[hist-analytic] On Steve's Comments on my Ch 2, part 2

Bruce Aune 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  
valid arguments.

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  
elementary level.

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  
on them.

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.

Bruce Aune
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