[hist-analytic] Notes on Conjunction, etc.

Bruce Aune aune at philos.umass.edu
Tue Feb 3 11:00:18 EST 2009

JL Speranza’s latest note, partly in response to an earlier note of  
mine, contains some observations on Quine’s and other logicians  
treatments of “and.”  I agree with many of JL’s observations, but I  
want to call attention to the fact that knowledgeable logicians are  
well aware of the fact that the truth-functional conjunction, often  
represented by an upside-down 'v' but sometimes by ‘&’, is a sentence  
or clause connective, not a true counterpart to the English ‘and.’  
The latter, unlike the former, can properly connect expressions from  
many categories other than clauses.  It can connect noun phrases, as  
in ‘Tom’s dog and Mary’s cat,’ verb phrases, as in ‘slipping and  
falling,’ adjectival phrases, as in ‘powerful and threatening,’ and  
adverbial phrases, such as ‘cautiously and prudently.’  (I speak of  
phrases here, instead of simply nouns, verbs, adjectives, and  
adverbs, because longer verbal, adverbial, etc. units can also be  
joined by ’and,’ ‘or,’ ‘neither,’ ‘nor,’ and other vernacular  
conjunctions.)  To say this is not to criticize the logician’s  
familiar symbols.  Sometimes ‘and’ and so forth are used as clause  
connectives for which operations such as commutation holds, as in ‘2  
+ 2 = 4 and 3 + 2 = 5.’  Of course, as J.L. observed, commutation  
sometimes fails for the truth-functional ‘and,’ as in ‘Tom sat down  
and started to eat,’ which might be written more perspicuously as  
‘Tom sat  down and then he started to eat.’  G.H. Von Wright once had  
a little calculus featuring an ‘and then’ connective. A moral of my  
observations here is that, to avoid error, we have to be very careful  
when putting vernacular inferences into symbolic notation.  If an  
'or' isn't used as a truth-functional clause-connective, it should  
not be represented by a logician's '&'.

J.L. also comments on Excluded Middle, saying he thinks it is really  
a law about the tilde rather than ‘or’ (or ‘v’).  I think it is  
“about” both symbols if it is about either.  Actually, it uses both  
and mentions neither.  But Excluded Middle holds only for sentences  
(or clauses) that have a determinate meaning and satisfy the  
principle of Bivalence: when they are either true or false but not  
both.  Sentences containing vague predicates such as ‘fat’ don’t  
(without regimentation) satisfy bivalence and so provide counter  
instances to Excluded Middle. Jack Sprat is clearly thin and his wife  
is clearly fat, but if Jack’s brother is a borderline case, neither  
clearly fat nor clearly thin, then the sentence ‘Jack’s brother is  
fat’ is (without regimentation) neither true nor false, and ‘Jack’s  
brother is fat v ~( Jack’s brother is fat)’ is not true and so is an  
exception to Excluded Middle.

For various reasons, Carnap thought that the meaning of some  
predicates, including vague ones, can usefully be clarified  
incompletely by ‘A-Postulates.’  If I wish to use the predicate ‘fat’  
in a discussion where I want my meaning to be relatively clear, I  
might offer a partial clarification of its meaning by offering two A- 

1)    (x)(x is fat -> ~(x is thin))

2)   (x)(x is obese -> x is fat).

Carnap regarded A-postulates as semantical rules, so the two formulas  
I have just given could be considered true by virtue of the  
semantical rules of a certain system.  As such they would count as  
analytic truths of that system. I defend Carnap on this matter in  
chapter 3 of my recent book; I can think of no enable objection to  
his procedure.

My thanks to JL for the comments.

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