[FOM] Re: FTGI1:Classical Propositional Calculus (Harvey Friedman)

Sandy Hodges SandyHodges at attbi.com
Wed May 7 16:44:26 EDT 2003

I thought I'd mention a couple of points about the connection between
paradoxes and propositional calculus.

I.   Curry paradox and substructural logic.

    A.   Sentence A implies that all donkeys are men.

This, naively treated, leads to a proof that all donkeys are men as

    1.  Suppose sentence A is the case
    2.  if so, it is the case that sentence A implies that all donkeys
are men.
    3.  If A is the case and A implies that all donkeys are men, then
all donkeys are men.   Thus if our supposition (1) is right, all donkeys
are men.
    4.  Our supposition (1) implies that all donkeys are men.
    5.  Our supposition (1) was that sentence A is the case; thus,
sentence A implies that all donkeys are men.
    6.  That sentence A implies that all donkeys are men, is just what
sentence A says, so sentence A is correct.
    7.   All donkeys are men.  From steps 5 and 6 by modus ponens.

More formally, assume there an A such that A = (A => Men(Donkeys))
   1.   A => A  ; [I forget what this rule is called]
   2.   A => ( A => Men(Donkeys) ) ; from 1 by substitution of
   3.   A => Men(Donkeys) ; from 2 by rule of contraction
   4.   A ; from 3 by substitution of identities
   5.   Men(Donkeys) ; from 3 and 4 by Modus Ponens

This paradox is sometimes called Curry's, although "Curry's paradox" is
more properly used, I would say, for something of which this is only one

The paradox has led some to consider logics without the rule of
contraction.   These are called "Linear logics."   A good reference is
Greg Restall - his book "Substructural Logics" and a paper "Costing
non-classical solutions".

    B.   Sentence B is not true.
    C.   Sentence E is true.
    D.   Sentence C is not true.
    E.    Sentence C is not true.

If someone takes C and E to be of the same status as a Liar sentence
(such as sentence B), and if she considers that sentences with Liar
status are not true, she would say of sentence C, that it is not true.
And if this person regards her own conclusion about C to be true, she
should regard D to be true.    Now consider:

    F.  Sentence J is true.
    G.  Sentence F is not true.
    H.  If two is less than three, then sentence F is not true.
    I.   Two is less than three.
    J.   Sentence F is not true.

The person who concluded that D was true, should on the same grounds
conclude that G is true, and therefore that H is.   I is true of
course.   But this person does not accept that J is true, even though J
follows from H and I by Modus Ponens.

If even Modus Ponens is not truth-preserving, then we can't hope to deal
with these paradoxes by giving up the rules of propositional calculus
one after another.   Our only hope (if we do take the line that C and E
are Liar-like, and that Liar-like sentences are not true) is to continue
to regard Modus Ponens and all the other rules as "valid", but to change
the notion of "valid rule" enough so that when a Liar-like sentence
follows from true premises by a "valid" rule, we don't therefore say
that the Liar-like conclusion is true.

If we are willing to accept this altered notion of validity, then we can
keep strong rules and call them "valid."    We don't have to weaken the
rules if we are already weakening the notion of validity.

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

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