Re: Godel's comments about the "true reason" for incompleteness

Newberry <newberryxy@xxxxxxxxx> writes:

On Mar 15, 4:37 am, "Jesse F. Hughes" <je...@xxxxxxxxxxxxx> wrote:

Out of curiosity, suppose that I tell you (A v ~A) -> B is true.  Now,
we would naturally conclude B, since A v ~A is the antecedent and we
all know that A v ~A is true.

Let me paraphrase it step by step:
1) I tell you that (A v ~A) -> B is true
2) (A v ~A) -> B is true
3) B is true

The error is in going from 1) to 2). Just because you think that
something is true it does not follow that it actually is. (A v ~A) ->
B is neither true nor false. BTW, modus ponens cannot be used in a
natural deduction system for t-relevant logic.

Fascinating!

Suppose that my lawn sprinkler is watering my lawn. It seems to be
that the following statement is true:

"My lawn is wet whether it is raining or not." (*)

It seems to me natural to symbolize this statement as:

(Raining v ~Raining) -> WetLawn.

It seems to me that from that claim, I should conclude that the lawn
is wet.

What does *not* seem to be true is your interpretation: that the above
sentence (*) is literally meaningless (right?).

Let me just make sure that I've got you right: If I told you (*) is
true and then asked whether my lawn was wet, would you hesitate to
answer yes? Would you balk and suggest that all I did was utter some
meaningless nonsense which is necessarily irrelevant to making
inferences about the state of my lawn?

(The one out I see is that you might balk at my translation, but it
still seems reasonable to me. The only qualm I have is the
translation of "whether" as "if", but I can see no other obvious
alternative.)

One last question, just to be sure I follow you. In classical logic,

(A v B) -> C

is equivalent to

(A -> C) & (B -> C).

I suppose this is true for your logic as well, for some A and B (for
atomic propositions, for instance), but what about

(A v ~A) -> C ?

This is meaningless (or neither true nor false), if I understand
correctly. Should I also conclude that

(A -> C) & (~A -> C)

similarly lacks a truth value?

What about a slightly different case? Suppose that we're considering

(A v B) -> C (**)

where, say, A stands for "it is raining", B stands for "the sprinklers
are on" and C stands for "the lawn is wet." This statement may be
true. Now suppose I tell you, in addition, that the sprinklers are
hooked up to a rain detector so that they come on whenever it is not
raining. In other words, independently of (**), it so happens that

~A -> B (***)

is also true. Would you say that once you learn (***), you would have
to revise your opinion about (**). The sentence (**) used to be true,
but now it's meaningless and so offers no support for the conclusion
C.

Is that right?

--
Jesse F. Hughes

"Yesterday was Judgment Day. How'd you do?"
-- The Flatlanders
.

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