[hist-analytic] On Steve's Comments on my Ch 2, part 2
aune1 at verizon.net
Sat Oct 24 10:36:11 EDT 2009
Steve takes issue with what I say about logic in chapter 2. He does
not describe my position accurately, though, and his objections call
for some clarification.
Rationalists typically claim that they can simply “see” the truth of
basic logical laws such as the principle of non-contradiction. I
argue against this, saying that such “laws” have a kind of generality
that makes them an inappropriate object of a supposed truth-
ascertaining act of mental “vision.” I will enumerate and state some
of the claims I was making.
The “laws” in question are normally stated in schematic form. Steve
uses quantifers, but they are not normally stated that way (unless
substitution-quantifiers are used), for a reason I mention in footnote
26. The actual content of the laws is that all instances of the
schemas are true.
I claim that the supposed acts of vision cannot survey the relevant
instances and therefore add credibility to the claim that all of them
are true. How do I know that the mental survey fails in this way?
For three distinguishable reasons. (A) Some sentences that have the
appearance of being appropriate instances can be shown to be false if
classical logical principles are applied to them; (B) Some such
sentences are arguably not true, though they are not false; and (C)
Arguments corresponding to conditional sentences supposedly expressing
logical truths are actually debatable and have convinced very
intelligent philosophers that they are counter instances to supposedly
The arguments given in my second chapter are expressed in language
that I have worked very hard to make transparently clear. I have read
them over many times, and I am convinced that I succeeded in making
them clear. I can’t expect to improve on them in a short post, but
knowing that they span a large chunk of text, I can at least highlight
a number of points that bring out crucial steps in my argument that
some readers, Steve and Danny Frederic included, seemed to miss. The
points in question shouldn’t be considered controversial. They are
actually quite familiar to all teachers of logic in any but the most
Here is a comment on my reason (A) stated in paragraph 2 above.
I identify two statements supporting reason (A). I describe one
sentence as appearing in a sphere and another appearing in a
rectangle. The sentence are, ‘The sentence in the rectangle is true’
and ‘The sentence in the circle is false’. From the two sentences I
derive a contradiction.
Steve says, “As for the derivation, what it shows is that one of the
two statements (or both), either the one in the rectangle is false or
the one in the circle is false. Which one is it?” This is wrong. My
derivation is valid, and its conclusion has the form of “p and not-p.
We do not “know” that one of these conjuncts is false and not true.
What we know is this: If standard logic applies to them, they are both
true and both false. The proof I give shows this.
Steve thinks that my claim can be refuted by using truth tables. He
is wrong. A proper truth table will have values for sentences that
are truth-functions of the elementary statements occurring in the
argument. Two such statements are the disjunctions from which a
conjunction, ‘A and not-A’, is inferred. This conjunction tops a
column containing Ts and only Ts. There will, of course, be another
column under another such conjunction containing only Fs.
Observation: I do not claim that the contradictions I derive should be
accepted as both true and false, as some contemporary logicians do
(e.g. Graham Priest). My claim is that the decision of how to
accommodate them requires argumentative considerations far more
discursive than anything plausibly supplied by immediate intuition.
Their status as true or false requires a kind of support not available
to the rationalist.
Here is my comment on reason B:
Certain vague statements are widely conceded to violate the principle
of bivalence, which requires statements to have the value T or F but
not both. If “Tom is thin” is such a statement, then if it is
abbreviated as “A”, the conjunction “A and not-A” and the disjunction
“A or not-A” should not have the value T, which classical logic
requires. Immediate intuition provides no counter-assurance that they
are true. It is philosophically useless here.
Steve offers some criticisms of my claims regarding B, but they do not
succeed. He says, for instance, “what Bruce seems to be denying by
interjecting [a third value] ‘IND’ is not the principle of excluded
middle but, rather bivalence.” Not true. If certain statements are
assigned a third value distinct from truth and falsity, the
conjunctions and disjunctions mentioned in the last paragraph will not
have the value T, which they should have if classical logic
accommodates them. The principle of excluded middle would thus have a
counter instance, not a false one but one not having the value T.
Steve says, “if a sentence is vague, it is reasonable to suppose that
there is no proposition it expresses. If the sentence does not express
a proposition, then it might be maintained that it has no truth value.
If it has no truth value it cannot provide the sort of counterexample
Aune is looking for.” Claiming that vague sentences don’t express
propositions and therefore don’t belong within the purview of standard
logic is a familiar move in philosophical logic. I don’t think it is
a very plausible move, because vague statements are often clearly true
or clearly false. “Fat” might be a vague predicate, but if Sally in
anorexic, the statement “Sally is fat” is clearly false. Vague
statements are neither true nor false when they are asserted of
borderline cases. But however this may be, the decision to regard
vague statements as not expressing propositions and therefore as not
being covered by classical logic principles is not incompatible with
anything I claim. I was arguing against the rationalist claim that
basic logical truths are known by immediate intuition. But this kind
of intuition does not identify vague statements as excluded instances
of classical logical principles. The decision to classify them in a
way that does plausibly exclude them fits in nicely with the position
I was defending.
I might mention here that Steve suggested a way of excluding
paradoxical self-referring statements from the scope of classical
logic; he said that Russell’s vicious circle principle or Zermelo’s
“approach” might work. Steve’s details aren’t right, since Russell’s
vicious circle principle succumbed to criticism by Frank Ramsey, and
Zermelo’s “approach” has nothing to do with the so-called semantic
paradoxes (it provided an alternative to Frege’s axiom of abstraction
in set theory, to which Russell had given a counter instance). But the
basic strategy Steve used was in full agreement with what I was
claiming. The paradoxical assertions are not surveyed, let alone
accommodated, by the rationalist’s appeal to immediate intuition. The
latter is powerless to discriminate proper from reasonably excluded
instances of logical laws and inference patterns.
Re reason [C]. In spite of what he said about Russell and Zermelo,
Steve says he still “takes a position something like the one Bruce
argues against. But my position is not that there is a special class
of propositions that intuition justifies, but rather that certain
inferences are justified by means of a rational intuition.” My
argument against knowing inferences are valid this way is based
specifically on arguable counter instances to modus ponens, which some
shrewd philosophers defend claim. The plausibility of these instances
undermines the supposed compelling evidence provided by rational
intuition. But Steve has not yet commented on these instances; in a
recent note to me, he says he is in the process of preparing comments
Final remark: I was not concerned to argue that classical logic (laws
or inference patterns) is defective in any way. Rather, I wanted to
identify statements and laws that appear to be contrary to such laws
or patterns and whose status as true, false, acceptable or
unacceptable rests on a basis clearly different from immediate
intuition. I think that the rationalist position offers an over-
simplified and erroneous picture of how logical laws and patterns of
inference—their acceptable and excluded instances—are reasonably
identified and justified. Showing this was perhaps my basic concern
in chapter 2.
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