Siddhant Volume 14, Issue 2, April-June, 2014, pp- 133-136 DOI: 10.5958/j.2231-6728.3.2.029
The Theory of Inference for Statement Calculus Jayesh K. Tiwari1*, Rajendra Tiwari2 ABSTRACT In this article we are trying to prove how to get success through logical approach of validity of argument using truth table and different logical connectives. It will be proved by tautology. KEYWORDS: Calculus, Inference Calculus, Logics, Principles, Truth Table
INTRODUCTION The main function of logic is to provide rules of inference, or principles of reasoning. The theory associated with such rules is known as inference theory because it is connected with the inferring of a conclusion from certain premises. When a conclusion is derived from a set of premises by using the accepted rules of reasoning, then such a process of derivation is called a deduction or a formal proof. An important difference between the reasoning used in any general discussion and that the premises used are believed to be true either from experience or from faith, and if proper rules are followed, then one expects the conclusion to be true. In mathematics, one is solely concerned with conclusion which is obtained by following the rules of logic. In any argument, a conclusion is admitted to be true provided that the premises (assumptions, axioms and hypotheses) are accepted as true and the reasoning used in deriving the conclusion from the premises follows certain accepted rules of logical inference. Such an argument is called sound. In any argument we are always concerned with its soundness. In logic the situation is slightly different, and we concentrate our attention on the study of the rules of inference by which conclusion are derived from premises. Any conclusion which is arrived by following these rules is called a valid conclusion, and the argument is called a valid argument.
Now we come to the questions of what we mean by the rules and theory of inference. The rules of inference are the criteria for determining the validity of an argument. These rules are stated in terms of the forms of the statements (premises and conclusions) involved rather than in terms of any specific statements. These rules are not arbitrary in the sense that they allow us to indicate as valid at least those arguments which we would normally expect to be valid. In addition, neither do they characterize as valid nor those arguments which we normally consider as invalid. The actual truth values of the premises do not play any part in the determination of the validity of argument. In short, in logic we are concerned with the validity but not necessarily with the soundness of an argument.
PRELIMINARIES Now we begin with some definitions. Definition 2.1- Sentences: Group of words with proper meaning is called a sentence. Definition 2.2- Statements: All the declarative sentences to which it is possible to assign one and only one of the two possible truth values are called statements. The symbols, which are used to represent statements, are called statement letters usually the letters P, Q, R…, p, q, r, etc. are used.
1
Associate Professor, Department of Computer Science, Shri Vaishnav Institute of Management, Davi-Ahilya University, Indore-452009, Madhya Pradesh, India. *Corresponding author Email:
[email protected] 2 Professor, Department of Mathematics, Government Madhav Science College, Vikram University, Ujjain-456010, Madhya Pradesh, India. Email:
[email protected] Siddhant
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Jayesh K. Tiwari, Rajendra Tiwari
Definition 2.3- Logical Connectives: Logical connectives or sentence connectives are the words or symbols used to combine two sentences or statements to form a compound sentence or compound statements. Table 1: Logical connectives with their symbols Connective word
Name of connective
Symbols
Rank
Not And
Denial or Negation
1
Conjunction
2
Or
Disjunction
3
If…then
Conditional
4
Iff
Bi-conditional
5
Definition 2.4- Tautology: A statement formula which is true regardless of truth values of the statements which replaces the variable in it is called a universally valid formula or a tautology or logical truth. Definition 2.5- Contradiction: A statement formula which is false regardless of the truth values of the statement which replace the variable in it is called a contradiction. Definition 2.6- Validity Using Truth Tables: Let A and B be two statement formulas. We say that “B logically follows from A” or “B is a valid conclusion of the premise A” iff A B is a tautology. Just as the definition of implication was extended to include a set of formulas rather than a single formula, we say that from a set of premises {H1, H2, H3,…, Hm} a conclusion C follows logically iff H1 H2 … Hm C. Given a set of premises and a conclusion, it is possible to determine whether the conclusion logically follows from the given premises by constructing truth table as follows. Let P1, P2, P3, …, Pn be the atomic variables appearing in the premises H1, H2, H3, …, Hm and the conclusion C. If all possible combinations of truth values are assigned to P1, P2, …, Pn and if the truth values of H1, H2, …, Hm and C are entered in a table, then it is easy to see from such a table whether (1) is true. We look for the rows in which all H1, H2, …, Hm have the value true (T). If, for every such row, C also has the value T, then (1) holds otherwise fallacy exist.
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Definition 2.7- Law of Syllogism: We know that the statement [(p q) (q r)] (P r) is tautology, and therefore the argument pq } premises qr ……………………………………………………… pr (conclusion) is a valid argument . This is called law of Syllogism. Definition 2.8- Rule of Detachment: We know that the statement [p (pq)] q is a tautology, and therefore the argument, p } premises pq ........………………………………………………… q (conclusion) is a valid argument . This is called rule of detachment. This rule is also known as Modus ponens.
MAIN RESULTS We take some premises, by the help of premises we derive an argument and then we verify the validity of argument by truth table. If it is tautology then our argument is valid argument. Let us consider a premise that “every human being has appreciable success.” As per the above given premise, all those human beings who have excelled in their fields have become a face in crowd, and today they are the well known personality in the world namely Bill Gates, Steve Jobs, Narayan Murthy, Kumar Mangalam, Ratan Tata, Sachin Tendulkar, Roger Fedrer, Ramanujan and others. In today’s world many such great personalities have gained appreciable success in their respective fields. Many young minds are eager to follow the foot prints of the great distinguished personalities of today’s world but a million dollar question arises, i.e., success comes from where? That is, how some people are able to achieve great heights? Why can’t all reach to the same level? The Volume 14, Issue 2, April-June, 2014
The Theory of Inference for Statement Calculus
most possible answer to the above query is that “success comes from hardwork”. This is also a correct premise. Now again the question arises, i.e., “from where the hard work comes?” The answer is, as all of you know, “hard work comes from involvement”. It is also a correct premise. Now the next question that strikes our brain is that. “involvement comes from where? The obvious answer is, the involvement comes from our “devotion”. It is also a correct premise. Hence, the next question that puzzles a curious mind is that where do the “devotion” comes from? We can rightly say that “devotion” comes from “our day-to-day life regularity and punctuality”. It can also be said to be a correct premise. In nutshell we can say that if our dayto-day life is regular, then we will be devoted and if we are devoted, we get involved in our target which make us hardworking then we will definitely hit the jackpot and achieve the much desired success. Test for validity of the arguments If our day-to-day life is regular and punctual then we will be devoted.
P
q
T T F F
T F T T
p q (p q) p T F T T
T F F F
[(pq) p] q T T T T
Since last column contains only T’s, hence the given argument is valid. If we will be devoted then we will get involved. If we will be devoted. ……………………………………………………… we will get involved Let
p: if we will be devoted q: we will be involved
we can express following argument in symbolic form as pq
…………………………………………………………………………………….
} premises p ………………………………………………………… q (conclusion)
we will be devoted.
We shall construct the truth table for the statement
If our day-to-day life is regular and punctual.
Let
p: if our day-to-day life is regular and punctual q: we will be devoted
we can express following argument in symbolic form as pq }
premises
p ………………………………………………………… q
(conclusion)
We shall construct the truth table for the statement [(p q) p] q Siddhant
[(p q) p] q P
q
T T F F
T F T T
p q (p q) p T F T T
T F F F
[(pq) p] q T T T T
Since last column contains only Ts, the given argument is valid. If we will be involved then we will be hard working. If we will be involved ……………………………………………………………. we will be hard working 135
Jayesh K. Tiwari, Rajendra Tiwari
Let
p: if we will be involved q: we will be hardworking
Let
p: if we are hardworking q: we will get success
we can express following argument in symbolic form as
we can express following argument in symbolic form as
pq }
pq
premises
p ………………………………………………………… q (conclusion) We shall construct the truth table for the statement
} premises p ………………………………………………………… q (conclusion) We shall construct the truth table for the statement
[(p q) p] q p q (p q) p
P
q
T T F F
T F T T
T F T T
T F F F
[(p q) p] q [(pq) p] q T T T T
Since last column contains only Ts, the given argument is valid.
P
q
T T F F
T F T T
p q (p q) p T F T T
T F F F
[(pq) p] q T T T T
If we are hardworking then we will get success.
Since last column contains only Ts, hence the given argument is valid. Hence the above-said argument is a valid argument.
If we are hardworking.
ACKNOWLEDGEMENT
……………………………………………………………
The authors are thankful to Dr. K.N. Rajeshwari Professor & HOD School of Mathematics DAVV, Indore for her useful suggestion.
we will get success
BIBLIOGRAPHY Anthony, A. and Michale, Z, eds. (2002). Logic, meaning and computation. In Essays in Memory of Alonzd Church. Springer. Barwise, J. (1985). Model theoretical logics; background and aims. Model Theoretic Logics. Springer-Verlag. David, B.P., Barwise, J. and Etchemedy, J. (2011). Language, Proof and Logic, second edition. Stanford center for the Study of Language and Information, Springer-Verlag 2011. Dinkines, F. (1964). Introduction to Mathematical Logic. New York: Appleton century-crafts, Inc.. Novikov, P.S. (1964). Elements of Mathematical Logic, translated by Leo F. Born. Reading, MA: Addison-Wesley Publishing Company, Inc.
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