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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
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Theoretical basis of GUHA Definition 1. A (simplified) observational predicate languageLn consists of (i) (unary) predicates P1,…,Pn, and an infinite sequence x1,…,xm,… 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 Q1,…,Qk to be specified later. Given an observational predicate language Ln 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(P1(x),P2(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
Assume you have a row data ... TRUE TRUE TRUE TRUE FALSE TRUE You have to make it Boolean e.g. in the following way... It becomes (automatically) a model in a simple way, namely...
Given an observational language Ln , consider all mn-matrices composed of “0”:s and “1”:s; the set M of modelsM of Ln. Fix such a model M. For example, a M54-matrix: Associate to each i = 1,…,m (row) and each j = 1,…,n (column) a function v such that I 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 Ln{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 = x1,…,xm and v(x(x)) = TRUE iff v((x)) = TRUE for some x = x1,…,xm.
Note. Given a model M, the value v() of any formulae of the language Ln can be calculated immediately. In some cases we consider several models M, N,…Thus, to avoid confusion, we write vM, vN,…. 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: =>Cx((x),(x)) [read: implies ]. (not to be confused with the logical implication ). Given a model M, we define v(=>Cx((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 mn 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
(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-implicationsp,n , where nN, 0<p1, p rational: p,nx((x),(x))[read: implies with confidence p and support n] Given a model M, we define v(p,nx((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 Ln, 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 Ln,M,VAL,V is an observational semantic system. We say that a sentence Ln is a tautology (note ) if v() = TRUE for all valuations vVAL (and, thus, in all models M). Moreover, Ln is a logical consequence of a finite set A of sentences of Ln 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
Definition 3. An observational semantic system Ln,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 Ln,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 Ln is a recursive set. The claim now follows by the fact that TAUSENT. 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 calleddeduction rules, too.
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)[Wx1(x)Wx2(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 Pi(x) =>CPj(x) such that v(Pi(x) =>CPj(x)) = TRUE, (ij). 9. sentences x(Pi(x) Pj(x)) such that v(x(Pi(x) Pj(x))) = TRUE, (ij). 10. sentences Pi(x) 0.9,10Pj(x) such that v(Pi(x) 0.9,10Pj(x) ) = TRUE, (ij). 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)?
