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A Revised Projectivity Calculus for Inclusion and Exclusion Reasoning

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Abstract

We present a Revised Projectivity Calculus (denoted RC) that extends the scope of inclusion and exclusion inferences derivable under the Projectivity Calculus (denoted C) developed by Icard (Stud Log 100(4):705–725, 2012). After pointing out the inadequacies of C, we introduce four opposition properties (OPs) which have been studied by Chow (in: Aloni et al (eds) Proceedings of the 18th Amsterdam Colloquium, Springer, Berlin, 2012; Beziau, Georgiorgakis (eds) New dimensions of the square of opposition, Philosophia Verlag GmbH, München, 2017) and are more appropriate for the study of exclusion reasoning. Together with the monotonicity properties (MPs), the OPs will form the basis of RC instead of the additive/multiplicative properties used in C. We also prove some important results of the OPs and their relation with the MPs. We then introduce a set of projectivity signatures together with the associated operations and conditions for valid inferences, and develop RC by inheriting the key features of C. We then show that under RC, we can derive some inferences that are not derivable under C. We finally discuss some properties of RC and point to possible directions of further studies.

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Notes

  1. Icard (2012) claimed that the quantifier most is multiplicative. But it will be shown below that more than 1/2 of is not multiplicative. Note that many GQT scholars, such as Peters and Westerståhl (2006), assume that most is equivalent to more than 1/2 of.

  2. Icard (2014) provided an axiomatization for the inclusion and exclusion reasoning and conjectured that the resulting calculus (called C2) is complete (but without proving it). Since C2 does not contain as much detail as the calculus C developed in Icard (2012), the discussion of this paper is mainly based on Icard (2012).

  3. The names and symbols of the relations are adapted from MacCartney (2009), MacCartney and Manning (2009) and Icard (2012). The definitions are taken from Icard (2012). The “exclusive” and “exhaustive” relations are the same as the “CC” (short for “contrary or contradictory”) and “SC” (short for “subcontrary or contradictory”) relations in Chow (2012, 2017). Moreover, in what follows, “iff” is short for “if and only if”. Note that we have chosen to use Δ and ∇ so that the shape of Δ reminds us of the “∧” in the definition of the exclusive property whereas the shape of ∇ reminds us of the “∨” in the definition of the exhaustive property.

  4. The following proposition can be easily proved by invoking the fact that x ≤ y iff x ∧ ¬y = 0 and de Morgan’s Law. The above fact can be found in Keenan and Faltz (1985).

  5. Note that the JOIN operation defined by Icard (2012) is different from the concept of “join” in a partial order. To highlight this difference, the symbol representing this special operation is capitalized in this paper.

  6. Icard (2012) used the terms “monotone” and “antitone” instead of “increasing” and “decreasing”.

  7. What Icard (2012) studied is the “complete” version of the properties.

  8. In this paper, we treat proportional quantifiers more than/less than/at least/at most r of the as equivalent to more than/less than/at least/at most r of.

  9. The truth conditions under GQT of these two exceptive quantifiers are (where s denotes Smith):

    ǁall except… Smithǁ(A)(B) is true iff A – B = {s}

    ǁno except… Smithǁ(A)(B) is true iff A ∩ B = {s}

  10. The truth conditions under GQT of these two identity comparative quantifiers are:

    ǁthe same… as….ǁ(A)(B1)(B2) is true iff A ∩ B1 = A ∩ B2

    ǁdifferent… than….ǁ(A)(B1)(B2) is true iff A ∩ B1 ≠ A ∩ B2

  11. For the purpose of discussion in this paper, we assume that any person is either asleep or awake but not both, and so treat asleep and awake as contradictory.

  12. In Chow (2012, 2017), the homo-exclusive, homo-exhaustive, anti-exclusive and anti-exhaustive properties are denoted “CC → CC”, “SC → SC”, “CC → SC” and “SC → CC”, respectively.

  13. Although the results in Chow (2017) are about generalized quantifiers, these results (together with their proofs) can be readily extended to general functions on Boolean algebras.

  14. The previous version of this paper did not discuss (sufficient) conditions associated with inclusion/exclusion reasoning. I am grateful to an anonymous reviewer for pointing out the need for clarifying these conditions. Without the reviewer’s advice, this paper would not have included this subsection.

  15. These conditions are not parts of the projectivity signatures but are additional information specified in the lexicon (the Appendix to this paper can be seen as a part of this lexicon). Therefore, the discussion in this subsection should be seen as a justification of the conditions specified in the Appendix rather than part of the calculus to be introduced in the next section.

  16. Here we have adapted definitions (86) and (88) in Keenan and Westerståhl (2011): let Q be an arity reducer and B an n-ary predicate, then Q(B) = {(x1,… xn−1): Q({xn: (x1,… xn) ∈ B})}.

  17. In GQT, a proper name can be seen as a generalized quantifier. Such quantifiers are called “Montagovian individuals” in Peters and Westerståhl (2006).

  18. To simplify matters, we treat “are noisy” as a unit.

  19. Icard (2012) called this the “Composition” Rule. To avoid confusing this rule with the composition of signatures introduced in subsection 2.4 above, we prefer to call this the JOIN Rule.

  20. Note that in Moss (2012), t2(x ← s) means the same thing as t2(s ← x) in this paper.

  21. In GQT, “Not everybody loves or hates everybody” can be represented as not everybody(everybody)(loveshates).

  22. To facilitate comparison, in what follows we modify the notations for the seven relations and projection used in Icard (2012).

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Appendix: Signatures and Conditions of Some Functions

Appendix: Signatures and Conditions of Some Functions

This Appendix provides a list of some important functions (including logical operators/generalized quantifiers) and their projectivity signatures and sufficient conditions for valid inferences. For each function, the signature listed in the middle column below represents the strongest MP and/or OP it possesses. For quantifiers with more than one argument, the signature in respect of each argument is specified. Otherwise, the signature in respect of that argument is understood to be ●. For a function with complex signature, different components of the signature may be associated with different conditions. For clarity, the components of a complex signature are listed on separate rows with their corresponding conditions given in the right column (where “–” represents no condition). In what follows, A and B represent the 1st (unary) argument and the 2nd (n-ary) argument of a determiner, while A, B1 and B2 represent the 1st (unary) argument and the 2nd and 3rd (n-ary) arguments of an identity comparative quantifier. For example, the information given below shows that every/all has complex signature “(↓ ∧ ∇ → Δ)” in respect of the 1st argument. Moreover, every/all is decreasing (i.e. ↓) with no additional condition and anti-exhaustive (i.e. ∇ → Δ) given the additional condition that ∀x1…xn−1.{xn: (x1,… xn) ∈ B} ≠ Ư.

Function

Signature

Condition

ID

Δ → Δ

∇ → ∇

not

Δ → ∇

∇ → Δ

Smith,

i.e. singular proper name

Δ → Δ

∇ → ∇

every/all

1st

∇ → Δ

∀x1…xn−1.{xn: (x1,… xn) ∈ B} ≠ Ư

2nd

Δ → Δ

A ≠ ∅

some

1st

∇ → ∇

∀x1…xn−1.{xn: (x1,… xn) ∈ B} ≠ ∅

2nd

∇ → ∇

A ≠ ∅

no

1st

∇ → Δ

∀x1…xn−1.{xn: (x1,… xn) ∈ B} ≠ ∅

2nd

∇ → Δ

A ≠ ∅

not every/not all

1st

∇ → ∇

∀x1…xn−1.{xn: (x1,… xn) ∈ B} ≠ Ư

2nd

Δ → ∇

A ≠ ∅

more than n, at least n

both ↑

fewer than n, at most n

both ↓

most

2nd

Δ → Δ

more than r of

(1/2 ≤ r < 1)

2nd

Δ → Δ

more than r of

(0 < r < 1/2)

2nd

∇ → ∇

at least r of

(1/2 < r < 1)

2nd

Δ → Δ

at least r of

(0 < r ≤ 1/2)

2nd

∇ → ∇

less than r of

(1/2 < r < 1)

2nd

Δ → ∇

less than r of

(0 < r ≤ 1/2)

2nd

∇ → Δ

at most r of

(1/2 ≤ r < 1)

2nd

Δ → ∇

at most r of

(0 < r < 1/2)

2nd

∇ → Δ

exactly r of (1/2 < r < 1)

2nd Δ → Δ

exactly r of (0 < r < 1/2)

2nd ∇ → Δ

between q and r of

(1/2 < q < r < 1)

2nd Δ → Δ

between q and r of

(0 < q < r < 1/2)

2nd ∇ → Δ

more than r or

less than q of

(1/2 < q < r < 1)

2nd Δ → ∇

more than r or

less than q of

(0 < q < r < 1/2)

2nd ∇ → ∇

allexcept Smith

1st

Δ → Δ

∇ → Δ

∀x1…xn−1.{xn: (x1,… xn) ∈ B} ∪ {s} ≠ Ư

2nd

Δ → Δ

A − {s} ≠ ∅

∇ → Δ

noexcept Smith

1st

Δ → Δ

∇ → Δ

∀x1…xn−1.{xn: (x1,… xn) ∈ B} − {s} ≠ ∅

2nd

Δ → Δ

∇ → Δ

A − {s} ≠ ∅

the sameas

1st

∇ → Δ

∀x1…xn−1.{xn: (x1,… xn) ∈ B1}

≠ {xn: (x1,… xn) ∈ B2}

2nd

Δ → Δ

∀x1…xn−1.A ∩ {xn: (x1,… xn) ∈ B2} ≠ ∅

∇ → Δ

∀x1…xn−1.A − {xn: (x1,… xn) ∈ B2} ≠ ∅

3rd

Δ → Δ

∀x1…xn−1.A ∩ {xn: (x1,… xn) ∈ B1} ≠ ∅

∇ → Δ

∀x1…xn−1.A − {xn: (x1,… xn) ∈ B1} ≠ ∅

differentthan

1st

∇ → ∇

∀x1…xn−1.{xn: (x1,… xn) ∈ B1}

≠ {xn: (x1,… xn) ∈ B2}

2nd

∇ → ∇

∀x1…xn−1.A − {xn: (x1,… xn) ∈ B2} ≠ ∅

Δ → ∇

∀x1…xn−1.A ∩ {xn: (x1,… xn) ∈ B2} ≠ ∅

3rd

∇ → ∇

∀x1…xn−1.A − {xn: (x1,… xn) ∈ B1} ≠ ∅

Δ → ∇

∀x1…xn−1.A ∩ {xn: (x1,… xn) ∈ B1} ≠ ∅

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Chow, Kf. A Revised Projectivity Calculus for Inclusion and Exclusion Reasoning. J of Log Lang and Inf 29, 163–195 (2020). https://doi.org/10.1007/s10849-019-09292-5

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