# Changes

## gadri: an unofficial commentary from a logical point of view

, 2 years ago
m
Link to Thomas McKay's homepage corrected
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[[User:Guskant|I (guskant)]] will make here a commentary on BPFK's gadri so that it is undersеood understood by them correctly. ([[gadri の論理学的観点からの解説|Japanese version/日本語版]]) <!--''Note dated 2016-09-12: [https://groups.google.com/d/msg/lojban/t2h3yV5_TIU/Y0WOTTRkBgAJ BPFK has approved interchange of some cmavo] including « '''su'o''' - '''su''' », « '''ce'u''' - '''ce''' », « '''ke'a''' - '''ke''' ». Those cmavo in the text below conform to the older definition.''-->
==Glossary==
==Plural quantification==
In order to understand arguments of Lojban from a logical point of view, it is essential to have a knowledge of '''plural quantification''' (see, for example, [http://thecollegeasfaculty.syr.edu/profilespages/pagesphi/mckay-thomas.html Thomas McKay]: ''Plural Predication'', Oxford University Press, 2006).
Plural quantification was invented in order to facilitate expression of proposition that is meaningful only when the referent of an argument is plural.
On the other hand, in the expression "people ate", although the constant "people" refers to plural people, the predicate "ate" is satisfied by each person. That is to say, each sentence such that "Alice ate", "Bob ate" and so on is meaningful.
When each referent referred to by a constant satisfies a predicate alone, we express it as "an argument satisfies an predicate'''distributively'''", or "the argument has '''distributivity'''".
Moreover, if the predicate "eat" means an act "put food in a mouth, bite it, let it pass through an esophagus and send it to a stomach", it is hardly considered that "people" satisfies "eat" collectively. Even if a person helps another to eat, the helper is not eater, and the eater is not collective people but an individual.
We introduce relations between plural constants and plural variables: {me} and {jo'u}.
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The property 3 means that the identity between referents of X and Y is represented with {me}, as a relation that {X me Y ijebo Y me X}.
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'''An individual''' is defined as follows:
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{ro da} (for all {da}) and {su'o da} (there is at least one {da}), which are officially defined in Lojban, are bound singular variables. They can be defined with bound plural variables as follows:
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: It is prefixed to selbri, and forms a plural constant that refers to what satisfies x1, the first place of the selbri. If a quantifier follows {lo}, the quantifier represents the count of all the referents of the plural constant. In the case that a quantifier follows {lo}, a sumti may follow it. In this case, it forms a plural constant that refers to what is/are among ''sumti''.
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: {le broda} refers '''specifically''' to a referent of {lo broda}, and '''explicitly express that the speaker has the referent in mind'''. Its logical property is the same as that of {lo}.
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: It is prefixed to selbri or cmevla, and forms a plural constant that refers to what is named the selbri or cmevla string. Its logical property is the same as that of {lo}.
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: {loi/lei/lai broda} refers to a referent of {lo/le/la broda}, and '''explicitly express that the referent satisfies a predicate collectively'''.
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: {lo'i/le'i/la'i broda} refers to a set or sets of individual(s) that constitute(s) a plural constant {lo/le/la broda}. Because {lo'i/le'i/la'i} forms a set or sets, it is defined only when its/their member(s) {lo/le/la broda} is/are an individual or individuals. A set itself is always an individual, and sets are always individuals: there is no set that is not an individual.
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According to [http://jbovlaste.lojban.org/dict/selcmi jbovlaste], {selcmi} is defined as follows:
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[[BPFK Section: gadri|BPFK defines inner quantification]] as follows:
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; Definition
:
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: (D2) is
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Applying (D1) to (S2),
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(D2) is therefore
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When N=1,
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Because of Axiom 1, it implies
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; Unofficial definition of {lo no broda}
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[[BPFK Section: gadri|Actually, piPA is defined only for outer quantification.]]
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As we can see in the definition, the body of outer quantification by {piPA} is plural constant {lo piPA si'e}, which is not a bound singular variable. However, x2 of {piPA si'e} is {pa me ''sumti''}, to which [[BPFK Section: gadri|the definition of PA broda]] is applied:
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; Unofficial definition of {piPA} of inner quantification
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It is possible, but [[BPFK Section: Numeric selbri|BPFK's current definition of {si'e}]] depends on {pagbu}:
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Those preparations of {ko'a, ko'e, ...} and (D2) imply only
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; Unofficial definitions under the condition that Axiom 1 is abandoned
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[[BPFK Section: gadri|BPFK defines outer quantification]] as follows:
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{PA lo broda} differs from {PA broda} in domain of referents of bound singular variable to be counted. The definitions of outer quantification are applied to them as follows:
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The definitions of inner and outer quantification imply the following interpretations:
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"Constants" in this meaning correspond to Skolem functions in Skolem normal forms of predicate logic. The table below shows comparison of interpretations between predicate logic, xorlo on which this commentary depends and implicit quantifier ([http://lojban.github.io/cll/6/1/ CLL Chapter 6]) which was abolished. The expressions that have the same truth value are aligned in the same column. Upper case Y represents a plural variable. The row of zo'u+xorlo shows unofficial suggestion of interpretation. In the gray part in the row of Prenex normal, unofficial expressions with an experimental cmavo {su'oi} are shown. (Click on the table to enlarge.)
===Relation between lu'a, lu'o, lu'i and gadri===
[[BPFK Section: Indirect Referers|BPFK defines]] {lu'a}, {lu'o}, {lu'i} of LAhE as follows:
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; Unofficial definition of {lu'a}
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[[BPFK Section: Non-logical Connectives|According to BPFK Section]], {jo'u}, {joi} and {ce} of selma'o JOI are defined as follows:
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==Notes==
This section consists of notes of the author [[User:Guskant|guskant]], and it is not at all important for understanding gadri.