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May
19
answered Logic: Can you drop parentheses in a conjunction?
May
12
comment Distance between theorems
@Peter You might want to see Robert Veroff's "Finding Shortest Proofs: An Application of Linked Inference Rules" which you can read in Postscript format via his website: cs.unm.edu/~veroff
May
12
answered Distance between theorems
May
10
comment Complete operator base logic
@learner Sorry, but no. I feel like I'm treading a thin line by posting this in the first place, because this seems suspiciously like homework. I'd suggest this short primer on Polish notation: plato.stanford.edu/entries/lukasiewicz/polish-notation.html Take the formation rules in Polish notation and build them up from their variables and connectives. Translate as you go along and I suspect you will manage to do so for yourself.
May
8
comment What is the set of propositional formulas?
What are the formation rules you have for propositional formulas?
May
8
comment What is the set of propositional formulas?
@BrunoBentzen " In fact it's not unique: the problem is because there are infinitely many sets of strings that satisfy the formation rules - rules such as (1), (2) and (3) in my answer below - but still contain some "garbage" like '→¬' or 'P¬' as elements of it." Alright, I see that given the way you've defined things. However, the way I usually understand things is more like "1. All lower case letters of the alphabet with or without subscripts are variables". And then we have rules for the connectives.
May
8
answered Complete operator base logic
May
8
comment What is the set of propositional formulas?
@BrunoBentzen Why are you emphasizing smallest? The set of all strings which satisfy the formation rules seems unique to me.
May
7
comment What is the set of propositional formulas?
The set of all strings which satisfy the formation rules.
Apr
25
comment Propositional Logic: Proof involving only the symbols $\{\rightarrow,F \}$
"D∨H∨Y" From your translation it's clear that you meant ((D∨H)∨Y). You didn't make this clear until you did the translation.
Apr
25
comment Propositional Logic: Proof involving only the symbols $\{\rightarrow,F \}$
"I think they called Lyndon Axioms.." On p. 218 of J. A. Kalman's Automated Reasoning With Otter he indicates that these three axioms appear in a 1939 paper of Wajsberg (the axioms are in Polish notation, but this is not material here).
Apr
19
comment How to demystify the axioms of propositional logic?
I've tried to give an intuitive explanation (in my answer below) as to how CCNpNqCqp tells us everything else about C, and everything about N too.
Apr
19
answered How to demystify the axioms of propositional logic?
Apr
15
comment Can't get Logical Error
(∼p)∨q)∧(∼q)=(∼p) is not a law. Let p=0, and q=1. Then the left hand side evaluates to 0, but the right evaluates to 1. ((∼p)∨q)∧p=q is also not a law. Let q=1, and p=0.
Apr
15
comment If there are Predicates before Predicate Calculus, why is it called such?
Propositional calculus doesn't require a reference to "True" or "False".
Apr
11
comment Why is the negation of $A \Rightarrow B$ not $A \Rightarrow \lnot B$?
"A⇒B says that knowing A alone is enough to know B." Yikes! What if detachment isn't a rule of inference? How does knowing "A" end up as enough to know B in such a case? Also, even if we have detachment as a rule of inference, if Epq is true also, then it seems fair to say that knowing "p" alone is enough to know "q" also. However, Epq is definitely different from Cpq. Alright, I guess if detachment is a rule of inference, and it is not presumed that knowing B is enough to know A, then A $\rightarrow$ B says that knowing A is enough to know B.
Apr
11
comment Why is the negation of $A \Rightarrow B$ not $A \Rightarrow \lnot B$?
"The OP is obviously not using whatever kind of logic it is you mentioned." Syntactically speaking the logic he is using probably is the same as Lukasiewicz 3-valued logic in that the formation rules are the same. You've tried to explain things here semantically, but the issue here is purely syntactical in terms of formation rules (and in truth semantics doesn't always explain much ... not all tautologies have the same power). To get the negation of something, we first have to have the entire wff. Then we negate it. (A$\rightarrow$$\lnot$B) doesn't negate the conditional, it negates "B".
Apr
11
answered Why is the negation of $A \Rightarrow B$ not $A \Rightarrow \lnot B$?
Apr
11
comment Why is the negation of $A \Rightarrow B$ not $A \Rightarrow \lnot B$?
This doesn't explain things. Suppose that (1) is False, as say it is in Lukasiewicz 3-valued logic. It still holds that NCab and CaNb differ.