Introduction to Philosophy/Logic/Some Properties of the Logical Connectives

Introduction to Philosophy > Logic > Some Properties of the Logical Connectives

&and; and &or; are commutative:


 * p &and; q gives the same result as q &and; p;
 * p &or; q gives the same result as q &or; p.

&and; and &or; are associative:


 * (p &and; q) &and; r gives the same result as p &and; (q &and; r);
 * (p &or; q) &or; r gives the same result as p &or; (q &or; r).

&and; is distributive over &or;:


 * p &and; (q &or; r) gives the same result as (p &and; q) &or; (p &and; r);
 * (p &or; q) &and; r gives the same result as (p &and; r) &or; (q &and; r).

&or; is distributive over &and;:


 * p &or; (q &and; r) gives the same result as (p &or; q) &and; (p &or; r);
 * (p &and; q) &or; r gives the same result as (p &or; r) &and; (q &or; r).

I say 'gives the same result as' since we have yet to talk about equality.

Those of you who know a little bit about abstract algebra will recognise that ({T, F}, &or;, &and;) is a ring - indeed it is a commutative ring with identity, and with only two elements, it is as simple a ring as you can get without being totally trivial or degenerate. To prove this, we need to observe, in addition to the commutative, associative and distributive properties above, that:


 * F acts as a zero: F &or; p is the same as p for any p &isin; {T, F};
 * T acts as a one: T &and; p is the same as p;
 * F is the &or;-inverse of all the elements of our ring: p &or; F is the same as p.

If you are not familiar with abstract algebra, just observe that &or; and &and; with T and F behave a bit like addition and multiplication with numbers. Note that &or; ('or') is the connective that corresponds to addition in this analogy, even though we often say 'and' when we mean 'plus' as in '3 and 4 equals 7'.

That our connectives &and; and &or; behave as a ring could be considered be an interesting result about the nature of reason - it shows that our propositional calculus has a structure similar to structures to be found elsewhere in mathematics.