[FOM] Weak truth and strong truth

[Sandy Hodges]

This may be too well-known or too obvious to be worth posting.

Accounts of truth can be divided into those based on weak truth and
those based on strong truth.   By any account, there are some items
which can neither be called true, nor called false.   This is true even
of so-called "two-value" accounts; for example, items that are not
sentences at all can't be called true, and can't be called false.   The
difference between strong and weak truth depends on what happens when
such an item is called true anyway.   If "x" names some item, let us say
a stone, which is neither true nor false, what is the status of
"True(x)"?   In a strong truth account, "True(x)" is false; the idea is
that x falls outside the extension of the truth predicate.   In a weak
truth account, "True(x)" is itself an item which is neither true nor
false; the idea is that it is ungrammatical or otherwise an error to
apply the predicate "True" to an item of the wrong type, so "True(x)" is
not a well-formed formula, so it is neither true nor false.

The two accounts can be summarized in truth tables:

Strong Truth:
X   |  True(X) | False(X)
True |  True |  False
neither | False | False
False | False | True

Weak Truth:
X   |  True(X) | False(X)
True |  True |  False
neither | neither | neither
False | False | True

The two accounts of truth, strong and weak, treat the Liar paradox in
very different ways:

(1)   ~True(1)

In a weak truth account, assume sentence (1) is neither true nor
false.   Then calculate the value of sentence (1) using the truth table
for weak truth.  (Also assume choice negation for "~", so ~X is neither
when X is neither).   The calculated value for (1) is "neither", the
same value that was assumed.   Thus "neither" is the value that 'solves'
sentence 1, and it is the only solution.

In contrast, assume (1) is neither true nor false in a strong truth
account.   Then calculate the value using tables for strong truth.   The
result is that (1) is true.   Thus "neither" does not 'solve' sentence
(1); nothing does.  The strong truth account claims that sentence (1) is
neither true nor false precisely because there is no solution.

This is the fundamental difference between the accounts: weak truth says
(1) is neither because that's the solution, strong truth says (1) is
neither because there is no solution.

In either account, (1) is called "neither."  There are many difficulties
I won't discuss here with assuming an infinite hierarchy of
meta-languages; but if we want talk about (1) in the object language
instead, we'll need a "Neither" predicate.    A "Neither" predicate
allows us to formulate the strengthened Liar:

(2)  ~True(2) & ~Neither(2)

The strengthened Liar can be a problem for weak truth accounts, at least
if those accounts hope to call their own claims "True."   The only
account I know of that can overcome this, was proposed by Gupta and
Martin in 1984; they proposed using weak Kleene truth tables for &, v,
etc.    With weak Kleene, (A & B) and (A v B) are both 'neither' when
either A or B is 'neither.'    With weak Kleene tables, 'neither' is a
solution for (2).

But weak Kleene tables are somewhat limiting.   Suppose an after dinner
speaker has a pile of blue cards, on which are written various
paradoxes, and a pile of green cards, on which are written various
falsehoods.    An observer sees the speaker pick up a card and read it,
but the observer is both hard of hearing and colorblind.   So the
observer says:

(3) If the card was blue, then the sentence the speaker read was neither
true nor false.
(4) If the card was green, then the sentence the speaker read was false.

(5) The card was either blue or green.
(6) The sentence the speaker read was either false, or it was neither
true nor false.

The observer hopes in saying these things, to say things which are true
whichever pile the card came from.   But in the Gupta-Martin account he
has not succeeded.   If the card was in fact blue, then (4) and (6) are
both of them 'neither'.      Notice that my own sentence "If the card
was in fact blue, then (4) and (6) are both of them 'neither'" is not
true but neither if the card was blue (and the same applies to this
sentence).   Thus the Gupta-Martin account means that a plausible
every-day situation, such as the one I've attempted to describe about
the speaker and her piles of cards, is un-describable.

Gupta-Martin is the only weak truth account which can call its own
conclusions true (no one level in a hierarchy can, nor can the 'level'
in which the whole hierarchy is described).

Strong truth also has some odd results, as are shown by the revenge
Liar:

(7) True(8)
(8)  False(7) v Niether(7)
(9)  Neither(7) & Neither(8)
(10) Neither(7)
(11) False(7) v Neither(7)

No truth value 'solves' 7 and 8, so they are both 'neither' in the
strong truth account.     Thus (9) expresses our own conclusion, so we
must call (9) true if we are to call our own conclusions true.  (10) and
(11) follow from (9), so we call them "True" also.  Thus we call (8)
'neither', and (11) 'true', even though they are the same formula.
Thus: two instances of the same formula can have different truth
values.      (8) expresses something we are willing to assert, and do
assert; nonetheless we do not call (8) true.    Thus we have a violation
of Tarski's adequacy condition.

In comparing the implausible results of the weak vs. the strong accounts
of truth, it's worth noting that in the strong truth account, the
implausibilities are confined to those statements that are
self-referential (or which refer to an infinite chain).   If you are in
the position of an observer, and can be certain that the sentences you
describe do not refer back to the ones you are about to say, you have
nothing to worry about.   But with weak truth (as in the Gupta-Martin
account) the color-blind observer was unable to describe the situation
he observed, even though there was no question of the observer's
sentences being referred to by any card.

------- -- ---- - --- -- --------- -----
Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.