Talk given at the Graduate Center of the City University of New York, USA.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. ‘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. ‘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. ‘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. ‘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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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