Research Article | Open Access
A Modern Syllogistic Method in Intuitionistic Fuzzy Logic with Realistic Tautology
The Modern Syllogistic Method (MSM) of propositional logic ferrets out from a set of premises all that can be concluded from it in the most compact form. The MSM combines the premises into a single function equated to 1 and then produces the complete product of this function. Two fuzzy versions of MSM are developed in Ordinary Fuzzy Logic (OFL) and in Intuitionistic Fuzzy Logic (IFL) with these logics augmented by the concept of Realistic Fuzzy Tautology (RFT) which is a variable whose truth exceeds 0.5. The paper formally proves each of the steps needed in the conversion of the ordinary MSM into a fuzzy one. The proofs rely mainly on the successful replacement of logic 1 (or ordinary tautology) by an RFT. An improved version of Blake-Tison algorithm for generating the complete product of a logical function is also presented and shown to be applicable to both crisp and fuzzy versions of the MSM. The fuzzy MSM methodology is illustrated by three specific examples, which delineate differences with the crisp MSM, address the question of validity values of consequences, tackle the problem of inconsistency when it arises, and demonstrate the utility of the concept of Realistic Fuzzy Tautology.
Fuzzy deductive reasoning has typically relied on a fuzzification of the Resolution Principle of Robinson  in first-order predicate calculus. This principle uses a set of premises to prove the validity of a single clause or consequent at a time via the refutation (REDUCTIO AD ABSURDUM) method. Lee  proved that a set of clauses is unsatisfiable in fuzzy logic if and only if it is unsatisfiable in two-valued logic. He also proved that if the least truthful clause of a set of clauses has a truth value , then all the logical consequents obtained by repeatedly applying the resolution principle have truth values that are never less than . Later, the so-called Mukaidono Fuzzy Resolution Principle, developed by a group of Japanese researchers [3–6], was used to establish a powerful fuzzy Prolog system. The introduction of this principle involved several new concepts, including that of the contradictory degree of a contradiction whose truth value equals the truth value of the contradiction itself. Recently, a new fuzzy resolution principle was introduced in [7–9], wherein refutation is achieved by the antonym not by negation, and reasoning is made more flexible thanks to the existence of a meaningless range, which is a special set that is not true and also not false. Other notable work on various aspects and techniques of fuzzy reasoning and inference is available in [10–23].
The purpose of this paper is to implement fuzzy deductive reasoning via fuzzification of a powerful deductive technique of propositional logic, called the Modern Syllogistic Method (MSM). This method was originally formulated by Blake , expounded by Brown , and further described or enhanced in [26–33] and has a striking similarity with the resolution-based techniques of predicate logic [1, 34, 35].
The MSM has the distinct advantage that it ferrets out from a set of premises all that can be concluded from it, with the resulting conclusions cast in the simplest or most compact form. The MSM uses just a single rule of inference, rather than the many rules of inference conventionally employed in propositional-logic deduction (see, e.g., [36, 37]). In fact, the MSM includes all such rules of inference as special cases [30, 31]. The MSM strategy is to convert the set of premises into a single equation of the form or and obtain = the complete sum of (or = the complete product of ). The set of all possible prime consequents of the original premises are obtained from the final equation (or ).
We describe herein a fuzzy version of the MSM that utilizes concepts of the Intuitionistic Fuzzy Logic (IFL) [38–48] developed mainly by Atanassov [38, 40, 41, 43–45]. This fuzzy MSM reduces to a restricted version in the Ordinary Fuzzy Logic (OFL) of Zadeh [34, 49–55]. The IFL version of the MSM is more flexible, while the OFL version is simpler and computationally faster. We managed to adapt the MSM to fuzzy reasoning without any dramatic changes of its main steps. In particular, our algorithm for constructing the complete product (or complete sum) of a logic function via consensus generation and absorption remains essentially the same. This algorithm was first developed by Blake  and later by Tison [56–59]. It is usually referred to as the Tison method, but we will name it herein as the Blake-Tison method or algorithm.
The organization of the rest of this paper is as follows. Section 2 briefly reviews the concept of Intuitionistic Fuzzy Logic (IFL) and asserts why it adds necessary flexibility to Ordinary Fuzzy Logic (OFL). Section 3 combines ideas from Lee  and Atanassov [38, 40, 41, 43–45] to produce a novel simple concept of a Realistic Fuzzy Tautology (RFT) and explains why such a new concept is needed. Section 4 outlines the steps of MSM in two-valued Boolean logic and then adapts it to realistic fuzzy logic, which is an IFL in which the new RFT concept is embedded. Formal proofs of the correctness of this adaptation are provided. Three examples are given in Section 5 to demonstrate the computational steps and to demonstrate how, similar to the result of Lee , the validity of the least truthful premise sets an upper limit on the validity of every logical consequent. Section 6 concludes the paper. The Appendix provides a description of an improved version of the Blake-Tison algorithm for producing the complete product of a logical function.
2. Review of Intuitionistic Fuzzy Logic
In Intuitionistic Fuzzy Logic (IFL), a variable is represented by its validity which is the ordered couple where and are degrees of truth and falsity of , respectively, such that each of the real numbers , , .
Note that when , then IFL reduces to Ordinary Fuzzy Logic (OFL), in which alone suffices as a representation for , since is automatically determined by . The necessity of allowing the condition is established on the grounds that it allows a degree of hesitancy, ignorance, or uncertainty when one can neither designate a variable as true nor label it as false.
Since IFL includes OFL as a special case, operations in IFL should be defined such that they serve as extensions to their OFL counterparts. However, this allows the existence of many definitions for pertinent operations, such as the negation operation  or the implication operation . We will stick herein to the most familiar definitions. We have a single unary operation, namely, the negation operation, which produces the complement of a variable . We define this operation as one that interchanges the truth and falsity of the variable, that is,
The most important binary operations are(i)the intuitionistic conjunction or meet operation defined by (ii)the intuitionistic disjunction or join operation defined by (iii)the intuitionistic implication operation defined herein by
With any three intuitionistic fuzzy variables , , and , the following pairs of dual theorems are satisfied:(1)idempotency: (2)commutativity: (3)associativity: (4)absorption: (5)distributivity: (6)identities:
Atanassov [38, 41] defined the notion of Intuitionistic Fuzzy Tautology (IFT) by the following: is an IFT if and only if . For comparison, will be a tautology in crisp Boolean algebra if and only if and .
A variable is said to be less valid (less truthful) than another variable (written ) if and only if and . Hence, the complement of an IFT is less valid than this IFT.
3. Realistic Fuzzy Tautology
Since our attempts to fuzzify the MSM using the concept of Intuitionistic Fuzzy Tautology (IFT) were not successful, we were obliged to introduce a new concept of tautology that we call Realistic Fuzzy Tautology (RFT). A variable in IFT is an RFT if and only if . Note that an RFT is a more strict particular case of an IFT. If , then the concept of an RFT reduces to the representation of Fuzzy Tautology given by Lee . A variable in IFT is a non-RFT (denoted by nRFT) if and only if . Hence, two complementary variables and cannot be RFTs at the same time. The conjunction of two complementary variables is nRFT. If the disjunction of a variable with an nRFT is an RFT, then this variable is an RFT. For convenience, we will call the Intuitionistic Fuzzy Logic (IFL) with the concept of RFT embedding in it a Realistic Fuzzy Logic (RFL). The introduction of the RFT concept is utilized herein in fuzzifying the MSM, but it might have other far-reaching consequences in fuzzifying other topics.
4. The Modern Syllogistic Method
In this section, we describe the steps of a powerful technique for deductive inference, which is called “the Modern Syllogistic Method” (MSM). The great advantage of the method is that it ferrets out from a given set of premises all that can be concluded from this set, and it casts the resulting conclusions in the simplest or most compact form [24–33].
First, we describe the steps of the MSM in conventional Boolean logic. Then, we adapt these steps to realistic fuzzy logic. Since the MSM has two dual versions, one dealing with propositions equated to zero and the other dealing with propositions equated to one, we are going herein to represent the latter version which corresponds to tautologies.
4.1. The MSM in Conventional Boolean Logic
The MSM has the following steps.
Step 1. Each of the premises is converted into the form of a formula equated to 1 (which we call an equational form), and then the resulting equational forms are combined together into a single equation of the form . If we have logical equivalence relations of the formthey are set in the equational form We may also have logical implication (logical inclusion) relations of the form These relations symbolize the statements “If then ” or equivalently “ if only ”. Conditions (14) can be set into the equational form
Step 2. The totality of premises in (13) and (15) finally reduces to the single equation , where is given byEquations (13) and (15) represent the dominant forms that premises can take. Other less important forms are discussed by Klir and Marin  and can be added to (16) when necessary.
Step 3. The function in (16) is rewritten as a complete product (a dual Blake canonical form), that is, as a conjunction of all the prime implicates of . There are many manual and computer algorithms for developing the complete product of a switching function . Most of these algorithms depend on two logical operations: (a) consensus generation and (b) absorption.
Step 4. Suppose the complete product of takes the form where is the th prime implicate of . Equation (17) is equivalent to the set of equationsEquations (18) are called prime consequents of and state in the simplest equational form all that can be concluded from the original premises. The conclusions in (18) can also be cast into implication form. Suppose is given by a disjunction of complemented literals and uncomplemented literals , that is,then (18) can be rewritten as
4.2. The MSM in Realistic Fuzzy Logic
A crucial prominent feature of realistic fuzzy logic is that it can be used to implement the MSM without spoiling any of its essential features. We just need to replace the concept of a crisp logical “1” by that of the realistic fuzzy tautology (RFT) introduced in Section 3. Now, a realistic fuzzy version of the MSM has the following steps.
Step 1. Assume the problem at hand is governed by a set of RFTs . Each of these RFTs might be assumed from the outset or be constructed from equivalence or implication relations. Let be described by
Step 2. The given set of RFT premises are equivalent to the single functionThe function is also an RFT. This equivalence is proved in Theorem 1.
Step 3. Replace the function by its complete product . The resulting is also an RFT since the operations used in going from to preserve the RFT nature. These operations are as follows:(i)absorption, which is known to be tautology-preserving in general fuzzy logic and intuitionistic fuzzy logic and hence in the current realistic fuzzy logic,(ii)consensus generation, which preserves RFTs in the sense that when the conjunction of two clauses is an RFT, then it remains so when conjuncted with the consensus of these two clauses. This is proved in the form of Theorem 2.
Step 4. Since is an RFT, then when it is given by the conjunction in (17), each clause , in (17) will be an RFT (again thanks to Theorem 1). The fact that each of the clauses is an RFT is all that can be concluded from the original premises. The procedure does not necessarily provide specific information about the validity of each consequent . However, as we show in the examples below, it is possible to obtain such information in specific cases.
Theorem 1. Each of the realistic fuzzy variables is an RFT if and only if their conjunction is an RFT.
Proof. Consider the following:
Theorem 2. The conjunction of two clauses with a single opposition retains the RFT property when augmented by a third clause representing the consensus of the two original clauses. Specifically, if is an RFT, then is also an RFT.
Proof. Let , By virtue of Theorem 1, the fact that is an RFT implies that is an RFT (i.e., ) and that is an RFT (i.e., ).
Now consider two cases.
Case 1. One has , and henceCase 2. One has Now each of , , and is an RFT. Hence, thanks to Theorem 1, their conjunction is an RFT.
One prominent difference between fuzzy MSM and ordinary MSM is that the complementary laws in ordinary logic do not hold in any fuzzy logic including OFL, IFL, or RFL. This means that in implementing our algorithm for generating the complete product of a switching function, a conjunction of the form might appear, and then it is left as it is, and not replaced by . This point will be clarified further in Example 2 of Section 5.
Table 1 employs the MSM to derive fuzzy versions of many famous rules of inference, including, in particular, the celebrated rules of MODUS PONENS and MODUS TOLLENS. The derivation shows that some of the rules have some intermediate consequences as well as a final particular consequence.
Example 1. A typical example of MSM presented by Brown , pp. 124–127, and taken from Kalish and Montague , has the following statements: (1)if Alfred studies, then he receives good grades ();(2)if Alfred does not study, then he enjoys college ();(3)if Alfred does not receive good grades, then he does not enjoy college ().
The MSM solution combines the above premises into a single equation and obtains the complete product of by adding consensus alterms or clauses  with respect to the biform variables and and absorbing subsuming alterms (see Appendix). Gradually, the formula for changes to end up as the complete product form:
The last expression for is and is still equated to . Hence, it asserts the not so-obvious conclusion of () beside the conclusion , which is just a reecho of one of the premises. These two conclusions are all that can be concluded from the premises in the simplest form. Any other valid conclusion must subsume one of these two conclusions. Now, suppose that our knowledge about the premises is fuzzy or uncertain, so that each of the premises is no longer a crisp tautology, but is weakened to the status of a realistic fuzzy tautology (RFT). To be specific, let us assign the following values for the validity of each premise:
The function in (27) is no longer a crisp tautology (=1), but rather an RFT with validityso inherits the validity of the first premise, which is the least-truthful premise. This validity is also inherited by in the last line of (28) and also by the novel consequent , that is,
This means that the consequent has a truth value of 0.6 and a falsity value of 0.3. The fact that leaves room for our uncertainty or ignorance about this fuzzy proposition.
Example 2. The MSM has a built-in capability of detecting inconsistency in a set of premises, since this produces as , and leads to which is unacceptable in two-valued logic [30, 31]. This feature is still enjoyed by the fuzzy MSM since an inconsistency will be revealed in the form of a variable and its complement being both RFT, which is a contradiction. For a specific example, consider the set of premises , , and . In equational form, these reduce to or equivalently to the single equation In two-valued logic, the complete product of is obtained via the Improved Blake-Tison Method (see Appendix) as which leads to the contradiction . However, in realistic fuzzy logic, we have being an RFT. This means that both and (and also both and and both and ) are RFTs, which is a contradiction. Hence, the original set of premises are inconsistent.
Example 3. Consider the set of premises [30, 37]: (1)Pollution will increase if government restrictions are relaxed .(2)If pollution increases, there will be a decline in the general health of the population .(3)If there is a decline in health in the population, productivity will fall .(4)The economy will remain healthy only if productivity does not fall .
These premises are equivalent to the propositional equation , where
The complete product of is obtained via the Improved Blake-Tison Method (see Appendix) as
The fact that means that there are six new consequents (that are not just a reecho of premises). The last of these consequents is or equivalently which means that if government restrictions are relaxed, then the economy will not remain healthy, an argument in favor of a stronger governmental regulatory role.
Now, suppose that the given premises are not crisp tautologies, but are just RFTs with respective validities
Hence, each of the new clauses in (37) is an RFT of a validity dependent on the validities of the clauses generating it. Table 2 lists these new clauses, identifies their generators, and hence assigns a validity to each of them. The issue of a stronger regulatory role for the government now has a validity of rather than . This validity is realistic in the sense that this issue can be viewed as supported by 60% of the voters and opposed by 30% of them, with 10% of them abstaining or undecided.
The Modern Syllogistic Method (MSM) is a sound and complete single rule of inference that encompasses all rules of inference. It extracts from a given set of premises all that can be concluded from it in the simplest possible form. It has a striking similarity with resolution-based techniques in predicate logic, but while these techniques chain backwardly from a given assertion seeking to refute it, the MSM chains forwardly from the set of premises seeking to prove all possible consequences .
This paper contributes a fuzzy version of MSM using a variant of Intuitionistic Fuzzy Logic (IFL) called Realistic Fuzzy Logic (RFL). Here, a propositional variable is characterized by 2-tuple validity expressing its truth and falsity. Automatically, a third dependent attribute for the variable emerges, namely, hesitancy or ignorance about the variable, which complements the sum of truth and falsity to 1. If Ignorance is 0, then IFL reduces to Ordinary Fuzzy Logic (OFL) and the RFL version of MSM reduces to a simpler but weaker OFL version. The slight restriction of IFL to RFL involves the replacement of the concept of an Intuitionistic Fuzzy Tautology (IFT), in which truth is greater than or equal to falsity, by a restricted concept of Realistic Fuzzy Tautology (RFT) in which truth is strictly greater than 0.5. The introduction of the RFT enabled us to fuzzify the MSM without making any significant changes in it and to formally prove the correctness of all the steps of the emergent fuzzy MSM. As an offshoot, the paper contributes an improvement of the main algorithm that constitutes the heart of the MSM, whether it is crisp, ordinary fuzzy, or realistic fuzzy. The improvement involves a matrix formulation of the typical step of consensus generation that minimizes the comparisons among pairs of alterms that might have consensus alterms. The following task of absorbing subsuming alterms is also reduced considerably via a set of novel observations that were formally proved. The concept of consensus used herein is exactly the one used in crisp two-valued propositional logic. There was no need herein to introduce a specific concept of fuzzy consensus. The only significant change is that relations (26) no longer hold.
The fuzzy MSM methodology is illustrated by three specific examples, which delineate differences with the crisp MSM, address the question of validity values of consequences, tackle the problem of inconsistency when it arises, and demonstrate the utility of RFL compared to ordinary fuzzy logic.
The current paper is one of several new papers by the authors which are intended to extend the utility and sharpen the mathematics of the MSM. One of these papers  presents an incremental version of the MSM, in which the core work of the MSM is not completely repeated but is slightly incremented when additional premises are added. Another paper  utilizes the MSM in the exploration of hidden aspects in engineering ethical dilemmas by investigating different scenarios describing the situation from various perspectives.
The Improved Blake-Tison Method (ITM)
The complete sum of a switching function , to be denoted by , is the all-prime-implicant disjunction that expresses , that is, it is a sum-of-products (SOP) formula whose products are all the prime implicants of . The complete sum is called the “Blake canonical form” by Brown  in honor of . Blake who was the first person to study this form in his thesis . Since is a disjunction of all the prime implicants of , and nothing else, it is obviously unique and hence stands for a canonical representation of the switching function . The dual quantity of the complete sum is the complete product of a switching function , denoted , which is the all-prime-implicate conjunction that expresses , that is, it is a product-of-sum (POS) formula whose alterms or sums are all the prime implicates of .
The concept of the complete product of a switching function is closely related to that of a dual syllogistic formula for . However, while is unique and canonical, there are infinitely many dual syllogistic formulas for . A dual syllogistic formula of can be defined as a POS formula whose alterms include, but are not necessarily confined to, all the prime implicates of , that is, it is the complete product of conjuncted (possibly) with alterms each of which subsumes some prime implicates of . The complete-product formula is minimal within the class of dual syllogistic formulas for , that is, the set of alterms in any dual syllogistic formula for is a superset of the set of alterms in . Hence, can be denoted by , where is any dual syllogistic formula for and denotes an equivalent absorptive formula of , that is, a formula obtained from by successive deletion of alterms absorbed in other alterms of . The complete-product formula may be generated by the following two-step iterative-consensus procedure: (a) Find a dual syllogistic formula for by continually comparing alterms and adding their consensus alterms to the current formula of and (b) delete absorbed alterms to obtain . Note that two alterms have a consensus if and only if they have exactly one opposition, that is, exactly one variable that appears complemented in one alterm and appears uncomplemented in the other. In such a case, the consensus is the ORing of the remaining literals of the two alterms, with idempotency of the OR operation being taken into consideration. The concept of a consensus of two alterms is illustrated in Figure 1.
Tison method (see, e.g., [56–59, 64–67]) is a systematic streamlined version of the iterative-consensus technique for obtaining the of a switching function , or dually the of a switching function . The original study of Tison appeared in , but a more readable exposition can be found in , and further proofs are available in [58, 59]. Related work and techniques are also available in [68–77]. Since Tison method is actually due to Blake , we will present it here under the name Blake-Tison Method. Its essence when used for obtaining the complete product is summarized as follows.
Blake-Tison Algorithm. Start with a set of alterms or sums of literals with biform variables and a Boolean function that is expressed by conjunction of the alterms in . Assume that any absorbable alterms in have been deleted, so that the conjunction of alterms in is an absorptive formula. For , repeat the following 2-part step that replaces an absorptive set of alterms by another :(1)For , if appears complemented in one of the two alterms and and appears uncomplemented in the other such that the two alterms have no other opposition, then they have a consensus with respect to . Form that consensus and add it to . Finally, is replaced by a superset of elements, where .(2)Consider every pair , , of (so far remaining) products in . If subsumes , then delete . Otherwise, if is subsumed by , then delete . Whenever all subsumptions (and subsequent deletions) are exhausted, let the remaining absorptive set be .
Blake  and later Cutler et al.  formally proved Theorem 3, asserting the success of the Blake-Tison algorithm in obtaining by merely applying the iterative-consensus procedure to each biform variable one by one.
Theorem 3. In the Blake-Tison algorithm above, (a)the conjunction of alterms in any of the sets , where is an expression of ,(b)the final set consists of all prime implicates of .
Rushdi and Al-Yahya  proposed an improvement of Blake-Tison’s Method in which the typical step starts by arranging a given expression for with respect to a biform variable , in the formwhere , , and are POS formulas that are independent of , and the symbols , , and denote alterms or sums of single literals. Thanks to intelligent multiplication [25, 64], the function takes the POS form
Next is augmented by all consensus alterms with respect to , which turn out to be the alterms which do not add to 1 in the expression
This is followed by absorbing or deleting alterms that subsume others. The method repeats this typical step for all biform variables ending with after the last step.
Table 3 suggests an economic layout  for implementing the typical step in the Improved Blake-Tison Method (IBTM) with a restricted number for the comparisons needed for implementing absorptions. This typical step, which performs consensus generation with respect to a specific biform variable , involves a rearrangement of the alterms whose conjunction constitutes the current formula of at this step. We construct a consensus-generation table with respect to that resembles a multiplication table or matrix. The vertical keys of this table are the alterms containing the uncomplemented literal and its horizontal keys are the alterms containing the complemented literal , while its entries are the consensus alterms generated by these keys with respect to . Alterms containing neither the uncomplemented literal nor the complemented literal are set aside and naturally not included in the consensus generation of the table but might absorb or be absorbed by the consensus alterms produced by the table. Table 3 shows typical keys and entries of the consensus-generation table, where we use the symbol to denote the consensus of the vertical key with the horizontal key , which is the ORing of the two alterms and after deleting any repeated literals (thanks to the idempotency of the logical operation “OR”). Of course, if the alterms and have at least one opposition, that is, one literal that appears complemented in one of them and uncomplemented in the other, then is 1 and hence it is ignored since it does not affect a POS formula when multiplied with it. Now, further benefit gained from the above construction is made apparent via the following novel theorem.