Notes and Quotes from Methods of Logic, 1.2

Notes:

A compound is called a truth function of its components if its truth value is determined in all cases by the truth values of the components. More precisely: a way of forming compound statements from component statements is truth-functional if the compounds thus formed always have matching truth values as long as their components have matching truth values.

Interestingly, all truth functions can be sufficiently notated using only negation and conjunction.

A neat routine to employ to this end is listing the schedule of truth values for a given compound, note those cases in which the compound is false, and then simply conjunct the denial of each of these cases. This new compound (using only negation and conjunction) will map on exactly to the aforesaid schedule, denying explicity just those cases in which it was to come out false; in all others cases it comes out true.

The only sort of situation that this routine cannot take care of are those in which one must deal with a compound that comes out true in all cases. Quine says this about it, cleverly: "These trivial exceptions call, then, for separate treatment; and a treatment is straightway forthcoming which is correspondingly trivial" (19).
His trivial treatment is hilarious. We need a compound that will come out true regardless of what truth values are assigned to its components. So, simply deny the conjunction of all the components with the denial of any one of the components. Such a conjunction will come out false in all cases, since it will simultaneously affirm and deny a given component, so that conjunction's denial will come out true in all cases.

The reason for keeping 'V' around is to facilitate certain technical manipulations yet to come, though it is unnecessary. It is handy, not essential. However, conjunction is no less superfluous than alternation. Should we want, we could sufficiently represent all compounds using only negation and alternation.

The single connective '|' is clever. It can single-handedly represent any compound, though uneconomically. 'p|q' is to be true if and only if 'p' and 'q' are not both true. This makes it equivalent to -(p&q). Negation could be represented 'p|p' since this would mean that not both 'p' and 'p' are true, which is just to say that 'p' is not true, and therefore false, or '-p'. Conjunction can be represented as well. Instead of 'pq' we can have '(p|q)|(p|q)', which would be to say '-(p|q)'.


Nice quotations/passages:

Nice way of putting truth-function description: "Any particular truth function can be adequately described by presenting a schedule showing what truth values the compound will take on for each choice of truth values for the components" (17).

"Given a description of a truth function--i.e., given simply a schedule showing what truth values the compound is to take on for each choice of truth values for the components--we can construct a truth function out of negation and conjunction which answers the description" (18).