1. Theoretical basis of GUHA Definition 1. A (simplified) observational predicate language L n consists of (i) (unary) predicates P 1,…,P n, and an infinite sequence x 1,…,x m,… of variables (ii) logical connectives  (falsehood), T (truth),  (negation),  (conjunction),  (disjunction),  (implication) and  (equivalence), (iii) classical (unary) quantifiers  (for all)  (there exists), (iv) non-standard (binary) quantifiers Q 1,…,Q k to be specified later. Given an observational predicate language L n the atomic formulae are the symbols , T and P(x), where P is a predicate and x is a variable. Atomic formulae are formulae and if ,  are formulae, then ,   ,   , , ,  x  (x),  x  (x) and Qx(  (x),  (x)) are formulae. Free and bound variables are defined as in classical predicate logic, e.g. in * P(x) and  yP(y)   P(x) x is a free variable but y is a bound variable * P(y)  Qx(P 1 (x),P 2 (x)) and  x[P(y)   P(x)] x is bound but y is free. Formulae containing free variables are open formulae, closed formulae do not contain free variables. Closed formulae are also called sentences. Exercises. Write by symbols the expression in Ex 1 and Ex 2. 1. Most x that are red are round, too. 2. Almost all x that are red are round and vice versa. 3. Is a) Qx(  (x)  (x)) b)  x(  (x)  (x)) a well-formed observational formula? I

2. Assume you have a row data... You have to make it Boolean e.g. in the following way... It becomes (automatically) a model in a simple way, namely... TRUE FALSE

3. Given an observational language L n, consider all m  n-matrices composed of “0”:s and “1”:s; the set M of models M of L n. Fix such a model M. For example, a M 5  4 -matrix: Associate to each i = 1,…,m (row) and each j = 1,…,n (column) a function v such that In particular, v(  ) = FALSE, v(T) = TRUE. Thus, in a model M, rows correspond to variables and columns correspond to predicates. Having in mind our previous example, if Olavi is allergic to tomato, them v(Tomato(Olavi)) = TRUE. We have now defined truth values {TRUE, FALSE} to all atomic formulae. Next we extend truth values to all formulae, that is, the set VAL of valuation functions v: M  L n  {TRUE, FALSE} Let v(  ) and v(  ) be defined. Then we define: Moreover, we define v(  x  (x)) = TRUE iff v(  (x)) = TRUE for all x = x 1,…,x m and v(  x  (x)) = TRUE iff v(  (x)) = TRUE for some x = x 1,…,x m. I

4. Note. Given a model M, the value v(  ) of any formulae  of the language L n can be calculated immediately. In some cases we consider several models M, N,…Thus, to avoid confusion, we write v M, v N,…. Till now we have defined only the truth values of classical quantifiers; they are not of particular interest in GUHA framework. The interesting ones are the non-standard quantifiers; here comes the first four: (1) Resher’s Plurality quantifier W: Wx  (x) [read: Most x have the property  ]. Given a model M, we define v(Wx  (x)) = TRUE iff #{x| v(  (x)) = FALSE}< #{x| v(  (x)) = TRUE}, where ‘#’ means ‘the number of elements’. (2) Church’s quantifier of implication => C : => C x (  (x),  (x)) [read:  implies  ]. (not to be confused with the logical implication  ). Given a model M, we define v(=> C x (  (x),  (x)) = TRUE iff there is no such x that v(  (x)) = TRUE but v(  (x)) = FALSE. Assume x is the only free variable in formulae  (x) and  (x). Let a = #{x| v(  (x)) = v(  (x)) = TRUE}’ b = #{x| v(  (x)) = v(¬  (x)) = TRUE}, c = #{x| v(¬  (x)) = v(  (x)) = TRUE}, d = #{x| v(¬  (x)) = v(¬  (x)) = TRUE}, in a model M m  n Then we have the following Four-fold contingency table M  ¬   a b a+b = r ¬  c d c+d = s a+c=k b+d=l m I

5. (3) Quantifier of simple association ~ : ~x (  (x),  (x)) [read: coincidence of  and  predominates over difference]. Given a model M, we define v(~x (  (x),  (x))) = TRUE iff ad>bc. (4) Quantifiers of founded p-implications  p,n, where n  N, 0<p  1, p rational:  p,n x(  (x),  (x)) [read:  implies  with confidence p and support n] Given a model M, we define v(  p,n x(  (x),  (x))) = TRUE iff a/(a+b)  p and a>n In general, a non-standard quantifier  x(  (x),  (x)) is written in the form  (x)  (x) [read:  and  are associated]. Thus, we shall write *  (x) => C  (x) for Church’s quantifier of implication *  (x) ~  (x) for quantifier of simple association *  (x)  p,n  (x) for quantifiers of founded p-implications Definition 2. Given an observational language L n, the set M of all models M, the set VAL of all valuations v and the set V = {TRUE, FALSE} of truth values, a system  L n,M,VAL,V  is an observational semantic system. We say that a sentence  L n is a tautology (note  ) if v(  ) = TRUE for all valuations v  VAL (and, thus, in all models M). Moreover,  L n is a logical consequence of a finite set A of sentences of L n if, whenever v(  ) = TRUE for all  A, then v(  ) = TRUE, too. Note. In theoretical considerations we are interested in tautologies (true in all models) while in practical GUHA- research we consider only one model, a given matrix M.. I

6. Definition 3. An observational semantic system  L n,M,VAL,V  is axiomatizable if the set TAU of all tautologies is recursively denumerable. Theorem 1. For any natural number n, the semantic system  L n,M,VAL,V  is axiomatizable. Proof. Since the set of all predicates is finite, the set of all variables is denumerable, the set of logical connectives is finite and the set of quantifiers is finite, one can show (details omitted ) that the set SENT of all sentences of L n is a recursive set. The claim now follows by the fact that TAU  SENT. Note. Axiomatizable means that there is a finite set of schemas of tautologies called axioms and a finite set of rules of inference such that all tautologies (and only them) can be reduced from axioms by means of rules of inference, i.e. that all tautologies  have a proof (noted by  —  ). Thus, axiomatizability means by symbols:  iff  — . For example, if  and  are axioms (or, if they have a proof), then one can infer  by means of a rule of inference called Modus Ponens. Thus,  has a proof, too. We are not interested in the general axiom system of GUHA. However, we will need rules of inference to decrease the amount of outcomes of practical GUHA procedures. Therefore, we give the definition of a (sound) rule of inference It has a form  1,…,  n such that, whenever v(  i )= TRUE for all i =1,…,n, then v(  )= TRUE, too  Rules of inference are called deduction rules, too.

7. Exercises. 4. Prove (i)     , (ii)     , (iii)   , (iv)   , (v)   (   )  (    ) , (vi)   (   )  (    ) , (vii)  (   T)  (   ) , (viii)  (   )  , (ix)  (   T )  T, (x)  (   )  (  )  (  ), (xi)   (   )  (    )  (   ), (xii)   (   )  (    )  (   ), (xiii)  ¬¬ , (xiv)  ¬(    )  (¬   ¬  ), (xv)  ¬(    )  (¬   ¬  ), (xvi)  (   ¬  )  , (xvii)  (   ¬  )  T. 5. Prove (i)  (  ), (ii)  x  (x)  Wx  (x), (iii)  Wx  (x)  x  (x), (iv)  [Wx  1 (x)  Wx  2 (x)]  x (  1 (x)   2 (x)), (v)  Wx  (x)  ¬(Wx¬  (x)), (vi)  {  x  (x)  [Wx(  (x)  (x)]}  Wx  (x), (vii)  {Wx  (x)  [  x(  (x)  (x)]}  Wx  (x). 6. Prove that Modus Ponens is a sound rule of inference. 7. Is  a logical consequences of a set {¬   , ¬    }? In problems 8 - 12, consider the ‘Allergic matrix’. Write down all 8. sentences P i (x) => C P j (x) such that v(P i (x) => C P j (x)) = TRUE, (i  j). 9. sentences  x(P i (x)  P j (x)) such that v(  x(P i (x)  P j (x))) = TRUE, (i  j). 10. sentences P i (x)  0.9,10 P j (x) such that v(P i (x)  0.9,10 P j (x) ) = TRUE, (i  j). 11. Does v(Apple(x) ~ Orange(x)) = TRUE hold true, where ~ is simple association? 12. Is v(Tomato(x) ~ ¬ Cheese(x)) = TRUE true ( ~ is simple association)?