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Erschienen in:
Cables, Trains and Types Kapitel lesen Erstes Kapitel lesen

2020 | OriginalPaper | Buchkapitel

Cathoristic Logic

A Logic for Capturing Inferences Between Atomic Sentences

verfasst von : Richard Evans, Martin Berger

Erschienen in: From Lambda Calculus to Cybersecurity Through Program Analysis

Verlag: Springer International Publishing

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Abstract

Cathoristic logic is a multi-modal logic where negation is replaced by a novel operator allowing the expression of incompatible sentences. We present the syntax and semantics of the logic including complete proof rules, and establish a number of results such as compactness, a semantic characterisation of elementary equivalence, the existence of a quadratic-time decision procedure, and Brandom’s incompatibility semantics property. We demonstrate the usefulness of the logic as a language for knowledge representation.

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Vorheriges Kapitel Cables, Trains and Types
Nächstes Kapitel A Type Theory for Probabilistic –calculus
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Fußnoten
1
Efficient handling of free/bound variables is an active field of research, e.g. nominal approaches to logic [23]. The problem was put in focus in recent years with the rising interest in the computational cost of syntax manipulation in languages with binders.
 
2
“Cathoristic” comes from the Greek \(\kappa \alpha \theta o \rho \acute{\i } \zeta \epsilon i \nu \): to impose narrow boundaries. We are grateful to Tim Whitmarsh for suggesting this word.
 
3
“Tantum” is Latin for “only”.
 
4
We will precisify this claim in later sections; (1) first-order logic’s representation of incompatibility is longer in terms of formula length than cathoristic logic’s (see Sect. 4.2); and (2) logic programs in cathoristic logic can be optimised to run significantly faster than their equivalent in first-order logic (see Sect. 5.3).
 
5
[8] pp. 47–48.
 
6
[7] pp. 88–89, our emphasis.
 
7
Compare Russell [24] p. 117: “A sentence is of atomic form when it contains no logical words and no subordinate sentence”. We use a broader notion of atomicity by focusing solely on whether or not it contains a subordinate sentence, allowing logical words such as “and” as long as they are conjoining noun-phrases and not sentences.
 
8
To see that “Jack loves Jill” is not a constituent of “Jack loves Jill and Joan”, observe that “and” conjoins constituents of the same syntactic type. But “Jack loves Jill” is a sentence, while “Joan” is a noun. Hence the correct parsing is “Jack (loves (Jill and Joan))”, rather than “(Jack loves Jill) and Joan”.
 
9
See [28] p. 282 for a spirited defence of predicate conjunction against Fregean regimentation.
 
10
Although natural languages are full of examples of inferences from dyadic to monadic predicates, there are certain supposed counterexamples to the general rule that a dyadic predicate always implies a monadic one. For example, “Jack explodes the device” does not, on its most natural reading, imply that “Jack explodes”. Our response to cases like this is to distinguish between two distinct monadic predicates \(explodes_1\) and \(explodes_2\):
  • \(X explodes_1\) iff X is an object that undergoes an explosion
  • \(X explodes_2\) iff X is an agent that initiates an explosion
Now “Jack explodes the device” does imply that “Jack \(explodes_2\)” but does not imply that “Jack \(explodes_1\)”. There is no deep problem here - just another case where natural language overloads the same word in different situation to have different meanings.
 
11
The application had thousands of paying users, and was available for download on the App Store for the iPad [12].
 
12
E.g. STRIPS [14].
 
13
Brandom [8] defines incompatibility slightly differently: he defines the set of sets of formulae which are incompatible with a set of formulae. But in cathoristic logic, if a set of formulae is incompatible, then there is an incompatible subset of that set with exactly two members. So we can work with the simpler definition in the text above.
 
14
[8] p. 123.
 
15
The converse of \((\lnot 2)\) follows from \((\lnot 1)\) and the general structural laws above.
 
16
\(\psi \) is the minimal incompatible of \(\phi \) iff for all \(\xi \), if \(\mathsf {Inc}(\{\phi \} \cup \{\xi \})\) then \(\xi \models \psi \).
 
17
The notion of incompatibility applies to all logics: two formulae are incompatible if there is no model which satisfies both.
 
18
We assume, in this discussion, that married is a many-to-one predicate. We assume that polygamy is one person attempting to marry two people (but failing to marry the second).
 
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Metadaten
Titel
Cathoristic Logic
verfasst von
Richard Evans
Martin Berger
Copyright-Jahr
2020
Buch
From Lambda Calculus to Cybersecurity Through Program Analysis

Print ISBN: 978-3-030-41102-2

Electronic ISBN: 978-3-030-41103-9

Copyright-Jahr: 2020

DOI
https://doi.org/10.1007/978-3-030-41103-9_2

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