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

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


∧ and ∨ are commutative:

  • p ∧ q gives the same result as q ∧ p;
  • p ∨ q gives the same result as q ∨ p.

∧ and ∨ are associative:

  • (p ∧ q) ∧ r gives the same result as p ∧ (q ∧ r);
  • (p ∨ q) ∨ r gives the same result as p ∨ (q ∨ r).

∧ is distributive over ∨:

  • p ∧ (q ∨ r) gives the same result as (p ∧ q) ∨ (p ∧ r);
  • (p ∨ q) ∧ r gives the same result as (p ∧ r) ∨ (q ∧ r).

∨ is distributive over ∧:

  • p ∨ (q ∧ r) gives the same result as (p ∨ q) ∧ (p ∨ r);
  • (p ∧ q) ∨ r gives the same result as (p ∨ r) ∧ (q ∨ 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}, ∨, ∧) 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 ∨ p is the same as p for any p ∈ {T, F};
  • T acts as a one: T ∧ p is the same as p;
  • F is the ∨-inverse of all the elements of our ring: p ∨ F is the same as p.

If you are not familiar with abstract algebra, just observe that ∨ and ∧ with T and F behave a bit like addition and multiplication with numbers. Note that ∨ ('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 ∨ 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.