Understanding "Sentential" Logic: Axioms, Notation & Binary Operations

[StephenPrivitera]

I came across this site: http://mally.stanford.edu/tutorial/sentential.html
It lists four axioms of "sentential" logic. I first would like to know if there are other axioms not listed here. Wouldn't you need some axiom like if P is true, then ~P is false? It seems difficult to prove the law of disjunctive inference with these axioms. Next, I need some help understanding the notation.

One axiom is:
P -> (Q -> P)
Can this be interpreted as

given P
therefore Q -> P

for any statement Q?

In other words, if I precede the antecedent with the word "given" and exchange the main operator (which is always the conditional in these cases) with the word "therefore", is the meaning of the statement the same? This issue is important to me as a matter of proving theorems.


Also, one source refers to "->", "V", "and", and "<->" as binary operations. I just recently learned about binary operations and abstract systems. Does this mean that we can study argument forms as an abstract system? Can we prove the associativity/commutivity/distributivity properties of the operations using the above axioms?
For a binary operation aOb=c
How does this relate to P->Q?
IOW P->Q=what? Is it merely the joining of the statements by the words "if" and "then"?

 

[Hurkyl]

Wouldn't you need some axiom like if P is true, then ~P is false?

Slow down, who said anything about "true" and "false"?

Note that there is nothing on the page that says anything about true and false! It just discusses rules of deduction.


You can, however, deduce things like

&phi; <=> ~~&phi;

from the given axioms

 

[StephenPrivitera]

Originally posted by Hurkyl
Slow down, who said anything about "true" and "false"?

Note that there is nothing on the page that says anything about true and false! It just discusses rules of deduction.

But why not? That makes no sense to me. Isn't the purpose of logic to study the validity of arguments? When you say you can deduce P<=>~~P from the axioms, aren't you implicitly saying that you can deduce that P<=>~~P is true?

 

[Hurkyl]

Isn't the purpose of logic to study the validity of arguments?

Yes... but "true" and "false" are not the fundamental concepts to the validity of an argument, deducibility is! Intuit, "true" merely means "this statement is an axiom or is deducible from the axioms", while "false" means "the negation of this statement is true".


More precisely, we can define a "truth assignment" to be a logical function v whose domain is all atomic formulas and whose range is {true, false} satisfying:

(v(~A) = T) <=> (v(A) = F)
(v(A=>B) = F) <=> (V(A) = T and V(B) = F)

(depending on your rules of inference, you may or may not need to add or change a couple requirements on the definition of a truth assignment)

And for any truth assignment v, we can deduce
v(P<=>~~P) = T
Or more trivially we can deduce v(P)=v(~~P)


But the point is that "true" and "false" are not the fundamental concepts of logic.