Propositional or truth functional calculus Statements and statement letters ## PlanSource: Quines Research other sources In process. - Need a template to render logical symbols and formatting.
- Write the essentials for truth functional schemata, Boolean term schemata, Boolean existence schemata, quantificationmonadic and general or polyadic, substitution, deduction, completeness, decidability, and soundness.
- Topics from Part 4 of Quines book: singular terms, identity, descriptions, elimination of singular terms, elimination of variables, classes, number, axiomatic set theory.
- When the section on implication is written place in it Quines example which is now in the section on use and mention.
## Propositional or truth functional calculus## The general truth function## Statements and statement lettersStatements, whose characteristic in logic, it is to be
true (T) or false (F or ^) are represented by letters ## Truth functional schemataA truth function of a number of statements is one whose
inputs are the truth values of the statements and whose value, for each
combination of input values, is a truth valuei.e., one of T and F. It is
convenient to work with letters Truth functions can have any number of inputs. We consider some common one place and two place functions (unary and binary), develop their properties, and show all truth functions can be built from them. Then the truth functional combinations of letters are called truth functional schemata (singular: schema). Letters S, _{2}S
stand for schemas. A truth
functional interpretation of a schema is an assignment of statements with
definite truth values or definite truth values to the letters._{3}## Use and mentionThe following true statement America is larger than Australia. The following true statement America is shorter than Australia. The following explains how this distinction between use and mention is significant in talking of linguistic objects, particularly logical ones. That a statement has a truth value T is syntactic and we might say: The truth value of p is T. On the other hand that p is true is semantici.e. of the meaning of p or, equivalently, of the relation between p and the world. And to say that a statement is true we would write: p is true. Quine gives the following example that concerns the difference between implication and the conditional ฎ. The conditional ฎ is a connective written p ฎ q, said if p then q but not p implies q, and is false only in the case that p is true and q is false. On the other hand a truth functional schema S_{2}but S._{2}
## IdentityI( ## NegationThe negation of (the common symbolization with a top bar as in pā
is not conveniently available on Word). p is true if and only if p
is true so there is no need to ever write the double negation## ConjunctionConjunction is a two place function or connective: it is
the logical and. Conjunction here uses juxtaposition as its implicit
symbol, it is said ## The general truth functionWe will show that the general truth function is a combination of negations and conjunctions. An n place truth function has n inputs; let them be p.
Each letter has two input truth values, so the number of combinations of
input values is 2_{n}^{n}. Of these
combinations the result will be true in some cases and false in others. An
example of a case in which the result is false might be p, _{1}p, _{2}p,
_{3}p, _{n-1}p; their conjunction is true when
the result is false; collect all such cases and form the conjunction of their
negations; provided that there are some cases when the n place function is
false, the compound is false when at least one factor is true but otherwise
true and is thus equivalent to the given n place truth function. If the n
place function is always true write (_{n}p_{1}p_{1}p_{2}p_{3}p_{n-1}p) which is always true since the
expression in brackets is always false. Note, by the way, the superfluity of
identity from the equivalence of I(_{n}p) to p.## AlternationThe non-exclusive (truth function) or uses the symbol ฺ or vel from Latin; it is a two place function,
is said p and q are false; so () is
true (precisely) when not both are falsei.e. precisely when pqp ฺ q is true. That is, p ฺ q is equivalent to ()
(the truth functions are identical). So, alternation can be written in terms
of negation and conjunction; this shows, as we also know from consideration
of the general truth function; alternation is superfluous. But similarly,
conjunction can be written in terms of alternation for pqpq is
equivalent to ( ฺ p).
So the general truth function can be written in terms of negation and
alternation; either alternation or conjunction can be taken as primitive and
the other superfluous; but it is convenient to retain both for their use and
convenience.qThis is a convenient place to introduce ## De Morgans lawsExpression of conjunction and alternation in terms of one another generalize to the following equivalences: ( and (
ฺ
ฺ
q).s
## Sheffers strokeThis connective written as pp
ฏ p and pq as
ฏ p, and hence the
rest.q |