Formal System of Categorical Syllogistic Logic Based on the Syllogism AEE-4 ()
1. Introduction
Syllogisms are important forms of reasoning in natural language and logic from Aristotle onwards. There are various syllogisms in natural language, such as categorical syllogisms (Moss, 2008) , modal syllogisms (Zhang, 2020a, 2020b) , generalized syllogisms (Murinová & Novák, 2012) , relational syllogisms (Pratt-Hartmann, 2009, 2014) , syllogisms with adjectives (Moss, 2011) , and so on. Among them, categorical syllogisms have a long history of research and are widely used in human reasoning (Chen, 2000) . Categorical syllogisms involve sentences of the following four forms: all xs are y, no xs are y, some xs are y, and not all xs are y.
This article focuses on the time-honored categorical syllogistic logic which has been discussed from different perspectives since Aristotle, for example by Łukasiewicz (1957) , Corcoran (1972) , van Benthem (1984) , Westerståhl (1989) , Martin (1997) , and Zhang (2016, 2020a, 2020b, 2021) , and so on. The reason why categorical syllogistic logic is widely studied is that it is a common form of reasoning in natural languages.
It is well known that merely 24 of 256 types of categorical syllogisms are valid (Chen, 2000) . When deriving all of the other valid syllogisms, at least two valid syllogisms were used as basic axioms in previous studies, for example by Łukasiewicz (1957) , Cai (1988) , Zhang (2016, 2018) and Zhou et al. (2018) . Adopting a different approach from the previous scholars, this article deduces the remaining 23 valid syllogisms taking just one syllogism (that is, AEE-4) as the basic axiom.
2. Relevant Preliminary Knowledge
In this article, Q represents one of the four Aristotelian quantifiers (that is, all, some, no, not all), x, y and z represent lexical variables, and D indicates the domain of lexical variables. In order to express concisely, D is omitted in contexts or without ambiguity.
An Aristotelian syllogism contains three categorical propositions, two of which are premises and one is conclusion. Categorical propositions include the following four types of propositions: A, E, I and O. The proposition A is a universal affirmative proposition, which means that all xs are y and can be formalized as all(x, y). The proposition E is a universal negative proposition, which means that no xs are y and can be denoted as no(x, y). The proposition I is a particular affirmative proposition, which means that some xs are y and can be formalized as some(x, y). The proposition O is a particular negative proposition, which means that not all xs are y and can be symbolized as not all(x, y). The definition of figures of syllogisms is as usual. The syllogism AEE-4 indicates the fourth figure of a syllogism which its major premise, minor premise and conclusion are respectively the proposition A, E and E. And then the syllogism AEE-4 is denoted as all(y, z) Ù no(z, x)®no(x, y). Other formal representations are similar.
3. The Structure of Axiomatic System of Categorical Syllogisms
This formalized axiom system is structured on the basis of the following four parts: initial symbols, formation rules for well-formed formulas, axioms, and rules of deduction.
3.1. Primitive Symbols
(1) lexical variables: x, y, z
(2) quantifier: all
(3) unary negative operator: Ø
(4) binary conjunction operator: Ù
(5) binary implication operator: ®
(6) brackets: (,)
3.2. Formation Rules
(1) If Q is a quantifier, x and y are lexical variables, then Q (x, y) is a well-formed formula.
(2) If p is a well-formed formula, then Øp is well-formed formula.
(3) If p and q are well-formed formulas, then pÙq and p®q are well-formed formulas.
(4) Only the formulas obtained by the above three rules are well-formed formulas.
3.3. Basic Axioms
(1) A1: if p is a valid formula in classical propositional logic, then ⊢p.
(2) A2: ⊢all (y, z)Ùno (z, x)®no (x, y) (that is, the syllogism AEE-4).
3.4. Rules of Deduction
The following deductive rules in classical propositional logic (c.f. Hamilton (1978) ) are also applicable in categorical syllogistic logic. In the following rules, p, q, r and s are well-formed formulas. ⊢p means that p is provable. The other notations are similar. And the replacement rule is used by default in this article.
(1) Rule 1 (antecedent interchange): From ⊢(pÙq®r) infer ⊢(qÙp®r).
(2) Rule 2 (subsequent weakening): From ⊢(pÙq®r) and ⊢(r®s) infer ⊢(pÙq®s).
(3) Rule 3 (anti-syllogism): From ⊢(pÙq®r) infer ⊢(ØrÙp®Øq).
3.5. Relevant Definitions
(1) Definition of connective «: (p«q) =def (p®q)Ù(q¬p)
(2) Definition of inner negative quantifier: (QØ)(x, y) =def Q (x, D-y)
(3) Definition of outer negative quantifier: (ØQ)(x, y) =def It is not that Q(x, y)
(4) Definition of dual quantifier: ØQØ(x, y) =def It is not that Q(x, D-y)
The categorical syllogisms characterize the semantic and inferential properties of the four Aristotelian quantifiers (that is, all, no, some and not all). The reason why this article only takes one Aristotelian quantifier (i.e., all) as the initial quantifier is that the other three Aristotelian quantifiers can be defined by this one. More specifically, no =def allØ, not all =def Øall, and some =def ØallØ by the above definitions.
3.6. Relevant Facts
The following four facts are the basic facts in the generalized quantifier theory (c.f. Peters & Westerståhl (2006) , and Zhang (2014) ), which can be easily proved by using the above definitions, axioms, and rules of deduction.
Fact 1 (inner negation):
(1) ⊢all(x, y)«noØ(x, y); (2) ⊢no(x, y)«allØ(x, y);
(3) ⊢some(x, y)«not allØ(x, y); (4) ⊢not all(x, y)«someØ(x, y).
Fact 2 (outer negation):
(1) ⊢Ønot all(x, y)«all(x, y); (2) ⊢Øall(x, y)«not all(x, y);
(3) ⊢Øno(x, y)«some(x, y); (4) ⊢Øsome(x, y)«no(x, y).
Fact 3 (symmetry):
(1) symmetry of some: ⊢some(x, y)«some(y, x); (2) symmetry of no: ⊢no(x, y)«no(y, x).
Fact 4 (assertoric subalternations):
(1) ⊢all(x, y)®some(x, y); (2) ⊢no(x, y)®not all(x, y).
4. The Reduction from the Syllogism AEE-4 to the Remaining 23 Valid Syllogisms
In the following theorem 1, AEE-4ÞAEE-2 means that the validity of the syllogism AEE-2 can be deduced from the validity of the syllogism AEE-4. In other words, the two syllogisms are reducible. Other notations are similar.
Theorem 1: The remaining 23 valid syllogisms can be deduced merely from the syllogism AEE-4. According to the order and steps of the proof, the following can be obtained:
(1) AEE-4ÞAEE-2
(2) AEE-4ÞAEE-2ÞEAE-2
(3) AEE-4ÞEAE-1
(4) AEE-4ÞAEO-4
(5) AEE-4ÞAEO-4ÞAEO-2
(6) AEE-4ÞAEE-2ÞEAE-2ÞEAO-2
(7) AEE-4ÞEAE-1ÞEAO-1
(8) AEE-4ÞAEE-2ÞAII-1
(9) AEE-4ÞAEE-2ÞAII-1ÞAII-3
(10) AEE-4ÞAEE-2ÞAII-1ÞAII-3ÞIAI-3
(11) AEE-4ÞAEE-2ÞAII-1ÞAII-3ÞIAI-3ÞIAI-4
(12) AEE-4ÞAEO-4ÞAEO-2ÞEAO-3
(13) AEE-4ÞAEO-4ÞAEO-2ÞEAO-3ÞEAO-4
(14) AEE-4ÞAEO-4ÞAEO-2ÞAAI-3
(15) AEE-4ÞEAE-1ÞAAA-1
(16) AEE-4ÞEAE-1ÞAAA-1ÞAAI-1
(17) AEE-4ÞEAE-1ÞAAA-1ÞAAI-1ÞAAI-4
(18) AEE-4ÞEAE-1ÞAAA-1ÞOAO-3
(19) AEE-4ÞEAE-1ÞAAA-1ÞOAO-3ÞAOO-2
(20) AEE-4ÞAEE-2ÞAII-1ÞEIO-1
(21) AEE-4ÞAEE-2ÞAII-1ÞEIO-1ÞEIO-3
(22) AEE-4ÞAEE-2ÞAII-1ÞEIO-1ÞEIO-3ÞEIO-4
(23) AEE-4ÞAEE-2ÞAII-1ÞEIO-1ÞEIO-3ÞEIO-4ÞEIO-2
Proof:
[1] ⊢all(y, z)Ùno(z, x)®no(x, y) (i.e. AEE-4, basic axiom A2)
[2] ⊢no(z, x)«no(x, z) (by (2) of Fact 3)
[3] ⊢all(y, z)Ùno(x, z)®no(x, y) (i.e. AEE-2, by [1] and [2])
[4] ⊢all(y, z)Ùno(x, z)®no(y, x) (i.e. EAE-2, by [3] and (2) of Fact 3 )
[5] ⊢all(y, z)Ùno(z, x)®no(y, x) (i.e. EAE-1, by [1] and (2) of Fact 3)
[6] ⊢no(x, y)®not all(x, y) (by (2) of Fact 4)
[7] ⊢all(y, z)Ùno(z, x)®not all(x, y) (i.e. AEO-4, by [1], [6] and Rule 2
[8] ⊢all(y, z)Ùno(x, z)®not all(x, y) (i.e. AEO-2, by [2] and [7])
[9] ⊢all(y, z)Ùno(x, z)®not all(y, x) (i.e. EAO-2, by [4], (2) of Fact 4 and Rule 2)
[10] ⊢all(y, z)Ùno(z, x)®not all(y, x) (i.e. EAO-1, by [2] and [9])
[11] ⊢Øno(x, y)Ùall(y, z)®Øno(x, z) (by [3] and Rule 3)
[12] ⊢some(x, y)Ùall(y, z)®some(x, z) (i.e. AII-1, by [11] and (3) of Fact 2)
[13] ⊢some(y, x)Ùall(y, z)®some(x, z) (i.e. AII-3, by [12] and (1) of Fact 3)
[14] ⊢some(y, x)Ùall(y, z)®some(z, x) (i.e. AII-3, by [13] and (1) of Fact 3)
[15] ⊢some(x, y)Ùall(y, z)®some(z, x) (i.e. IAI-4, by [14] and (1) of Fact 3)
[16] ⊢Ønot all(x, y)Ùno(x, z)®Øall(y, z) (by [8], Rule 1 and Rule 3)
[17] ⊢no(x, z)Ùall(x, y)®not all(y, z) (i.e. EAO-3, by [16], (1) and (2) of Fact 2, and Rule 1)
[18] ⊢no(z, x)Ùall(x, y)®not all(y, z) (i.e. EAO-4, by [2] and [17])
[19] ⊢Ønot all(y, x)Ùall(y, z)®Øno(x, z) (by [9] and Rule 3)
[20] ⊢all(y, x)Ùall(y, z)®some(x, z) (i.e. AAI-3, by [19], (1) and (3) of Fact 2)
[21] ⊢all(y, z)ÙallØ(z, x)®allØ(y, x) (by [5] and (2) of Fact 1)
[22] ⊢all(y, z)Ùall(z, D-x)®all(y, D-x) (by [21] and (2) of Definition (3.5))
[23] ⊢all(y, z)Ùall(z, x)®all(y, x) (i.e. AAA-1, by [22])
[24] ⊢all(y, z)Ùall(z, x)®some(y, x) (i.e. AAI-1, by [23], (1) of Fact 4 and Rule 2)
[25] ⊢all(y, z)Ùall(z, x)®some(x, y) (i.e. AAI-4, by [24] and (1) of Fact 3)
[26] ⊢Øall(y, x)Ùall(y, z)®Øall(z, x) (by [23] and Rule 3)
[27] ⊢not all(y, x)Ùall(y, z)®not all(z, x) (i.e. OAO-3, by [26] and (2) of Fact 2)
[28] ⊢Ønot all(z, x)Ùnot all(y, x)®Øall(y, z) (by [27] and Rule 3)
[29] ⊢all(z, x)Ùnot all(y, x)®not all(y, z) (i.e. AOO-2, by [28], (1) and (2) of Fact 2)
[30] ⊢some(x, y)ÙnoØ(y, z)®not allØ(x, z) (by [12], (1) and (3) of Fact 1)
[31] ⊢no(y, D-z)Ùsome(x, y)®not all(x, D-z) (by [30] and (2) of Definition (3.5))
[32] ⊢no(y, z)Ùsome(x, y)®not all(x, z) (i.e. EIO-1, by [31])
[33] ⊢no(y, z)Ùsome(y, x)®not all(x, z) (i.e. EIO-3, by [32] and (1) of Fact 3)
[34] ⊢no(z, y)Ùsome(y, x)®not all(x, z) (i.e. EIO-4, by [33] and (2) of Fact 3)
[35] ⊢no(z, y)Ùsome(x, y)®not all(x, z) (i.e. EIO-2, by [34] and (1) of Fact 3)
5. Conclusion and Future Work
The basic idea of this study is as follows: Firstly, make full use of the trichotomy structure of categorical proposition to formalize categorical syllogisms. Then, taking advantage of the deductive rules in classical propositional logic and the basic facts in the generalized quantifier theory, we can deduce the other 23 valid categorical syllogisms by taking just one syllogism (that is, AEE-4) as the basic axiom. This article not only reveals the reducible relations between the syllogism AEE-4 and the other 23 valid syllogisms, but also establishes a concise formal axiomatic system for categorical syllogistic logic. The research methods and results are concise, clear, and enlightening.
The basic steps for computer to process a statement in natural language are as follows: first, formalize the statement; then give the algorithm of its formal expression; finally, compile the program according to the algorithm. In other words, formalizing sentences in natural language is the first step of natural language information processing. This paper makes a formal study of categorical syllogisms from the perspective of mathematical structuralism and generalized quantifier theory. This study not only provides a universal mathematical paradigm for studying other kinds of syllogisms, but also provides theoretical support for natural language information processing, knowledge representation and knowledge reasoning in computer science.
How to integrate the research results of generalized quantifier theory and categorical syllogistic logic to further improve their role in the intersection of logic, natural language processing and computer science, and how to make the best of the spillover effects in theoretical research to deal with practical problems and promote computer context awareness and knowledge reasoning? These issues need to be explored in depth.
Acknowledgements
This work was supported by Science and Technology Philosophy and Logic Teaching Team Project of Anhui University under Grant No. 2022xjzlgc071.