The Modern Syllogistic Method (MSM) of propositional logic ferrets out from a set of premises
Fuzzy deductive reasoning has typically relied on a fuzzification of the Resolution Principle of Robinson [
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 [
The MSM has the distinct advantage that it ferrets out from a set of premises
We describe herein a fuzzy version of the MSM that utilizes concepts of the Intuitionistic Fuzzy Logic (IFL) [
The organization of the rest of this paper is as follows. Section
In Intuitionistic Fuzzy Logic (IFL), a variable
Note that when
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 [
The most important binary operations are the intuitionistic the intuitionistic the intuitionistic
With any three intuitionistic fuzzy variables idempotency: commutativity: associativity: absorption: distributivity: identities:
Atanassov [
A variable
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 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 [
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.
The MSM has the following steps.
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
We may also have
These relations symbolize the statements “If
The totality of premises in (
Equations (
The function
Suppose the complete product of
Equations (
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
Assume the problem at hand is governed by a set of RFTs
The given set of RFT premises are equivalent to the single function
Replace the function absorption, which is known to be tautology-preserving in general fuzzy logic and intuitionistic fuzzy logic and hence in the current realistic fuzzy logic, 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
Since
Each of the realistic fuzzy variables
Consider the following:
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
Let
Now consider two cases.
Now each of
One prominent difference between fuzzy MSM and ordinary MSM is that the complementary laws
Table
MSM derivation of fuzzy versions of famous rules of inference
Rule name | Fuzzy RFT antecedents (premises) | Premises as separate fuzzy equations |
Premises as a single fuzzy equation |
Conclusions as a single fuzzy equation |
Conclusions as separate fuzzy equations |
Fuzzy RFT consequence (conclusion) |
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MODUS PONENS |
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MODUS TOLLENS |
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HYPOTHETICAL SYLLOGISM |
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SIMPLIFICATION |
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CONJUNCTION |
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CONSTRUCTIVE DILEMMA |
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DISJUNCTIVE SYLLOGISM |
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ADDITION |
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ABSORPTION |
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CASES |
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CASE ELIMINATION |
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REDUCTIO AD ABSURDUM (CONTRADICTION) |
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A typical example of MSM presented by Brown [ if Alfred studies, then he receives good grades ( if Alfred does not study, then he enjoys college ( if Alfred does not receive good grades, then he does not enjoy college (
The MSM solution combines the above premises into a single equation
The last expression for
The function
This means that the consequent
The MSM has a built-in capability of detecting inconsistency in a set of premises, since this produces
In two-valued logic, the complete product of
Consider the set of premises [ Pollution will increase if government restrictions are relaxed If pollution increases, there will be a decline in the general health of the population If there is a decline in health in the population, productivity will fall The economy will remain healthy only if productivity does not fall
These premises are equivalent to the propositional equation
The complete product of
The fact that
Now, suppose that the given premises are not crisp tautologies, but are just RFTs with respective validities
Hence, each of the new clauses in (
Validities of consequences obtained in Example
New clause | Nature | Validity |
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Consensus of |
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Consensus of |
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Consensus of |
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Consensus of |
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Consensus of |
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Consensus of |
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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
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
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 [
In future work, we hope to combine the contributions of the current paper with those of [
The complete sum of a switching function
The concept of the complete product of a switching function
(a) The alterms
Tison method (see, e.g., [
For Consider every pair
Blake [
In the Blake-Tison algorithm above, the conjunction of alterms in any of the sets the final set
Rushdi and Al-Yahya [
Next
This is followed by absorbing or deleting alterms that subsume others. The method repeats this typical step for all biform variables ending with
Table
The general layout of the consensus generation table of the Improved Blake-Tison Method when producing consensus alterms with respect to
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Set-aside alterms | ||||||
(alterms containing neither |
In the consensus-generation table of Table there are no absorptions among vertical keys, horizontal keys, and set-aside alterms; a table entry cannot be absorbed by a table key, but it could be absorbed by another table entry or a set-aside alterm. A set-aside alterm could be absorbed by a table entry; if a table entry if a table vertical key if a table horizontal key
In the following, we outline a proof and reflect on the ramifications of Theorem Each of the conjunctions of vertical keys, that of horizontal keys, and that of set-aside alterms constitutes an absorptive formula. Therefore, there are no absorptions among alterms of such a formula. A table entry cannot be absorbed by a table key because the former cannot subsume the latter since the former lacks the literal Suppose that the table entry Now, suppose that the vertical table key Likewise, it can be shown that if a table horizontal key
To change the conjunction of alterms in the whole table (including keys, entries, and set-aside alterms) into an absorptive formula, there is no need to compare every alterm with all other alterms in the whole table. Instead, every remaining table entry not equal to 1 is either absorbed in another in the same row or column of the table or in one of the set-aside alterms or it stays unabsorbed. A vertical table key is either absorbed in a table entry in the same column of the table or it stays unabsorbed. A horizontal table key is either absorbed in a table entry in the same row of the table or it stays unabsorbed. A set-aside alterm is either absorbed in one of the remaining (not equal to 1) table entries or it stays unabsorbed.
In summary, the number of comparisons needed to implement the absorption operation comparing each remaining table entry not equal to 1 to the alterms with fewer or the same number of literals in ( comparing each vertical table key to the table entries not equal to 1 with fewer or the same number of literals in its column of the table; comparing each horizontal table key to the table entries not equal to 1 with fewer or the same number of literals in its row of the table; comparing each of the set-aside alterms to the remaining table entries not equal to 1 with fewer or the same number of literals.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.