After presenting Sommers and Englebretsen's Term Functor Logic, and Thompson's statistical syllogistic, we produce some tableaux for a fragment of Thompson's syllogistic. Term logics are interesting logics. They are Aristotelian in principle, rather than Fregean, and maybe because of that, they have been disparaged in various ways, particularly since the late 19th and the early 20th; however, nowadays, far from being superseded (contra [1, 2, 3]), they are in a path of revision and revival (v.gr. [4, 5, 6, 7, 8, 9, 10]). In this contribution we follow this path and so we ofer a tableaux proof method for statistical reasoning by using a particular term logic. More specifically, after presenting Sommers and Englebretsen's Term Functor Logic, and Thompson's statistical syllogistic, we produce some tableaux for a fragment of Thompson's syllogistic.
Term Functor Logic [4, 11, 12, 6, 13] is a plus-minus algebra that employs terms and functors, in Aristotelian fashion, rather than Fregean, first order language elements such as individual variables or quantifiers. According to this algebra, the four categorical statements of syllogistic, ℒℒ, can be represented by the following syntax [6]:
All S is P := − S + P All S is not P := − S − P
Some S is P := +S + P
Some S is not P := +S − P
Given this representation, Term Functor Logic, ℱ ℒ, 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 [6, p.167]. Thus, for instance, if we consider a valid syllogism, we can see how the application of this rule produces the right conclusion (Table 1).
In this example, we can clearly see how the rule works: i) if we add up the premises we obtain the algebraic expression (− C + A) + (− L + C) = − C + A − L + C= − L + A, so that the sum of the premises is algebraically equal to the conclusion and the conclusion is of the form − L + A, rather than +A − L, 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).1 In contrast, just for the sake of comparison, consider an invalid syllogism that does not add up (Table 2).
Statement
ℱ ℒ 1. All computer scientists are animals. − C + A 2. All computer scientists are logicians. − C + L ̸⊢ All logicians are animals. − L + A
Now, as exposed elsewhere [14, 15], we can develop a tableaux proof method for ℱ ℒ. So, let us say a tableau for ℱ ℒ is an acyclic connected graph determined by nodes and vertices. 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 expansion rules that allow us to extend the initial list (Figure 1).
− S ± P − S
± P (a) +S ± P +S ± P (b)
Figure 3a depicts the rule for universal statements, while Figure 3b shows the rule for particular statements. After applying a rule we introduce some index ∈ {1, 2, 3, . . .}. For universal statements the index may be any natural number; for particular statements the index has to be a new natural number if they do not already have an index. Also, following ℱ ℒ tenets, we assume the following rules of rejection: − (± T) = ∓ T, − (± T ± T) = ∓ T ∓ T, and − (− − T − − T) = +(− T) + (− T).
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 ± A and ∓ A 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 ∞. A tableau is closed if and only if every branch is closed; otherwise it is open. So, as usual, ± T is a logical consequence of the set of terms Γ (i.e. Γ ⊢ ± T) if and only if there is a complete closed tableau whose initial list includes the terms of Γ and the rejection of ± T (i.e. Γ ∪ {∓ T} ⊢ ⊥). As an example, consider Figure 2, which shows the inferences exposed in Tables 1 and 2. 1Although we are exemplifying this logic with syllogistic inferences, this system is capable of representing relational, singular, and compound inferences with ease and clarity. Furthermore, ℱℒ is arguably more expressive than classical first order logic [12, p.172].
− C + A − L + C ⊢ − L + A − (− L + A) +L − A +L1 − A1
+C1 − L1 ⊥ − C1 +A1
⊥ ⊥ (a) A valid syllogism − C + A − C + L ⊢ − L + A − (− L + A) +L − A +L1 − A1 − C1 +A1
⊥ − C1 +L1 ∞ ∞ (b) An invalid syllogism
Peterson [16] and Thompson [17] developed an extension of ℒℒ by adding three intermediate quantifiers: “few” (for predominant statements), “many” (for majority statements), and “most” (for common statements). The result was an intermediate syllogistic, ℒℒ+, with which we can model inference between universal, particular, predominant, majority, and common statements. Thompson’s Statistical Syllogistic, ℒℒ, is an extension of ℒℒ+ that models inference between statements using statistical quantifiers [18]. To observe the diferences among these logics, consider Table 3.
100, for all the quantifiers that receive a minimal interpretation: – is a significance level. Given some context, is the value such that “much more than % of S are P” is true when the actual percentage of S that are S is ( + ) or more. By the way in which “much more than %” is defined, is also the value such that “almost % of S are P” is false when the actual percentage of S that are P is ( − ) or less. is thus arbitrarily defined, but if it works with its usual meaning, it cannot be less than or equal to 0 or greater than 100, and, like the significance level of statistical tests, it is rarely greater than 5. – denotes an infinitesimal positive magnitude with two properties: 2As explained in [17], a quantifier receives a minimal interpretation when it means at least a certain amount or more; a quantifier receives a maximal interpretation when it means no more than a certain amount or less. Thus, for example, “25% of S is P” is true if the percentage of S that are P is exactly 25%, 50%, or even 100%. each term in a given statement: 1. Distribution by quality. 2. Distribution by quantity.
of 0. index of ( − ). tion index of . distribution index of . index of (100 − ) .
∗ ( + ) > , and
real numbers.
∗ if < , then < − ( × ), where is a positive infinitesimal and , , and are Being greater than 0, is a value such that “more than % of S are P” is true when the actual percentage of S that are P is ( + ) or more. Consequently, is also a value such that nearly % of S are P is true when the percentage of S that are P is greater than or equal to ( − ) = + ( − ).
With these assumptions, the next rules of distribution allow us to associate a distribution index to a) For positive statements, the predicate term has a distribution index of 0 . b) For negative statements, the predicate term has a distribution index of 1000. a) For statements with a quantifier of the form “ %” the subject term has a distribution index b) For statements with a quantifier of the form “almost %” the subject term has a distribution c) For statements with a quantifier of the form “more than %” the subject term has a distribud) For statements with a quantifier of the form “Many more than %” the subject term has a e) For statements with a quantifier of the form “Less than %” the subject term has a distribution if:
Given these preliminary considerations, Thompson ofers the following rules of validity, where 1 and are the distribution indices of the terms of the major premise (the middle term and the major term, respectively); 2 and are the distribution indices of the minor premise (the middle term and the minor term, respectively); and are the distribution indices of the terms of the conclusion (the minor term and the major term, respectively); and finally, is the distribution index of the predicate of the major premise and is the distribution index of the predicate of the minor premise. The maximum distribution value that an occurrence of a term can receive is 1000, such that a term with a distribution index of 100 is maximally distributed. Thus, a syllogism is valid in ℒℒ if and only 1. The middle term is more that maximally distributed in the premises, i.e., 1 + 2 > 1000. 2. The minor term in the premises is distributed at least to the same degree as in the conclusion, i.e., ≥ . ≥ .
+ 0 . 3. The major term in the premises is distributed at least to the same degree as in the conclusion, i.e., 4. The number of negative premises is equal to the number of negative conclusions, i.e., + = Thus, for example, the syllogisms in Tables 4, 5 are valid in ℒℒ , while the syllogism in Table 6 is invalid. The syllogism in Table 4 is valid because it follows all the rules. It satisfies rule 1, because ( 1 + 2) = (1000 + 0 ) = (100 + 0) = 100 , and 100 > 1000, since (100 − 100) = 0 > − = 0 − . It also satisfies rule 2, since 37, 20 ≥ 37, 20; and rule 3, because 0 ≥ 0 . Also, vacuously, it satisfies rule 4. The example shown in Table 5 also satisfies rule 1 insofar as ( 1 + 2) = (27 − + 73 ) = (27 + 73)(( − )+ ) = 100 . Clearly, the other rules are also satisfied. The example in Table 6 is invalid because the middle term is not more than maximally distributed (i.e. 5 + 0 < 1000) and the major term in the premises is not distributed to at least the same degree as the major term in the conclusion (i.e. < ). 1. Almost 27% of philosophers are not friendly. 2. Much more than 73% of philosophers are strage. ⊢ Some strange people are not friendly. 1. More than 5% of philosophers are vegan. 2. Less than 100% of philosophers are not smart. ̸⊢ Almost 95% of smart people are vegan. ℒℒ
1 = 27 − , = 0 2 = 73 , = 0 = 0 , = 0 As can be seen up to this point, ℒℒ ofers an interesting approach to model statistical syllogistic; however, it does not ofer a more general algebraic model. Given this state of afairs, in this section we propose the logic ℱ ℒ in order to unify the virtues of ℒℒ with those of ℱ ℒ. To achieve this goal we follow two steps: first, we propose an adaptation of the ℱ ℒ syntax to include the statistical quantifiers of ℒℒ, and then we modify the ℱ ℒ rules.
In order to accommodate the statements of ℒℒ within the signature of ℱ ℒ, consider Table 7. 3.2. Rules Now, we say that a syllogism is valid in ℱ ℒ if and only if i) the sum of the premises is algebraically equal to the conclusion, ii) the number of conclusions with particular quantity (i.e., zero or one) is equal to the number of premises with particular quantity; iii) the sum of the distribution indices of the middle terms is greater than 1000; and iv) the distribution indices of the conclusion do not exceed the distribution indices of the premises. To illustrate this definition, let us reconsider the previous examples (Tables 8, 9 and 10). 1. Almost 27% of philosophers are not friendly. 2. Much more than 73% of philosophers are strage. ⊢ Some strange people are friendly. 1. More than 5% of philosophers are vegan. 2. Less than 100% of philosophers are not smart. ̸⊢ Almost 95% of smart people are vegan.
Given these ideas, we would like to ofer some tableaux for expansion rules (Figure 3): ℱ ℒ. So, consider the following − S ± P
− S ± P (a) +S ± P +S
± P (b)
These rules work as expected. After applying a rule we introduce some subindex ∈ {1, 2, 3, . . .} as in ℱ ℒ, but also, we use a superindex that represents the distribution index of a given term according to ℒℒ. With these assumptions, we say tableau is complete if and only if every rule that can be applied has been applied. A branch is closed if and only if i) there are terms of the form ± A and ∓ A on two of its nodes or ii) there are terms of the form ± A and ∓ A and the sum of the distribution indexes is greater than 1000; otherwise it is open. A closed branch is indicated by writing a ⊥ at the end of it; an open branch is indicated by writing ∞. A tableau is closed if and only if every branch is closed; otherwise it is open. Thus, again as usual, ± T is a logical consequence of the set of terms Γ if and only if there is a complete closed tableau whose initial list includes the terms of Γ and the rejection of ± T. As an example, consider Figure 4, which shows the inferences exposed in Tables 8, 9, and 10. +P137,20
Now, before we continue with some formal results, let us consider a couple of features of this proposal. First, we have to point out that our proposal difers from Thompson’s insofar as ℒℒ allows wuneivhearvsealtostaatdedmaennottshteorernutlaeiltopathrteicu lℱarℒstatemfraemntesw,bourtks:iinfcteheouprrepmroipseossahlafvoelloawsusbthjeecttetneremts owfithℱthℒe, functor “− ”, then the conclusion cannot have a subject term with the functor “+”. This consideration causes inferences such as those in Table 11 to be conditionally or enthymematically correct, as in Figure 5. 0. There are more than 63% of philosophers. 1. Every Greek is human. 2. 37% of philosophers are Greek. ⊢ More than 0% of philosophers are human. +P163 +P10
Second, it seems that the expressive power of ℱ ℒ for dealing with relations can be used to produce statistical syllogisms with relations, for example, as in Table 12 and Figure 6. 1. 37% of philosophers hate some logicians. 2. More than 89% of philosophers are cynical. ⊢ Some cynical hates some logician.
− P370 + (+H0 +L0 ) +P89 +C
0 ⊢ +C0 + (+H0 +L0 ) − (+C0 + (+H0 +L0 )) − C − (+H0 +L0 ) 0 +P189 +C10 0 − C
1 ⊥
− (+H0 +L0 )1
Finally, before we close this contribution, we would like to ofer some formal results: Proposition 1 (Soundness). If a syllogism is valid in ℒℒ, then its tableau is closed complete. Proof. Notice that when the statements have indices 1000 and 0 only, the proof is trivial: the syllogisms have closed complete tableaux because in that case ℒℒ collapses with ℱ ℒ. For the rest of syllogisms let us suppose, for reductio, that s is an arbitrary syllogism that is valid in ℒℒ but its
If we develop tableaux for each of these syllogisms, we will see that all them are closed complete, but since s is a syllogism built after the rules of ℒℒ, it must be included in Table 13, and hence its tableau must be closed complete, which contradicts our assumption.
Proposition 2. If a syllogism is valid in ℱ ℒ, then it is also valid in ℒℒ. Corollary 1. If a syllogism is valid in ℱ ℒ, then its tableau is closed complete.
After presenting Sommers and Englebretsen’s Term Functor Logic, and Thompson’s statistical syllogistic, we have ofered some tableaux for a fragment of Thompson’s syllogistic. This result, albeit humble, updates the research on term logics with the purpose of dealing with non-deductive inference, namely, inductive and abductive inference, in a terministic, Aristotelian fashion. Our future work consists in studying the formal properties of this proposal and developing fine-tuned implementations.
This research has been developed thanks to a UPAEP University Grant for Research w./no. and to a visiting scholar position granted by the Instituto Promotor del Bien Común and the Universidad Francisco de Vitoria during the summer of 2024.
[1] R. Carnap, Die alte und die neue logik, Erkenntnis 1 (1930) 12–26. URL: http://www.jstor.org/ stable/20011586. [2] B. Russell, A Critical Exposition of the Philosophy of Leibniz: With an Appendix of Leading
Passages, Cambridge University Press, 1900. [3] P. T. Geach, Reference and Generality: An Examination of Some Medieval and Modern Theories,
Contemporary Philosophy / Cornell University, Cornell University Press, 1962. [4] F. Sommers, The Logic of Natural Language, Clarendon Press; Oxford: New York: Oxford University
Press, 1982. [5] E. Mozes, A deductive database based on aristotelian logic, Journal of Symbolic Computation 7 (1989) 487 – 507. URL: http://www.sciencedirect.com/science/article/pii/S0747717189800306. doi:https://doi.org/10.1016/S0747-7171(89)80030-6. [6] G. Englebretsen, Something to Reckon with: The Logic of Terms, University of Ottawa Press, 1996. [7] P. Wang, Return to term logic (1998). [8] M. Correia, La lógica aristotélica y sus perspectivas, Pensamiento. Revista de Investigación e Información Filosófica 73 (1) 5–19. URL: https://revistas.comillas.edu/index.php/pensamiento/ article/view/7832. doi:10.14422/pen.v73.i275.y2017.001. [9] P. Simons, Term logic, Axioms 9 (2020). URL: https://www.mdpi.com/2075-1680/9/1/18. doi:10.
3390/axioms9010018. [10] G. Englebretsen (Ed.), New Directions in Term Logic, College Publications, London, 2024. [11] F. Sommers, G. Englebretsen, An Invitation to Formal Reasoning: The Logic of Terms, Ashgate, 2000. [12] G. Englebretsen, The New Syllogistic, Peter Lang, 1987. [13] G. Englebretsen, C. Sayward, Philosophical Logic: An Introduction to Advanced Topics, Bloomsbury Academic, 2011. [14] J.-M. Castro-Manzano, A tableaux method for term logic, in: LANMR, 2018. [15] J.-M. Castro-Manzano, Distribution tableaux, distribution models, Axioms 9 (2020). doi:10.3390/ axioms9020041. [16] P. L. Peterson, On the logic of "few", "many", and "most", Notre Dame J. Formal Log. 20 (1979) 155–179. [17] B. Thompson, Syllogisms using “few”, “many”, and “most”., Notre Dame J. Formal Logic 23 (1982) 75–84. URL: https://doi.org/10.1305/ndjfl/1093883568. doi: 10.1305/ndjfl/1093883568. [18] B. Thompson, Syllogisms with statistical quantifiers, Notre Dame J. Formal Log. 27 (1986) 93–103. [19] J.-M. Castro-Manzano, Silogística estadística usando términos, Universitas Philosophica 38 (2021) 171–187. doi:10.11144/javeriana.uph38-76.seut.