Applied Science and Innovative Research ISSN 2474-4972 (Print) ISSN 2474-4980 (Online) Vol. 7, No. 2, 2023 www.scholink.org/ojs/index.php/asir 45 Original Paper How to Obtain Valid Generalized Modal Syllogisms from Valid Generalized Syllogisms Jing Xu1,2 & Xiaojun Zhang3 1 School of Philosophy, Anhui University, Hefei, China 2 School of Marxism, Anhui Medical University, Hefei, China 3 School of Philosophy, Anhui University, Hefei, China Received: March 9, 2023 Accepted: March 19, 2023 Online Published: March 21, 2023 doi:10.22158/asir.v7n2p45 URL: http://doi.org/10.22158/asir.v7n2p45 Abstract Making full use of the truth value definitions of sentences with quantification, possible world semantics and/or fuzzy logic, one can prove the validity of generalized modal syllogisms. This paper shows that the proof of the validity of a generalized modal syllogism can be transformed into that of its corresponding generalized syllogism, and that the generalized syllogism obtained by removing all modalities in any valid generalized modal syllogism is still valid. Therefore, the simplest way to screen out valid generalized modal syllogisms is to add modalities to valid generalized syllogisms, and then to delete all invalid syllogisms by means of the basic rules with which valid generalized modal syllogisms should meet. And then the remainders are valid. This paper illustrates how to obtain 12 valid generalized modal syllogisms by adding necessary modalities and/or possible modalities to any valid generalized syllogism. The two kinds of syllogisms discussed in this paper are composed of sentences with quantification which is the largest number of sentences in natural language. Hence, this innovative research can provide theoretical support for linguistics, logic, artificial intelligence, and among other fields. Keywords generalized modal syllogisms, generalized syllogisms, validity, truth value definition 1. Introduction There are many kinds of syllogisms in natural language, such as Aristotelian syllogisms (Hui, 2023), generalized syllogisms (Xiaojun, 2020a), Aristotelian modal syllogisms (Johnson, 2004), generalized modal syllogisms, and so on. This paper shall restrict attention entirely to how to obtain valid generalized modal syllogisms from valid generalized syllogisms. www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 7, No. 2, 2023 46 Published by SCHOLINK INC. There are some studies on generalized syllogisms, such as Peterson (2000), Novák (2008a, 2008b), Moss (2010), Novák (2012), Endullis and Moss (2015), Xiaojun (2016), Cheng (2023), and so on. Nevertheless, up to now, there are no relevant literature of generalized modal syllogisms found at home or abroad. Therefore, this study is a pioneering research. The largest number of sentences in natural language is sentences with quantification (Cheng, 2022). The two kinds of syllogisms discussed in this paper are composed of sentences with quantification (Long, 2023). And they play an important role in natural language information processing, which involves interdisciplinary research fields such as linguistics, logic, artificial intelligence, and among other fields. It is hoped that this study can provide theoretical support for these fields. 2. Preliminaries In the paper, let S, M and P be the lexical variables in sentences with quantification, and D the domain of lexical variables. Syllogisms often contain sentences in the following forms: All Ss are P, No Ss are P, Some Ss are P, Not all Ss are P. The four sentences can be symbolized as all(S, P), no(S, P), some(S, P) and not all(S, P), and abbreviated as the proposition A, E, I and O, respectively.  is the symbol of a necessary modality,  is that of a possible modality.S represents the cardinality of the set composed of the variable S. A generalized modal syllogism can be obtained by adding necessary modalities and/or possible modalities to a generalized syllogism. Example 1: Major premise: All middle-aged healthy hens are egg-laying hens. Minor premise: Most chickens in this farm are middle-aged healthy hens. Conclusion: Some chickens in this farm are egg-laying hens. Let S be the chickens in the domain, M the middle-aged healthy hens in the domain, and P the egg-laying hens in the domain. Then the generalized syllogism in Example 1 can be formalized by all(M, P)most(S, M)some(S, P). The following generalized modal syllogism in Example 2 can be obtained by adding modalities to Example 1. Example 2: Major premise: All middle-aged healthy hens are necessarily egg-laying hens. Minor premise: Most chickens in this farm are middle-aged healthy hens. Conclusion: Some chickens in this farm are possibly egg-laying hens. Similarly to Example 1, Example 2 can be formalized by all(M, P)most(S, M)some(S, P). In fact, a sentence with quantification discussed in this paper can be formalized into a tripartite structure like Q(S, P). Let Q1, Q2, Q3 be generalized quantifiers, and take the first figure syllogism as an example. It can be formalized as Q1(M, P)Q2(S, M)Q3(S, P). There are four situations as follows: (1) If Q1, Q2 and Q3 only range over the four Aristotelian quantifiers (that is, all, some, no and not all), one can obtain Aristotelian syllogisms (Long, 2023). (2) If Q1, Q2 and Q3 range over all the generalized quantifiers including Aristotelian quantifiers, one can obtain generalized syllogisms (Xiaojun, 2021). (3) www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 7, No. 2, 2023 47 Published by SCHOLINK INC. If Q1, Q2 and Q3 range over the following 12 Aristotelian modal quantifiers: all, some, no, not all, all, some, no, not all, all, some, no and not all, one can obtain Aristotelian modal syllogisms (Cheng, 2023). (4) If Q1, Q2 and Q3 range over all generalized modal quantifiers including the above 12 Aristotelian modal quantifiers, one can obtain generalized modal syllogisms. Definition 1 (truth value definitions of non-modal sentences with quantification): (1.1) all(S, P)SP; (1.2) no(S, P)S∩P=; (1.3) some(S, P)S∩P; (1.4) not all(S, P)S⊈P; (1.5) most(S, P)S∩P0.6S. The truth value definitions of modal sentences with quantification can be given as follows in line with Definition 1 and possible world semantics (Chellas, 1980). Definition 2 (truth value definitions of modal sentences with quantification): (2.1) all(S, P) is true if and only if SP is true in any possible world ; (2.2) all(S, P) is true if and only if SP is true in at least one possible world ; (2.3) no(S, P) is true if and only if S∩P= is true in any possible world ; (2.4)no(S, P) is true if and only if S∩P= is true in at least one possible world ; (2.5) some(S, P) is true if and only if S∩P is true in any possible world ; (2.6)some(S, P) is true if and only if S∩P is true in at least one possible world ; (2.7) not all(S, P) is true if and only if S⊈P is true in any possible world ; (2.8) not all(S, P) is true if and only if S⊈P is true in at least one possible world ; (2.9) most(S, P) is true if and only if S∩P0.6S is true in any possible world ; (2.10) most(S, P) is true if and only if S∩P0.6S is true in at least one possible world . Fact 1 (a necessary proposition implies an assertoric proposition): (1.1) all(S, P)all(S, P); (1.2) no(S, P)no(S, P); (1.3) some(S, P)some(S, P); (1.4) not all(S, P)not all(S, P); (1.5) most(S, P)most(S, P). Fact 2 (an assertoric proposition implies a possible proposition): (2.1) all(S, P)all(S, P); (2.2) no(S, P)no(S, P); (2.3) some(S, P)some(S, P); (2.4) not all(S, P)not all(S, P); (2.5) most(S, P)most(S, P). Fact 3 (a necessary proposition implies an possible proposition): (3.1) all(S, P)all(S, P); (3.2) no(S, P)no(S, P); (3.3) some(S, P)some(S, P); (3.4) not all(S, P)not all(S, P); (3.5) most(S, P)most(S, P). The above facts are the basic knowledge of generalized quantifier theory (Peters and Westerståhl, 2006) or classical modal logic (Chagrov & Zakharyaschev, 1997). Generalized modal syllogistic is an extension of classical first-order logic, so the basic rules of the latter also hold in the former. For instance, Let p, q, r and s be propositions, if ⊢(pqr) and ⊢(rs), then ⊢(pqs). www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 7, No. 2, 2023 48 Published by SCHOLINK INC. 3. How to Screen out Valid Generalized Modal Syllogisms The study of generalized modal syllogisms in this paper is inspired by that of Aristotelian modal syllogisms. It is shown that the Aristotelian syllogism obtained by removing all modalities in any valid Aristotelian modal syllogism is still valid (Xiaojun, 2020b), and the latter can be obtained by adding modalities to the former. Therefore, all valid Aristotelian modal syllogisms can be screened out from all this kind of syllogisms in the light of some basic rules with which valid Aristotelian modal syllogisms should meet (Xiaojun, 2020c). Generalized syllogisms are extensions of Aristotelian syllogisms, and generalized modal syllogisms are extensions of Aristotelian modal syllogisms. It is found that the study of generalized modal syllogistic is along similar lines to that of Aristotelian modal syllogistic. In other words, it can be shown that the generalized syllogism obtained by removing all modalities in any valid generalized modal syllogism is still valid. Therefore, the simplest way to screen out valid generalized modal syllogisms is to add modalities to valid generalized syllogisms, and then to delete all invalid syllogisms by means of the basic rules with which valid generalized modal syllogisms should meet, and then all the rest syllogisms are valid. For every valid generalized modal syllogism, the assertive proposition of the conclusion cannot be stronger than that of two premises. Specifically, if the weakest proposition in two premises is a possible one, then the conclusion can only be a possible one; similarly, if the weakest proposition in two premises is an assertoric one, then the conclusion can only be an assertoric one or a possible one. Theorem 1: Let Q1, Q2 and Q3 be generalized quantifiers. If the generalized syllogism Q1(M, P)Q2(S, M)Q3(S, P) is valid, then the following 12 valid generalized modal syllogisms can be obtained by adding modalities to the generalized syllogism: (1) Q1(M, P)Q2(S, M)Q3(S, P); (2) Q1(M, P)Q2(S, M)Q3(S, P); (3) Q1(M, P)Q2(S, M)Q3(S, P); (4) Q1(M, P)Q2(S, M)Q3(S, P); (5) Q1(M, P)Q2(S, M)Q3(S, P); (6) Q1(M, P)Q2(S, M)Q3(S, P); (7) Q1(M, P)Q2(S, M)Q3(S, P); (8) Q1(M, P)Q2(S, M)Q3(S, P); (9) Q1(M, P)Q2(S, M)Q3(S, P); (10) Q1(M, P)Q2(S, M)Q3(S, P); (11) Q1(M, P)Q2(S, M)Q3(S, P); (12) Q1(M, P)Q2(S, M)Q3(S, P); In terms of Theorem 1, the following will use several examples to illustrate how to obtain the corresponding 12 valid generalized modal syllogisms from a valid generalized syllogism. Before doing this, it is necessary to prove the validity of the following generalized syllogism. www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 7, No. 2, 2023 49 Published by SCHOLINK INC. Theorem 2: The generalized syllogism all(M, P)most(S, M)some(S, P) is valid. Proof: Suppose that all(M, P) and most(S, M) are true, then all(M, P)MP and most(S, M) S∩M0.6S are true in the light of the clause (1.1) and (1.5) in Definition 1, respectively. It is easily seen that MP and S∩M0.6S. It follows that S∩P0.6S. And it is clear that S∩P. Thus, some(S, P) is true in line with the clause (1.3) in Definition 1, just as desired. According to Theorem 1 and Theorem 2, one can obtain the following Theorem 3: Theorem 3: The following 12 generalized modal syllogisms are valid: (3.1) all(M, P)most(S, M)some(S, P); (3.2) all(M, P)most(S, M)some(S, P); (3.3) all(M, P)most(S, M)some(S, P); (3.4) all(M, P)most(S, M)some(S, P); (3.5) all(M, P)most(S, M)some(S, P); (3.6) all(M, P)most(S, M)some(S, P); (3.7) all(M, P)most(S, M)some(S, P); (3.8) all(M, P)most(S, M)some(S, P); (3.9) all(M, P)most(S, M)some(S, P); (3.10) all(M, P)most(S, M)some(S, P); (3.11) all(M, P)most(S, M)some(S, P); (3.12) all(M, P)most(S, M)some(S, P). Proof: The proof of the validity of a generalized modal syllogism can be transformed into that of its corresponding generalized syllogism. For (3.1). Suppose that all(M, P) and most(S, M) are true, in line with the clause (2.1) in Definition 2, all(M, P) is true if and only if MP is true in any possible world . Similarly, with the help of the clause (2.9) in Definition 2, most(S, M) is true if and only if S∩M0.6S is true in any possible world . Thus it follows that MP and S∩M0.6S are true in any possible world . It follows that S∩P0.6S. And it can be seen that S∩P is true in any possible world . Therefore, some(S, P) is true in accordance with the clause (2.5) in Definition 2. The proof of (2.1) has been completed. The others can be similarly proved on the basis of the above theorems, facts and rules. Theorem 4: The generalized syllogism no(P, M)most(S, M)not all(S, P) is valid. Proof: Suppose that no(P, M) and most(S, M) are true, according to the clause (1.2) and (1.5) in Definition 1, one can obtain that no(P, M)P∩M= and most(S, M)S∩M0.6S are true. It can be seen that P∩M= and S∩M0.6S. Therefore, S⊈P. This can be prove by reductio ad absurdum. Suppose that S⊈P is not true. It is clear that SP is true, and it has been proved that P∩M=. Thus, it follows that S∩M= which contradicts S∩M0.6S. So SP is not true. In other words, S⊈P is true. According to definition (1.4), not all(S, P) is true, as required. According to Theorem 1 and Theorem 4, one can obtain the following Theorem 5: www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 7, No. 2, 2023 50 Published by SCHOLINK INC. Theorem 5: The following 12 generalized modal syllogisms are valid: (5.1) no(P, M)most(S, M)not all(S, P); (5.2) no(P, M)most(S, M)not all(S, P); (5.3) no(P, M)most(S, M)not all(S, P); (5.4) no(P, M)most(S, M)not all(S, P); (5.5) no(P, M)most(S, M)not all(S, P); (5.6) no(P, M)most(S, M)not all(S, P); (5.7) no(P, M)most(S, M)not all(S, P); (5.8) no(P, M)most(S, M)not all(S, P); (5.9) no(P, M)most(S, M)not all(S, P); (5.10) no(P, M)most(S, M)not all(S, P); (5.11) no(P, M)most(S, M)not all(S, P); (5.12) no(P, M)most(S, M)not all(S, P). The proof of Theorem 5 is similar to that of Theorem 3. That is to say that its proof can be transformed into that of Theorem 4. 4. Conclusion and Future Work The research conclusions of this paper are as follows: (1) Making full use of the truth value definitions of sentences with quantification, possible world semantics and/or fuzzy logic, one can prove the validity of generalized modal syllogisms. (2) The proof of the validity of a generalized modal syllogism can be transformed into that of its corresponding generalized syllogism. And it can be shown that the generalized syllogism obtained by removing all modalities in any valid generalized modal syllogism is still valid. So the simplest way to screen out valid generalized modal syllogisms is to add modalities to valid generalized syllogisms, and then to delete all invalid syllogisms by means of the basic rules with which valid generalized modal syllogisms should meet. And then the remainders are valid. (3) One can obtain 12 valid generalized modal syllogisms by adding necessary modalities and/or possible modalities to any valid generalized syllogism. This study provides an unified mathematical paradigm for discussing the other generalized modal syllogisms including different generalized quantifiers (such as a minority of, few, at most n, at least half of the, a large part of, etc.). There are an infinite number of instances in natural language corresponding to any valid generalized modal syllogism. Hence, this study has important theoretical value and practical significance. Acknowledgement This work was supported by the National Social Science Foundation of China under Grant No.22&ZD295. www.scholink.org/ojs/index.php/asir Applied Science and Innovative Research Vol. 7, No. 2, 2023 51 Published by SCHOLINK INC. References Chagrov, A., & Zakharyaschev, M. (1997). Modal Logic. Oxford: Clarendon Press. Chellas, B. F. (1980). Modal Logic: an Introduction. 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