The Journal of the Indian Mathematical Society

S. Pankajam ¹

Affiliations
1 University of Madras, IN

Abstract

In the previous paper with the same title, it was shown that with the accepted meanings of 'and', 'or', the totality of propositions form a distributive lattice P, in which the 'sum' and 'product' correspond to 'and', 'or'; we specifically examined the nature of negation with respect to these operations and showed that if we took the minimal meaning of negation, then the negation of any proposition turns out to be its product-complement in P. Lastly, it was shown that this meaning of negation is conformable to its meaning in Intuitionistic Logic (as for instance that formulated by Heyting).

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S. Pankajam

Abstract

In the review of my first paper 'On the Formal Structure of the Propositional Calculus I' in the Journal of Symbolic Logic.

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S. Pankajam ¹

Affiliations
1 University of Madras, IN

Abstract

The prepositional calculus is the study of the set P of all elementary (or unanalysed propositions) under the three binary operations denoted by and, or, implies and the unary operation of negation. The simplest of these are probably the operations 'and', 'or' (in symbols x, +); the other two operations depend upon each other, and different views of them can be taken.

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S. Pankajam ¹

Affiliations
1 Madras, IN

Abstract

The type of argument given in text-books to derive the form of Φ(n)-the number of numbers less than n and prime to n, is as follows. If n be the given number and p₁, p₂,...,p_q be the different prime factors of n, then there are n/Pi numbers divisible by pi, n/pj numbers divisible by pj, and so on. Generally there are n/pipjpk..... numbers divisible by pi, pj, pk,......, simultaneously.

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S. Pankajam ¹

Affiliations
1 University of Madras, IN

Abstract

The elements of a Boolean Algebra may be interpreted either as classes or as attributes. We shall as a rule adopt the former interpretation in this paper. The operations of the Boolean Algebra are +, X, and negation. The sum A + B of two classes A, B is defined as the class of elements belonging either to A or to B.

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S. Pankajam

Abstract

The Logical Calculus contains two operations ⊕ and ⊗, having respectively the sense of 'or' and 'and'. They are connected by the following laws:

(α) Commutative x⊕y=y⊕x.

x⊗y=y⊗x.

(β) Distributive x⊗(y⊕z) = (x⊗y)⊕(x⊗z).

x⊕(y⊗z) = (x⊕y)⊗(x⊕z).

(γ) Associative x⊕(y⊕z) = (x⊕y)⊕z.

x⊗(y⊗z) = (x⊗y)⊗z.

We see that in these axioms, ⊕ and ⊗ occur symmetrically.

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