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The semantics of a symbolic logic consists
of analyses of the concepts of semantic entailment, semantic validity, and semantic
consistency. For a propositional or sentenial logic, such as the sentence language, SL, of
SYMLOG, this involves the notions of a truth value assignment and truth on an assignment.
| A truth value assignment is a function.
Like any function, it consists of arguments and values that it takes on such arguments. The
arguments are atomic sentences of SL--the SL sentence letters.
The values are the truth-values T and F.
Hence, any given truth value assignment assigns either a T or an F to every
atomic sentence of SL.
In addition, no truth value assignment assigns both a T and an F to any atomic
sentence of SL.
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| Note that it looks like there is more than
one truth-value assignment; there are, in fact, infinitely many such assignments; for SL
has infinitely many atomic sentences. In virtue of what, then, is are two
distinct truth-value assignments distinct?
Two assignments, A and A', are distinct in
that with respect to at least one atomic sentence of SL, P, A(P)
is not identical to A'(P).
Hence, if A is distinct from A', then, for at
least one SL-atomic, P, either A(P) = T and
A'(P) = F or A(P)= F and A'(P)
= T.
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| The SL-sentence connectives are used to
generate molecular sentences from SL-atomics. Theses connectives are associatd with truth
functions, and the truth of any molecular sentence of SL is a function of the truth-values
of its atomic components. Let's consider the sentence,
L & M
It is generated by using the operation of & on the atomic sentences L and M.
And the truth-value of the molecular sentence is generated by applying the
truth-function associated with the & on the truth values assigned to the atomics.
The truth-function associated with & is given by the following truth table
for &:
| P |
Q |
P&Q |
| T |
T |
T |
| T |
F |
F |
| F |
T |
F |
| F |
F |
F |
Now let A be a truth-value assignment, and let
us suppose that A(L )= T and A(M) = F. The table for the ampersand
makes it easy to see what truth-value assignment A assigns to the
conjunction L & M.
Look at the table's second row. It tells us that for every
assignment that assigns a conjunction's left conjunct T and right
conjunct F, assigns F to the conjunction.
Therefore, A(L & M) = F.
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| Now, let's see what the same truth-value
assignment assigns to the somewhat more complex sentence, L v (L & M).
Here we need to make reference to the truth-function associated with v.
That truth- function is represented by the following table:
| P |
Q |
PvQ |
| T |
T |
T |
| T |
F |
T |
| F |
T |
T |
| F |
F |
F |
more to come |
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