: B = {w → a, s → ¬a, 3s} Question: Is α the pertinent cause of β? M : ¬s, a, w w3 ¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w |< ¬w w2 ¬s, a, ¬w s, ¬a, ¬w w4 a ∧ 2¬s |< a ; a ∧ 23s |< a s |< ¬a ; s ∨ ¬a |< ¬a w1 ¬s, ¬a, ¬w Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
82. 82. Pertinence and Causation Example s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking” Background assumption: B = {w → a, s → ¬a, 3s} Question: Is α the pertinent cause of β? M : ¬s, a, w w3 ¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w |< ¬w w2 ¬s, a, ¬w s, ¬a, ¬w w4 a ∧ 2¬s |< a ; a ∧ 23s |< a s |< ¬a ; s ∨ ¬a |< ¬a w1 ¬s, ¬a, ¬w Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
83. 83. Pertinence and Causation Example s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking” Background assumption: B = {w → a, s → ¬a, 3s} Question: Is α the pertinent cause of β? M : ¬s, a, w w3 ¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w |< ¬w w2 ¬s, a, ¬w s, ¬a, ¬w w4 a ∧ 2¬s |< a ; a ∧ 23s |< a s |< ¬a ; s ∨ ¬a |< ¬a w1 ¬s, ¬a, ¬w Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
84. 84. Pertinence and Causation Example s: “the turkey is shot”; a: “it is alive”; w : “the turkey is walking” Background assumption: B = {w → a, s → ¬a, 3s} Question: Is α the pertinent cause of β? M : ¬s, a, w w3 ¬a ∧ ¬w |< ¬a ; ¬a ∧ ¬w |< ¬w w2 ¬s, a, ¬w s, ¬a, ¬w w4 a ∧ 2¬s |< a ; a ∧ 23s |< a s |< ¬a ; s ∨ ¬a |< ¬a w1 ¬s, ¬a, ¬w Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 27 / 29
85. 85. Conclusion Contributions Semantic approach to the notion of pertinence Pertinence captured in a simple modal logic Whole spectrum of pertinent entailments, ranging between ≡ and |= We restrict some paradoxes avoided by relevance logics |< possesses other non-classical properties Ongoing and Future Work Other infra-modal entailment relations Supra-modal entailment: prototypical and venturous reasoning Relationship with contexts such as obligations, beliefs, etc Pertinent subsumptions in Description Logics Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 28 / 29
86. 86. Conclusion Contributions Semantic approach to the notion of pertinence Pertinence captured in a simple modal logic Whole spectrum of pertinent entailments, ranging between ≡ and |= We restrict some paradoxes avoided by relevance logics |< possesses other non-classical properties Ongoing and Future Work Other infra-modal entailment relations Supra-modal entailment: prototypical and venturous reasoning Relationship with contexts such as obligations, beliefs, etc Pertinent subsumptions in Description Logics Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 28 / 29
87. 87. Reference K. Britz, J. Heidema, I. Varzinczak. Pertinent Reasoning. Workshop on Nonmonotonic Reasoning (NMR), 2010. Thank you! Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 29 / 29
88. 88. Reference K. Britz, J. Heidema, I. Varzinczak. Pertinent Reasoning. Workshop on Nonmonotonic Reasoning (NMR), 2010. Thank you! Britz, Heidema, Varzinczak (Meraka,Unisa) Pertinence Construed Modally 29 / 29