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PROfiling and PROgramming BEhaviour : Part II

Quantifers  in natural language refer to the positions  in  statements 
taken by pronouns. They allow us to keep track of who, or what, we are 
talking about. In formal, predicate (functional) logic, they demystify 
the  whole  process of deductive inference. In a  finite  universe  of 
discourse  (which is what we have in PROBE as a database of  inmates), 
existential  (y) quantification can be replaced by a finite series  of 
disjunctions  (V ie ORs), just as universal (z) quantification can  be 
replaced  by a finite series of conjunctions (U ie ANDs). Our  queries 
can  be  conceived as clauses written in  disjunctive  or  conjunctive 
normal  form. When applied to the database, cases either do or do  not 
meet the specified conditions for class membership, and this serves as 
our  basis for actuarial analysis of behaviour. Since our database  is 
finite,  we  can in fact venture into considering the  application  of 
probability quantifiers (e.g. Vickers 1988), ie the relative frequency 
of individuals which meet the conditions of our logical conditions. 


    'The major motivating principle of probability quantifiers is 
    the  development of probability within pure or general  logic 
    to the extent that this is possible. The great difficulty  of 
    precisely  defining general logic can perhaps be  avoided  by 
    agreeing that however it is defined, the semantics of  first-
    order  logic  as  developed by Frege and  Tarski  fall  quite 
    within its confines. Then, as the above remarks suggest,  the 
    question  is  just  to  what  extent  such  notions  as  "the 
    proportion  of  objects  falling under  a  concept"  or  "the 
    proportion of assignments satisfying a formula" can be  given 
    a meaning in general logic.'

    J. M. Vickers (1988)
    Chance  & Structure: An Essay on the Logical  Foundations  of 
    Probability
    Probability quantifiers:principles and semantics p.153

Once  this way of working with a data base becomes familiar, it  is  a 
simple  set of steps from relative frequencies to joint  probabilities 
and  correlations, regressions and the rest of descriptive  statistics 
which all Prison Psychologists receive systematic training in as  part 
of  their  induction MSc training. It should also be clear  that  work 
within  the  PROBE system is work in the  application  of  extensional 
logic  to  a specific domain within which the Prison  Service  employs 
Behaviour  Scientists  to  provide a technical  service  within.  That 
specificity,  or specialism is defined by the selection, analysis  and 
use  of  the behaviour predicates and functions  within  a  particular 
universe of discourse. 

In  the  case  of  breaches  of prison  rules,  each  instance  of  an 
infraction is identified by the paragraph of the prison rules  broken. 
In turn, further aspects of the infraction can be recorded such as the 
date, time and location, resulting in an n-place predicate. Individual 
names, or identifiers are syntactically referred to as constants, with 
arbitrary  individuals  being  represented  as  variables.  These  are 
jointly referred to as terms. A term, without variables is known as  a 
ground  term.  When  we describe an individual, the  descriptor  is  a 
predicate of order n, where n is the number of terms which follow. The 
predicate  and  its terms, together, are known as an  atom.  A  ground 
atom,  is an atom without variables. Semantically, the set  of  ground 
terms is known as the Herbrand Universe, ie the cases within our  CASE 
based data base which are inmates. The Herbrand base for any retrieval 
we may write is the set of ground atoms that we can construct from the 
predicates  available  (often confusingly called  variables)  and  the 
ground term in the Herbrand Universe. This encompasses all we can  say 
about  the  inmates in our database. A Herbrand  Interpretation  is  a 
subset of the Herbrand base, i.e. those assigned the value true. 

Ultimately,  these  notions will be important when we come to  use  AI 
techniques   such  as  the  resolution  principle  to  milk   implicit 
inferences  from  within  our database. Note, that  according  to  the 
thesis  being  developed in these volumes, it is only the  failure  of 
Leibniz's Law within epistemic (intensional) contexts which makes  any 
of this seem remotely difficult. What we are generally concerned  with 
is the creation of well formed formulae, simple atomic propositions or 
predicates,  which  when combined by logical  connectives,  amount  to 
compound  propositions,  or complex predicates.  Such  predicates  are 
generally  described as one-place, two-place or higher indicating  how 
many  argument positions they require. For example, age is  one  place 
predicate,  whilst associate_of, is a two place predicate.  The  PROBE 
data   dictionary  (Volume  5)  lists  one-place,  unary  or   monadic 
predicates.  Two place predicates are also referred to  as  relations. 
Two  place predicates, known as binary relations do not exist  in  the 
schema.  Individual  inmates  can be regarded as  unary  relations,  a 
strange  notion,  but  one which allows one to  treat  individuals  as 
classes. 

An  example  or two may help to make the above more concrete  at  this 
point. Age, NIC score, report rate and index offences will suffice  to 
illustrate  the value of working solely extensionally  with  relations 
and classes. Comparison of the distribution of inmates by age group is 
one  of the population measures provided to the field week as part  of 
weekly  analysis  of  the Long Term  prison  system.  Such  population 
parameters  readily highlight unplanned discrepancies  in  allocation. 
That there is a functional relationship between NIC score and age,  or 
age  and  rate  of disciplinary infractions was used in  Volume  2  to 
highlight how such relations can be used by management in the interest 
of maintaining control.  

Relations are clearly basic to relational data bases, and it should be 
noted that one of the great changes brought about by relativity theory 
was  that  Newtonian  monadic predicates were  replaced  by  relations 
(Churchland 1989). The logic of relations with quantifiers is  perhaps 
the  greatest breakthrough in human thought to date, and is still  one 
of  the most difficult to fully appreciate. Frege's  'Concept  Writing 
Script'  (his  'Begriffsschrift' or  Predicate  Calculus)  effectively 
introduced  for the first time, cognition or reasoning, as  a  formal, 
mechanical process. Here is how Carnap (1933) introduced the notion of 
the new logic:


    'The new logic is distinguished from the old not only by  the 
    form  in  which  it  is presented but  chiefly  also  by  the 
    increase   of  its  range....The  only  form  of   statements 
    (sentences)  in  the  old logic  was  the  predicative  form: 
    "Socrates  is  a  man," "All (or some)  Greeks  are  men."  A 
    predicate-concept  or  property is attributed to  a  subject-
    concept.  Leibniz  had already put forward  the  demand  that 
    logic  should  consider sentences of relational  form.  In  a 
    relational sentence such as, for example, "a is greater  than 
    b," a relation is attributed to two or more objects, (or,  as 
    it might be put, to several subject-concepts). Liebniz's idea 
    of  a  theory  of relations has been worked out  in  the  new 
    logic.  The  old  logic  conceived  relational  sentences  as 
    sentences  of  predicative  form.  However,  many  inferences 
    involving relational sentences thereby become impossible.  To 
    be sure, one can interpret the sentence "a is greater than b" 
    in  such  a  way  that the  predicate  "greater  than  b"  is 
    attributed to the subject a. But the predicate then becomes a 
    unity;  one  cannot  extract  b by  any  rule  of  inference. 
    Consequently,  the sentence "b is smaller than a"  cannot  be 
    inferred from this sentence. In the new logic, this inference 
    takes place in the following way: The relation "smaller than" 
    is defined as the "converse" of the relation "greater  than." 
    The  inference  in  question  then  rests  on  the  universal 
    proposition:  If  a  relation  holds between  x  and  y,  its 
    converse  holds  between  y and x. A  further  example  of  a 
    statement that cannot  be proved in the old logic:  "Wherever 
    there  is a victor someone is vanquished." In the new  logic, 
    this follows from the logical proposition: If a relation  has 
    a referent, it also has a relatum. Relational statements  are 
    especially  indispensable for the mathematical sciences.  Let 
    us  consider  as an example the geometrical  concept  of  the 
    three-place  relation "between" (on an open  straight  line). 
    The geometrical axioms "If a lies between b and c, b does not 
    lie between c and a" can be expressed only in the new  logic. 
    According to the predicative view, in the first case we would 
    have  the  predicates  "lying between b  and  c"  and  "lying 
    between  c and a". If these are left unanalyzed, there is  no 
    way of showing how the first is transformed into the  second. 
    If  one takes the objects b and c out of the  predicate,  the 
    statement  "a  lies  between b and c"  no  longer  serves  to 
    characterise  only one object, but three. It is  therefore  a 
    three-place relational statement....

    Restriction to predicate-sentences has had disastrous effects 
    on  subjects outside logic. Perhaps Russell is right when  he 
    made   this   logical   failing   responsible   for   certain 
    metaphysical  errors.....Above all, we may well  assume  that 
    this logical error is responsible for the concept of absolute 
    space.  Because the fundamental form of a proposition had  to 
    be predicative, it could only consist in the specification of 
    the  position  of a body. Since Leibniz  had  recognized  the 
    possibility of relational sentences, he was able to arrive at 
    a  correct  conception of space: the elementary fact  is  not 
    position  of a body but its positional relations relative  to 
    other bodies. He upheld the view on epistemological  grounds: 
    there  is  no way of determining the absolute position  of  a 
    body,  but  only its positional relations.  His  campaign  in 
    favor  of  the  relativistic view of space,  as  against  the 
    absolutistic views of the followers of Newton, had as  little 
    success as his program for logic.

    Only after two hundred years were his ideas on both  subjects 
    taken  up  and carried through: in logic with the  theory  of 
    relations (De Morgan 1858; Pierce 1870), in physics with  the 
    theory  of  relativity  (anticipatory  ideas  in  Mach  1883; 
    Einstein 1905).'

    R. Carnap
    The Old and the New Logic (1930)
    In A.J. Ayer (ed) Logical Positivism (1959)


Throughout these volumes, the case is made that, for PROBE to be  used 
as  an effective system, it will require users to analyse  and  manage 
inmate   behaviour   exclusively  according  to  an   inmate's   class 
membership, which in turn only makes sense relative to other  classes. 
The monadic predicate calculus (the calculus of classes), it should be 
understood:

    '..  consists  in  characterizing  the  predicates  by  their 
    extension  instead  of according to their  content.  To  each 
    predicate   corresponds   a  certain  "class"   of   objects, 
    consisting of all objects for which the predicate holds.  The 
    case  of  a  class  containing no object  is  of  course  not 
    excluded  here. Classes are now to be taken as  the  entities 
    dealt with by the calculus, which in this interpretation will 
    be called the calculus of classes.

    D. Hilbert & W. Ackermann (1950)
    The Principles of Mathematical Logic p.46
    

As stated above, a list of individuals which can occupy the  positions 
of an n-place, or n-ary, or degree n predicate, is known as an ordered 
n-tuple  (n-membered  sequence), and this is ultimately  what  we  are 
concerned  with as behaviour scientists. Date (1992), who  along  with 
E.F. Codd is a major spokesman for relational theory, had this to  say 
about predicates:

    'It is convenient to assume that the predicates "=", ">", "r" 
    etc,  are builtin (i.e they are part of the formal system  we 
    are  defining)  and that the expressions using  them  can  be 
    written  in  the  conventional manner, but  of  course  users 
    should  be able to define their own additional predicates  as 
    well.  Indeed,  that is the whole point, as we  will  quickly 
    see: The fact is, in database terms, a user-defined predicate 
    is nothing more nor less than a user-defined relation.'
    ...
    'The suppliers relation S, for example, can be regarded as  a 
    predicate with four arguments (S#, SNAME, STATUS, and  CITY). 
    Furthermore   the  expressions  S(S1,  Smith,20,London)   and 
    S(S6,Green,45,Rome)  represent "instances" or invocations  of 
    that predicate that evaluate to true and false respectively.'

    C. J. Date (1992)
    Logic Based Database Systems: A Tutorial Part II p.378
    Relational Database Writings 1989-1991  


The  import of this statement marks an important step on the route  to 
widescale  practise   of logical and  actuarial  behaviour  management 
rather than ad hoc clinicalism which as we have seen in Volume 1,  can 
only  be less precise instances of the former, acceptance of this  may 
be limited solely by the fact that it is all so relatively new:


    'Research  on  the relationship between database  theory  and 
    logic  goes back at least to the late 1970s, if not  earlier. 
    However,  the principal stimulus for the recent  considerable 
    expansion  of interest in the subject seems to have been  the 
    publication  in 1984 of a landmark paper by  Raymond  Reiter, 
    "Towards  a  Logical Reconstruction  of  Relational  Database 
    Theory,"  which  appeared in a book  entitled  On  Conceptual 
    Modelling:   Perspectives   from   Artificial   Intelligence, 
    Databases,   and   Programming   Languages   (eds.    Brodie, 
    Mylopoulos,  and  Schmidt;  Spinger-Verlag,  1984).  In  that 
    paper,  Reiter  characterised the traditional  perception  of 
    database  systems as model theoretic - by means of  which  he 
    meant, speaking very loosely, that:

    (a)  The  database is seen as a set of explicit  (i.e.  base) 
    relations, each containing a set of explicit tuples, and

    (b)  Executing  a query can be regarded  as  evaluating  some 
    specified  formula  (ie truth-valued expression)  over  those 
    explicit relations and tuples.

    Reiter  then  went  on to argue that  an  alternative  proof-
    theoretic view was possible, and indeed preferable in certain 
    respects.  In  that alternative view -  again  speaking  very 
    loosely  - the database is seen as a set of axioms  ("ground" 
    axioms,  corresponding  to  tuples in  base  relations,  plus 
    certain "deductive" axioms, to be discussed), and executing a 
    query is regarded as proving that some specified formula is a 
    logical consequence of those axioms - in other words, proving 
    that   it  is  a  theorem....Consider  the  following   query 
    (expressed in relational calculus)....
    
    SPX 
    WHERE SPX.QTY > 250

    Here  SPX  is  a tuple variable ranging  over  the  shipments 
    relation  SP.  In  the  traditional  (i.e.   model-theoretic) 
    approach,  we examine the shipment (SPX) tuples one  by  one, 
    evaluating the formula "SPX.QTY > 250" for each one in  turn; 
    the query result then consists of just those shipment  tuples 
    for  which  the  formula  evaluates to  true.  In  the  proof 
    theoretic  approach,  by contrast, we consider  the  shipment 
    tuples  (plus  certain other items) as axioms  of  a  certain 
    "logical theory"; we then apply theorem-proving techniques to 
    determine  for which possible values of the variable SPX  the 
    formula  "SPX.QTY  > 250" is a logical consequence  of  those 
    axioms within that theory. The query result then consists  of 
    just those particular values of SPX.'

    ibid p.267-368

Although  there is a degree of confusion in terminology in  the  area, 
Date (1992) suggests that a Deductive Database Management System is:

    'a  database  that  supports the proof-theoretic  view  of  a 
    database, and in particular is capable of deducing additional 
    facts   from  the  "extensional  database"  (i.e.  the   base 
    relations) by applying specified deductive axioms or rules of 
    inference  to  those facts. The deductive  axioms,  together, 
    together  with the integrity constraints  (discussed  below), 
    form  what  is sometimes called  the  "intensional  database" 
    (IDB),  and  the  extensional database  and  the  intensional 
    database  together  constitute  what is  usually  called  the 
    deductive  database  (not a very good term, since it  is  the 
    DBMS, not the database, that carries out the deductions).

    As just indicated, the deductive axioms form one part of  the 
    intensional  database. The other part consists of  additional 
    axioms that represent integrity constraints (i.e. rules whose 
    primary purpose is to constrain updates, though actually such 
    rules  can also be used in the deduction process to  generate 
    new  facts)....it now becomes more important than  ever  that 
    the  extensional  database not violate any  of  the  declared 
    integrity constraints! - because a database that does violate 
    any  such  constraints  represents  (in  logical  terms)   an 
    inconsistent  set  of  axioms,  and it  is  well  known  that 
    absolutely  any  statement  whatsoever can be  proved  to  be 
    "true"   from  such  a  starting  point  (in   other   words, 
    contradictions  can be derived. For exactly the same  reason, 
    it  is  also  important  that the  stated  set  of  integrity 
    constraints be consistent.'

    ibid p.394-5 

One might profitably read the above with the failure of Leibniz's  Law 
within  intensional contexts clearly in mind. Similarly, neophyte  PQL 
programmers  soon find that the reason why most of what they  want  to 
achieve  fails to materialize is due to errors in  their  programming, 
which  invariably  come down to them not specifying step by  step  the 
logical  and procedural steps of their query. Here again,  the  actual 
user, rather than the casual reader will appreciate the didactic force 
of the imperative "stay out of your head, and look at the screen". The 
experienced  user  should  appreciate that  the  keyboard  and  screen 
comprise  a  very  effective system of  'virtual'  reality,  which  is 
improved by a mouse. 

One  of  the main advantages of a formal database system  is  that  as 
updates  are  made to the overall data  structure,  cross  referencing 
maintains  database  integrity  constraints  by  only  making  updates 
according  to well established update rules. We have seen  at  length, 
the  problems  which  results from failure  of  substitutivity  within 
intensional  contexts  -  namely,  that  deductive  inference  is  not 
possible. Within PROBE, deductively driven updates are currently quite 
minimal, restricted essentially to PQL 'retrieval updates' which cross 
update inmate cell location and prison location across relations 3 and 
11.   Where  further  updates  are  possible,  implementation   beyond 
providing  quality  control  reports has been refrained  from  in  the 
interests of maintaining a degree of user input to maintaining overall 
system integrity. 

Returning  to  the  terminology  of  relational  technology,  where  a 
predicate  is  a  two-place predicate, it is an  ordered  2-tuple,  or 
ordered pair. A tuple is a row, and a relation is a set of  predicates 
comprising  a  record type (sometimes called a table). In  almost  all 
instances, whether a retrieval generates a simple list of inmates,  or 
a  multivariate statistical analysis (with post-processing using  SPSS 
for  multiple or logistic regression for example), we are  practically 
interested  in  value  distributions (Kerlinger  and  Pedhazur  1973). 
Carnap (1959) summarised the situation as follows (although it  should 
be  appreciated  that Quine's austere,  wholly  extensionalist  system 
developed  in  Word and Object (1960) was largely a  critique  of  the 
intensionalism which remained within Carnap's "Meaning and  Necessity" 
program):

Intensions and Extensions of the Chief Types of Expressions


Expression               Intension           Extension
Sentence                 Proposition         Truth-value
Individual constant      Individual concept  Individual
One-place predicate      Property            Class of individuals
n-place predicate (n>1)  n-place relation    Class of ordered n-tuples of 
                                             individuals
Functor                  Function            Value-distribution

Carnap (1958)
Introduction to Symbolic Logic and its Applications

In an annex to a short paper entitled 'What is a Relation' Date (1992) 
put the situation as follows:

    'In  the body of this paper, I gave the mathematician's  view 
    of  a relation as "An n-ary relation is a set of  ordered  n-
    tuples."  In  this  appendix,  I would  like  to  mention  an 
    alternative view very briefly - namely, the logician's  view. 
    In  logic,  an  n-ary  relation  is  simply  that  which   is 
    designated  by  an n-place predicate in what  is  called  the 
    first  order predicate calculus. For example, the  expression 
    ">(A,b)  is a 2-place predicate that designates the  "greater 
    than"   relation,  and  "SP(S#,P#,QTY)"  is   a   three-place 
    predicate  that  designates the "shipments" relation  in  the 
    usual  suppliers and parts database. In general,  an  n-place 
    predicate can be thought of as a truth-valued function with n 
    arguments;  a  given  tuple  appears  in  the   corresponding 
    relation  if and only if the function evaluates to  true  for 
    the argument values represented by that tuple.
    ..
    When  we talk about the foundations of the relational  model, 
    we  usually  talk  in  terms  of sets  and  set  theory  -  a 
    mathematical foundation, in fact. But the forgoing  indicates 
    that  it is at least equally possible to talk in terms  of  a 
    foundation  in  logic  - specifically,  in  the  first  order 
    predicate calculus - instead. And this alternative perception 
    does have certain arguments in its favor....some people would 
    argue  that  the true foundation of the relational  model  is 
    really  the first order predicate calculus, not  set  theory, 
    and  moreover  that  there is no real  need  to  invoke  set-
    orientated  ideas  at all in developing  and  discussing  the 
    model.'

    C. J. Date (1992)
    What is a Relation? A Logician's View
    Relational Database Writings 1989-1991 p.54-5

Whilst  initially  unfamiliar,  this logical notation,  basic  to  the 
predicate  or  functional calculus, provides an  invaluable  framework 
when  designing and managing data base management system's  structure, 
when  planning  analyses  and programming  automated  reports.  It  is 
certainly  easier  to  deal with in the author's view  than  the  more 
commonly encountered set theoretic terminology, and renders the  links 
with  work in theoretical logic (e.g. Quine 1960, 1992)  much  easier. 
All database systems must be reduced to 'normal form' in the interests 
of  being able to analyse the modelled domain at its most  fundamental 
levels. Through Quine's critique of analyticity (1951, 1960),  coupled 
with  the  axiomatic nature of Leibniz's Law the language  of  science 
(Quine 1954) has little choice but to dismiss intensional notions such 
as 'sense' (Frege 1883), or 'individual concept' (attribute, property, 
meaning, content etc; Carnap 1947; Church 1951). Intensional  contexts 
are  indeterminate,  and thereby unable to occupy positions  of  bound 
variables  (Quine  1943;1956)  in  any  form  of  scientific  analysis 
(computer or otherwise). 

In  1994, we simply do not know how to use formal  logic  (Information 
Technology)  to quantify reliably into intensional contexts  (such  as 
the  propositional attitudes), and attempts to do so using  techniques 
such  as  Repertory Grids (the 'Fragmentation  Corollary'  aside)  and 
Factor  Analysis may prove to be creative rather than analytical as  a 
consequence.  Less formally, we do not know how to reason within  such 
contexts  without  falling into rhetoric and sophistry. Until  we  are 
shown otherwise, extensional systems render us incapable of  analysing 
inmates  by anything other than the classes which they fall  into.  We 
can  do  no  more  than use  quantification  theory  to  extensionally 
identify  the functional relations which exist between  such  classes, 
and manage behaviour according to such functions.

Compound predicates, or n-ary relations e.g. Governor's reports can be 
created     such    as     'Rule_Paragraph',     'Date_of_Infraction', 
'Time_of_Infraction',     'Location_in_Prison'    and     a     unique 
'Inmate_Case_Identifier' (the constant, or when quantified, a variable 
x). Each predicate returns one, and only one value, and together  they 
comprise a vector which can be analysed like the values of any  simple 
or  atomic  predicate. In this way, it is possible,  using  relational 
technology,  to define the arity of relations or predicates using  the 
logical connectives within a fourth generation retrieval language  and 
thereby  expand or restrict relations or predicates to certain  times, 
dates,  places, or to inmates with certain classes of  index  offence, 
ages, or whatever the algorithm written, actually 'satisfies'  (Tarski 
1931)  through the tuples meeting the specified value criteria of  the 
well-formed formula (wff). That is, an instance (or instantiation)  of 
a  clause is obtained by applying a substitution to the clause, and  a 
substitution  is an assignment of terms to variables (Kowalski  1979). 
An example to illustrate the above should clarify the terminology  and 
illustrate  the potential of working within this framework, given  our 
understanding of Leibniz's Law. 

We will take record three of PROBE, Behavior  at  The  Current  Prison 
(CURPRIS). There are (34 Records in all, several one-many (eg. reports
movements, segregation periods, attainment assessments).

Key

     'Variable'     'Variable Label'
   01 a NATNUM      NATIONAL NUMBER                      
   02 b PRESCAT     PRESENT SECURITY CATEGORY            
a  03 c EDRCPRIS    EDR or NPD CURRENT PRISON            
r  04 d PRISON      CURRENT ESTABLISHMENT                
g  05 e DOR         DATE OF RECEPTION                    
u  06 f WINGINST    CURRENT WING                         
m  07 g TPPSYC      PSYCHIATRIC DIAGNOSIS AT CURRENT PRISON
e  08 h TPDRUGS     EVIDENCE OF DRUGS THIS PRISON        
n  09 i ELIST       PLACED ON E LIST THIS PRISON         
t  10 j NEWHOST     HOSTAGE TAKER AT THIS PRISON         
   11 k TPR43OR     RULE 43(OR) SEGREGATIONS THIS PRISON 
p  12 l TPR43GO     RULE 43(GOAD) SEGREGATIONS THIS PRISON
l  13 m TPC1074     CI1074/3790 TRANSFER FROM THIS PRISON
a  14 n TPSTVIO     (PROVEN) STAFF ASSAULTS THIS PRISON  
c  15 o TPINVIO     (PROVEN) INMATE ASSAULTS THIS PRISON 
e  16 p TPADJ       (PROVEN) ADJUDICATIONS THIS PRISON   
   17 q PSYMON3     PSYMON vs F1150 FLAG(3)              
   18 r DATMOD03    MODIFIED                             



Relation Name              = Curpris
Argument Positions (arity) = 18 

As an 18-ary relation:

            A R G U M E N T   P O S I T I O N S
                                                   1        1    1
               1   2    3      4     5     6 7 8 9 0        7    8 
 T  Curpris(113386,2,01011700,LLC,05041991,A,0,0,0,0,..... ,0 10041991)
 U  Curpris(119085,1,01011700,LLC,14111991,Z,0,0,0,0,..... ,0 01061993)
 P  Curpris(122004,2,01011700,LLC,14101988,B,1,0,0,0,..... ,0 30011989)
 L  Curpris(132016,1,01111988,LLC,01021979,E,0,0,0,0,..... ,1 01051988)
 E  Curpris(132687,1,01011700,LLC,30101989,S,0,0,0,0,..... ,0 29031990)
 S  Curpris(133616,2,01011700,LLC,11051982,F,0,0,0,0,..... ,0 01061993)

Or as a series of binary predicates:


   01  Natnum(113386,Curpris)   
   02  Natnum(119085,Curpris) 
P  03  Natnum(122004,Curpris) 
R  04  Natnum(132016,Curpris) 
E  05  Natnum(132687,Curpris) 
D  06  Natnum(133616,Curpris) 
I  07  Prescat(113386,2) 
C  08  Prescat(119085,1) 
A  09  Prescat(122004,2) 
T  10  Prescat(132016,1) 
E  11  Prescat(132687,1) 
S  12  Prescat(133616,2) 
   13  Etc., etc.  
   14  Etc., etc.  

Queries can then be expressed in 'clausal form' as:


  Answer(x)
  Answer(x)  Inmate(x, Curpris) AND Prescat(x,1)

  or 
  
  Answer(x)
  Answer(x)  Curpris(x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o)
    AND
  Curpris(x,1,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o)

Here, the value '1' is substituted for the variable b in order to list 
all inmates with a value of 1 for Present Security Category (Prescat). 
This  presentation  should make it graphically clear  why  some  query 
languages  are given the name 'Query By Example'. The same  format  is 
followed  of course when instantiating queries with  predicates  drawn 
from other relations such as Person, Utadata, Curpris, Reports and  so 
on.  As  covered  at length in Volume 1 and the early  parts  of  this 
volume,  the  fundamental value of relational,  deductive  technology, 
lies in the failure of effective substitutivity of identicals,  'salva 
veritate', within intensional contexts. The  failure of Leibniz's  Law 
within  epistemic and other intensional contexts renders anything  and 
everything inferable given the violation of the law of  contradiction, 
or failure of truth-functionality within such contexts. 

Comprehensive relational modelling and extensional deductive  analysis 
within a domain, or universe of discourse comprises a science of  that 
domain.  The  application of the theorems derived from  analysis  back 
into  the  domain,  comprises  a  technology.  There  can  be  nothing 
controversial  about this claim once the logical basis  of  relational 
theory and scientific method are clearly understood in conjunction and 
the  significance  of the failure of Leibnitz Law  within  intensional 
contexts is fully appreciated. 

At  the  end  of  Volume 1, and certainly within  Volume  2,  we  used 
functional  notation  rather  than  the  language  of  relations   and 
predicates,  so  before leaving the subject, we  show  how  functional 
notation  expresses  predicates  or  relations.  Recall  that  in  his 
discovery  of the Predicate Calculus (his 'Begriffsschrift') in  1879, 
Frege wrote that his discovery of the quantifiers was in large part  a 
consequence  of  rejecting the old  Predicate-Argument  notation,  and 
selecting instead an extended concept of the mathematicians notion  of 
function-argument,  at the same time, we will deal with the  important 
issue of equality or identity.

We can express  Times(x,y,z)
                    as        x * y = z
or              Father(x,y)   
                    as        x = father(y)

Relational calculus query language uses function symbols and equality:

prescat(x)     = y                                                   
                in place of Prescat(x,y)
winginst(x)    = y
                in place of Winginst(x,y)

The following is taken from Kowalski (1979), and effectively brings us 
full  circle  to the leitmotif of these volumes,  beginning  with  the 
quote at the beginning of Volume 1: the identity of indiscernibles and 
the failure of Leibniz's Law within intensional contexts.


    'Equality  is necessary whenever an individual has more  than 
    one name. For example:
                    
                 Jove = Jupiter .

    It  is  also necessary, even in the relational  notation,  to 
    express that one argument of a relation is a function of  the 
    others. For example:

                   x = y  Father(x,z) , Father(y,z)

    To  show  that  a set of clauses S  containing  the  equality 
    symbol  is inconsistent, the set of clauses needs to  contain 
    the  following axioms characterising the  equality  relation, 
    for  every  function symbol f and every  predicate  symbol  P 
    occurring in S, (including the equality symbol).

E1 x = x 
E2 P(x1,.....,xm)  P(y1,......,ym), x1=y1, ..., xm=ym
E3 f(x1,.....,xm) = f(y1,......,ym)  x1=y1, ..., xm=ym

for example, to demonstrate that the assumptions

J1 Jekyl = Hyde 
J2 father(John) = Hyde
J3 Member(father(John), birthday club) 

imply the conclusion

member(Jekyl, birthday club) 
it is necessary to deny the conclusion
J4 Member(Jekyl, birthday club)

and add the appropriate axioms for the equality relation:

J5 x = x 
J6 Member(x1,x2)  Member(y1,y2), x1=y1,x2=y2
J7 x1 = x2  y1 = y2, x1 = y1, x2 = y2
J8 father(x) = father(y)  x = y

The following set of clauses J1-8 is inconsistent because J1-3 are 
"obviously" inconsistent with the instances

Hyde = Hyde 
birthday club = birthday club 
Member(Jekyl, birthday club)  Member(father(John), birthday club),
                   Jekyl = father(John),
                   birthday club = birthday club
Jekyl = father(John)  Hyde = Hyde, Jekyl = Hyde, father(John) = Hyde

of J5-7. Clause 8 in this example does not contribute to the inconsistency.

R. Kowalski (1979)
Representation in Clausal Form: Equality
Logic for Problem Solving

In  managing  and designing and developing the PROBE  system,  special 
care  has  been  taken to ensure that  these  extensional,  ie  truth-
functional  principles  are followed and  that  referential  integrity 
constraints  or rules on data entry are built in to  optimize  quality 
control.  Training has emphasised the pitfalls of  clinical  judgement 
(Volume  1)  as  evidence  of the  failure  of  quantification  within 
intensional  contexts.  At times this has  been  extremely  difficult, 
since  many  users  still regard databases as 'nothing  more  than'  a 
research  data  storage  medium.  This  can  only  stem  from  a  poor 
conception  of the technology behind record (table) design, the  power 
of  'normal form' or 'clausal form' as an artificial language,  and  a 
very limited practical use of the such systems, e.g. the production of 
simple  lists  rather than full relational analyses.  This  functional 
specification  is  designed to suggest how the PROBE system  might  be 
used in support of an applied behaviour science and technology,  which 
in turn supports effective inmate management, not, it must be said, as 
an all purpose MIS. Any failure to fully appreciate these points  will 
inevitably lead to great financial investments with very little in the 
way of productive returns. Without a sound appreciation of logic, such 
systems  simply will not be used effectively. This is a simple  lesson 
from research in descriptive (folk) psychology (Volume 1).

For  those  who  are  sceptical about  the  value  of  checklists  for 
instance,  it is important perhaps to point out that the prison  rules 
can  be  listed  as 21 paragraphs under Rule 47, ie  as  a  series  of 
observation statements. An inmate will always be charged under one, or 
another  paragraph of the Rule (each as a separate event or  offence). 
The  paragraphs  serve as a set of declarative statements  (32  binary 
predicates or a 32-ary relation if the circumstances such as date  and 
time,  place  etc.  are included in the tuple).  The  Rule  47  system 
effectively operate as a behaviour checklist, or criterion referencing 
system. Where no offences have occurred it is as if null entries  were 
entered  for  each inmate, date, time and place - something  which  is 
made graphically clear when actual offences are plotted against  time. 
Construed  from the perspective of relational theory, this removes  in 
one  move, any objections to 'box ticking' as a means  of  assessment, 
since it can readily be seen that all inmate management must be  based 
on  such  predicate or relational systems, albeit sometimes  of  quite 
high  arity,  and therefore for memory capacity constraints,  quite  a 
bewildering  i.e.  impossible task for working memory as  outlined  in 
Volume  1  and elsewhere (Miller 1956; Attneave 1959;  Cherniak  1986, 
Stich 1990).

Based  on  this conception of a Data Base Management  System,  PROBE's 
second  phase  of development work between 1991 and 1994  enabled  the 
system  to  map entire prison regimes using  the  relational  concepts 
outlined above (and as illustrated in Volume 2). A system of  Sentence 
Management was designed whereby staff are able to continuously  define 
(and up to a point, dynamically refine) the regime functions they  are 
responsible for supervising, be these elements of wing routines or the 
requirements (performance criteria) of specific inmate activities such 
as education courses, periods in prison industries, special programmes 
etc.  Within the PROBE Sentence Management system, staff are  required 
to  define  declarative  statements  (predicates/regime  propositional 
functions/relations) or 'Attainment Criteria' which can be assessed as 
being true or false of an inmate, at specific stages of programmes, on 
specified  dates. Just as the truth or falsehood (guilt or  innocence) 
of a prison rules infraction is ascertained by an expert on the prison 
rules  (a  Governor), so too, the level of attainment  an  inmate  has 
attained is ascertained by, ideally, an accredited, expert supervisor. 
This system allowed us to expand the arity of the relations  available 
within  the  PROBE  relational  Data  Base  Management  System  almost 
infinitely  without  having to make physical changes to  the  system's 
data dictionary (the schema - Volume 5). Such a criterion  referencing 
system  can  develop flexibly, with individuals  being  profiled  with 
reference  to  such  criteria at any stage  of  their  prison  career. 
Together,  therefore,  the  predicates/relations/functions  and  truth 
values  within PROBE serve as a Knowledge Base for the  production  of 
comprehensive   inmate   career  profiles   which   are   descriptive, 
declarative  reports of inmate behaviour relative to  fixed  reference 
criteria.  Such extensional reports have clear reference criteria  and 
are produced by algorithms written using the 4GL (PQL) provided within 
the DBMS. The skilled work within such a system lies in the writing of 
retrievals. 

Furthermore, such retrievals can be written to incorporate  parameters 
of  the  population from which the inmate is  drawn,  such  parameters 
thereby  serving  as  reference  classes.  PROBE  routinely   provides 
profiles  which  provide information at both the  individual  (Section 
3.2)  and  group  (Section  3.3) levels.  As  the  technical  work  is 
primarily on the design and use of PQL algorithms in the management of 
inmate's  as  a  function  of  the classes  they  fall  into  and  the 
characteristics  of  those classes (e.g. age group and  report  rate), 
PROBE  is basically an actuarial system (Dawes, Faust and Meehl  1989, 
1993), as well as an application of Artificial Intelligence  research. 
Risk  assessment in all areas of inmate management becomes  largely  a 
matter of ascertaining what classes an individual belongs to, and  the 
characteristics   of  such  classes.  Providing  that  all   concerned 
appreciate  that  individual  assessment  must  always,  albeit  often 
implicitly, be assessment relative to some class or another, and  that 
class  membership  is  a dynamic function  of  ongoing  behaviour,  it 
becomes clear that the PROBE technology amounts essentially to no more 
than an MIS to support effective inmate management based on  actuarial 
rather than clinical judgment. 

As outlined above in the context of quantification, Vickers (1988) and 
Lukasiewicz  (1909)  have  generalized the Fregian  concept  of  truth 
function:

    'The  truth  value of an indefinite  proposition  is  "...the 
    ratio between the number of values of the variables for which 
    the  proposition yields true judgements and the total  number 
    of   values   of   the  variables"   (p.17).   The   relative 
    (conditional)  truth value of indefinite propositions is  the 
    quotient of the truth value of their conjunction and that  of 
    the  antecedent.  Lukasiewicz then argues  that  these  truth 
    values provide an adequate account of probability, free  from 
    many  of  the  difficulties that  plague  subjectivistic  and 
    empirical views.'

    J. M. Vickers (1988)
    Chance and Structure: 
    An Essay on the Logical Foundations of Probability p.149

Statistical  technology is covered little in these volumes since  most 
readers  will  have already undertaken the  course  which  complements 
these  volumes. However, for the sake of what follows it is  important 
that  the reader appreciates that we are, at least in part,  following 
Vickers (1988) in his treatment of Fregian quantification:


    'The major motivating principle of probability quantifiers is 
    the  development of probability within pure or general  logic 
    to the extent that this is possible. The great difficulty  of 
    precisely  defining general logic can perhaps be  avoided  by 
    agreeing that however it is defined, the semantics of  first-
    order  logic  as  developed by Frege and  Tarski  fall  quite 
    within its confines. Then, as the above remarks suggest,  the 
    question  is  just  to  what  extent  such  notions  as  "the 
    proportion  of  objects  falling under  a  concept"  or  "the 
    proportion of assignments satisfying a formula" can be  given 
    a meaning in general logic.'

    J. M. Vickers (1988)
    Chance  & Structure: An Essay on the Logical  Foundations  of 
    Probability
    Probability quantifiers:principles and semantics p.153


-- 
David Longley

