Tableaux for Murphree's numerical term logic

Abstract

Murphree's Numerical Term Logic is a logic capable of representing and performing inference with numerical quanti ers by modifying Sommers Term Functor Logic: in this contribution we o er a tableaux method for it. In other places we have developed tableaux for Sommers' Term Functor Logic [19,8,21], Intermediate Term Functor Logic a la Thompson [23], and Englebretsen's Modal Term Logic [7]. In this contribution we continue our exploration into the realm of term logics and, by using Murphree's Numerical Term Logic [13], we present tableaux for a family of term logics comprising Term Functor Logic, Intermediate Term Functor Logic, and Numerical Term Logic. Hence, for a more detailed exposition of Term Functor Logic and its tableaux, we refer the reader to our previous work [4]. Meanwhile, in order to achieve our present goal, we rst provide a summary of some preliminary concepts (syllogistic, term functor logic, and term functor logic tableaux), then we explain the basics of Murphree's logic (namely, its syntax and rules) and, nally, we develop tableaux for it (including some metalogical results).

Keywords

• Semantic trees
• term logic
• numerical term logic

Introduction

2.1

Preliminaries

Syllogistic

Syllogistic is a term logic that has its origins in Aristotle's Prior Analytics [ 1 ] and deals with inference between categorical propositions. A categorical proposition is a proposition composed by two terms, a quantity, and a quality. The subject and the predicate of a proposition are called terms: the term-schema S denotes the subject term of the proposition and the term-schema P denotes the predicate. The quantity may be either universal (All ) or particular (Some) and the quality may be either a rmative (is) or negative (is not ). These categorical propositions have a type denoted by a label (either a (universal a rmative, SaP), e (universal negative, SeP), i (particular a rmative, SiP), or o (particular negative, SoP)) that allows us to determine a mood, that is, a sequence of three categorical propositions ordered in such a way that two propositions are premises and the last one is a conclusion. A categorical syllogism, then, is a mood with three terms one of which appears in both premises but not in the conclusion. This particular term, usually denoted with the term-schema M, works as a link between the remaining terms and is known as the middle term. According to the position of this middle term, four gures can be set up in order to encode the valid syllogistic moods (Table 1).1

First Second Third Fourth gure gure gure Figure aaa eae aii eio eae aee eio aoo iai aii oao eio aee iai eio Term Functor Logic (TFL) [ 18,19,21,6,8,9 ] is basically a plus-minus algebra that employs terms rather than rst order language elements such as individual variables or quanti ers (cf. [ 17,14,10,19,20,11 ]). According to this algebra, the four categorical propositions can be represented by the following syntax [ 8 ]: a. SaP := S + P b. SeP := S P c. SiP := +S + P d. SoP := +S P

Given this representation, this plus-minus algebra provides a simple rule for syllogistic inference: a conclusion follows validly from a set of premises if and only if i) the sum of the premises is algebraically equal to the conclusion and ii) the number of conclusions with particular quantity (viz., zero or one) is the same as the number of premises with particular quantity [8, p.167]. Thus, for instance, if we consider a valid syllogism, say the mood aaa from the rst gure (i.e. aaa-1), we can see how the application of this method produces the right conclusion (Table 2).

In the previous example we can clearly see how the method works: i) if we add up the premises we obtain the algebraic expression 1 For sake of brevity, but without loss of generality, here we omit the syllogisms that require existential import.

TFL

1. All dogs are animals. D + A

2. All German Shepherds are dogs. G + D

` All German Shepherds are animals. G + A ( D + A) + ( G + D) = D + A G + D = G + A, so that the sum of the premises is algebraically equal to the conclusion and the conclusion is G + A, rather than +A G, because ii) the number of conclusions with particular quantity (zero in this case) is the same as the number of premises with particular quantity (zero in this case). 2.3

TFL tableaux

As in [ 4 ], we say a tableau is an acyclic connected graph determined by nodes and vertices [ 5,16 ]. The node at the top is called root. The nodes at the bottom are called tips. Any path from the root down a series of vertices is a branch. To test an inference for validity we construct a tableau which begins with a single branch at whose nodes occur the premises and the rejection of the conclusion: this is the initial list. We then apply the rules that allow us to extend the initial list (Diagram 1.1).

A Ai

B Bi +A

B +Ai

Bi

Diagram 1.1: TFL tableaux rules

In Diagram 1.1, from left to right, the rst rule is the rule for a (e) propositions, and the second rule is the rule for i (o) propositions. Notice that, after applying a rule, we introduce some index i 2 f1; 2; 3; : : :g. For propositions a and e, the index may be any number; for propositions i and o, the index has to be a new number if they do not already have an index. Also, following TFL tenets, we assume the followings rules of rejection: ( A) = A, ( A B) = A B, and ( A A) = +( A) + ( A).

As usual, a tableau is complete if and only if every rule that can be applied has been applied. A branch is closed if and only if there are terms of the form Ai and Ai on two of its nodes; otherwise it is open. A closed branch is indicated by writing a ? at the end of it; an open branch is indicated by writing 1. A tableau is closed if and only if every branch is closed; otherwise it is open. So, again as usual, A is a logical consequence of the set of terms (i.e. ` A) if and only if there is a complete closed tableau whose initial list includes the terms of and the rejection of A (i.e. [ f Ag ` ?). Accordingly, up next we provide some examples for basic syllogistic (Diagram 1.2).

M + P

S + M ` S + P ( S + P) +S P +S1

P1 M1 ? M P

S + M ` S ( S +S + P

P

P) +S1 +P1 S1 ? S1 ? +M1 +P1

? M1 ? +M1

P1 ?

M + P +S + M ` +S + P (+S + P)

S P +S1 +M1

S1 ?

M P +S + M ` +S (+S

S + P

P

P) +S1 +M1 M1 ? M1 ? +P1

P1 ? S1 ?

P1 +P1 ?

Diagram 1.2: Moods aaa-1, eae-1, aii-1, and eio-1 3

Murphree's Numerical Term Logic

In Szabolcsi's view [22, p.3], quanti cation within rst order logic systems is \extremist" in the sense that it is limited to represent two extremes: either everything or something.2 Indeed, it is usual to nd logical treatments that assume that intermediate or subjective quanti ers [22, p.26 ] such as \many", \most", 2 This is not completely true because, for example, we have the proposals developed by [ 12,24 ] (however, since these are not part of rst order logic prima facie and or \few" (cf. [ 15,23 ]) have to be treated as simple cases of existential quanti ers, and numerical or objetive quanti ers usually su er the same fate. However, such a treatment is far from being justi ed. There are several reasons to support this claim, but in the interest of time, consider that reasoning in natural language includes more quanti ers than the usual \all" or \some". For instance, regard that both \most" and \some" are particular (i.e. non-universal) quanti ers, but they do not express the same meaning: the former implies the latter, but the converse is false; and the same occurs when reasoning with numerical quanti ers. Given this situation, Murphree [ 13 ], and Szabolcsi [ 22 ] developed numerical extensions of Sommers' Term Functor Logic: in this work, however, we focus only on Murphree's [ 13 ].

Murphree's Numerical Term Logic (NTL) is a term logic capable of representing and performing inference with numerical quanti ers by modifying TFL's syntax and rule of inference. Thus, NTL o ers the following syntax for numerically quanti ed propositions: e. All but n S are P := nS + P f. At most n S are P := nS P g. At least n S are P := +nS + P h. At least n S are not P := +nS

P

To better explain this representation, consider the next examples: 1. At least 15 Americans are republicans := +15A + R 2. All but 120 students understand logic := 120S + U 3. At most 150,000 Europeans are male := 150;000E 4. There are at least 33 mathematicians := +33M + M 5. At least 66 logicians are not democrats := +66L D A

Now, at this point it is important to notice that when n = 0 (n = 1), the traditional universal (particular) categorical propositions are included within NTL: 6. All S is P = All but 0 S are P := 0S + P 7. No S is P = At most 0 S are P := 0S P 8. Some S are P = At least 1 S is P := +1S + P 9. Some S are not P = At least 1 S is not P := +1S P

Also, by adapting some ideas by Szabolcsi, in NTL we can represent exact (10), fractionary (12), and subjective (13) quanti ers. This last example shows in what sense the intermediate propositions of Intermediate Term Functor Logic (TFL+) are included within NTL: 10. Exactly n S are P := +(+n 1S + P) + ( nS 11. At most r=q of S are P := r=qS P P) they are developed by and for rst order languages, we omit their treatment in this contribution) and, in all fairness, we can charge TFL with the same o ense. 12. Many S are P := +mS + P

Given these syntactical tweaks, NTL modi es TFL's rule of inference as follows: a conclusion follows validly from a set of premises if and only if i) the sum of the premises is algebraically equal to the conclusion, ii) the number of conclusions with particular quantity (viz., zero or one) is the same as the number of premises with particular quantity, and iii) either (a) the value of a universal conclusion is equal to the sum of the values of the universal premises, or (b) the value of a particular conclusion is equal to the di erence of the universal premise minus the particular.3 Hence, TFL and TFL+ are sublogics of NTL. To exemplify NTL, let us consider some valid inferences (Tables 3-6).

Proposition NTL

1. All but 6 philosophers are logical. 6P + L

2. All but 20 smart people are philosophers. 20S + P

` All but 26 smart people are logical. 26S + L

Before we move on, and in order to reach our goal, we need to make a little syntax modi cation that will come in handy in due time: we add the predicate term of a proposition an arbitrary natural number e > n for any other natural n. This results in the following syntax for NTL: e'. All but n S are P := nS +e P f'. At most n S are P := nS e P g'. At least n S are P := +nS +e P h'. At least n S are not P := +nS

e P

Proposition NTL

1. All but 11 anarchists are logicians. 11A +e L

2. At least 30 Mexicans are anarchists. +30M +e A

` At least 19 Mexicans are logicians. +19M +e L 3 This last condition is di erent from Szabolcsi's, who considers that the value of the premises need be equal or greater than the value of the conclusion (cf. [22, p.45].

1. Most guys are friendly. mG +e F

2. All those who are friendly are tolerant. 0F +e T

` Most guys are tolerant. mG +e T NTL

With these elements, we now introduce the tableaux rules for NTL in Diagram 1.3: from left to right, the rst rule is for universal propositions, the second rule is for particular propositions. Notice that after applying a rule we introduce some index i 2 f1; 2; 3; : : :g just as in TFL tableaux; but also, notice that after applying a rule we create a vector v by keeping track of the numerical value n of the proposition. The third diagram, on the other hand, is a rule for ordering atomic terms with a \+" attached. Now, given these rules, we say A is a logical consequence of the set of terms if and only if there is a complete closed tableau whose initial list includes the terms of and the rejection of A, and v = 0. Accordingly, up next we show some examples of valid inferences (Diagrams 1.4 and 1.5) and then we argue that a valid inference in NTL produces a closed complete tree with v = 0 and vice versa.

nA e B nAi eBi v = n +nA e B +nAi

eBi v = n Diagram 1.3: NTL tableux rules +nA +k nA 6C +e L 20S +e C ` 26S +e L ( 26S +e L) +26S e L +26S1

eL1 +20S1 20S1 ? Proof. Consider that a valid inference in NTL may be of two basic general forms (Table 7).

Form 1

Form 2

Now, besides these two general forms, we must consider two cases. For the base case let us consider each basic form. For form 1, take n = m = 0, and for form 2 take n = 0 and m = 1. Hence, NTL collapses with TFL, in which case we just obtain closed complete trees in TFL. For the inductive case, take n = k and m = j. Such a replacement yields a valid inference in NTL, since all the conditions of a valid inference in NTL are preserved. Now, suppose that each form is also valid in NTL for n = k + 1 and m = j + 1 for k; j > 0, then the trees would look as follows: 3T + (+(+eG +4 B) 5 S) 2T +e U 0B +e E 7S +e I +50S +e S +10T +e T ` +5U + (+(+eG +4 E) +38 I (+5U + (+(+eG +4 E) +38 I) 5U (+(+eG +4 E) +38 I +50S1 +eS1 +10T2 +eT2 +2T2 7S1 ? 2T2 ? +eU2 +7S1 +eI1 +38I1 +3T2 0B2 ? 3T2 +(+eG +4 B) 5 S2 ? +eG2 +4B2

5S2 +0B2 +eE2 +4E2 +5U2 38I1 ? v = +50 + 10 2 7 3 + 4 ? 5 0 5 ? 38 4 = 0 Diagram 1.5: Tableau for Table 6 eG2 4E2 +k+j+2>j+1;8k;j>0Z1 Proof. Suppose there is a closed complete tree with v = 0 that is not a valid inference in NTL. Then there is a closed complete tree whose initial list includes a set of terms , the rejection of the conclusion and v = 0, but from we cannot develop a proof of the conclusion by following the rules of NTL. Without loss of generality, consider the trees corresponding to the valid basic forms of Table 7. aTnhden+wmeXhaveeYt.wNooawlt,ersninacteiveeascwhhtorseeeciosnccolumsipolnestea,rteh,erersupleecstifvoerlyg,enen+ramtXing ethYe, trees have been applied; and since each tree is closed, each tree has to have the following forms: ` n+mX e Y ( n+mX e Y) +n+mX e Y +n+mXi eYi

Suppose, then, that we have an instance of the rst tree but that the resulting proof is not valid, that is to say, where + = [ f+n+mX e Yg, + ` ?, but the application of the rule of inference to does not allow us to produce n+mX e Y. Now, since the tree of the diagram is complete, all the tips are closed, hence the previous nodes have to include something of the form m+nX e Y or mZ e Y and nX +e Z, that is to say, = f m+nX e Yg or = f: : : ; mZ e Y; nX +e Z; : : :g. But then, in any case, if we apply condition i) to we obtain something of the form X e Y, and not the other way around due to condition ii); and last, by condition iii)(a), the conclusion has to be of the form m+nX e Y. But that contradicts the assumption that we cannot develop a proof of such conclusion by using the rule of NTL. The same occurs, mutatis mutandis, for the second tree. Thus, = f+mX e Yg or = f: : : ; nZ e Y; +m+nX +e Z; : : :g. In any case, if we apply condition i) to we obtain something of the form +mX e Y, and not the other way around due to condition ii); and last, by the condition iii)(b), the conclusion has to be something of the form +mX e Y. But that contradicts the assumption that we cannot develop a proof of such conclusion. 4

Conclusions

In this contribution we have attempted to o er a tableaux method for a family of term logics comprising Term Functor Logic, Intermediate Term Functor Logic, and Numerical Term Logic by using and adapting Murphree's system. This is possible because TFL and TFL+ are sublogics of NTL and so, in some sense, the method we have developed for NTL represents this family of Sommersian term logics. Now, we have to mention that for the purposes of this paper we have focused only on the terministic features of NTL, but further comparison is required with the algebraic proof systems introduced by [ 3,2 ]; and alternative term logics like Szabolcsi's [ 22 ], for they may have more or less the same motivation but not the same rules.

Acknowledgments

We would like to thank the anonymous reviewers for their precise corrections and useful comments. Financial support given by UPAEP Research Grant.

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