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CMPSCI 601: Recall From Last Time Lecture 6
Boolean Syntax:
Boolean variables:
A boolean variable represents an atomic statement that may be either true or false. There may be infinitely many of these available.
Boolean expressions:
atomic: , (“top”), (“bottom”)
, , ! , " # , $ , for
Boolean expressions
Note that any particular expression is a finite string, and thus may use only finitely many variables.
A literal is an atomic expression or its negation:
As you may know, the choice of operators is somewhat arbitary as long as we have a complete set , one that suf- fices to simulate all boolean functions. On HW#1 we argued that
is already a complete set.
CMPSCI 601: Boolean Logic: Semantics Lecture 6
A boolean expression has a meaning, a truth value of true or false, once we know the truth values of all the individual variables.
A truth assignment is a function
^
true
false
, where
is the set of all variables. An as-
signment is appropriate to an expression if it assigns a
value to all variables used in .
The double-turnstile symbol
(read as “models”) de- notes the relationship between a truth assignment and an
expression. The statement “
” (read as “ models
”) simply says “ is true under ”.
Definition 6.1 A boolean expression is satisfiable iff
there exists
is valid iff for all appropriate to ,
VALID SAT UNSAT
Proposition 6.2 For any boolean expression ,
UNSAT VALID
UNSAT VALID VALID UNSAT
Proposition 6.3 is unsatisfiable iff .
is satisfiable iff .
is valid iff .
CMPSCI 601: The Fitch Proof System Lecture 6
A Fitch proof is a sequence of expressions, each one of which is justified in terms of previous ones. There are twelve proof rules that tell us when a statement is justi- fied.
Fitch has no axioms (statements assumed to be true with- out proof) but we typically start with some premises and reach a conclusion that follows from those premises.
If from a set of premises we can derive , we write
, read as “ proves ”. This single turnstile symbol
is not to be confused with the double turnstile symbol
CMPSCI 601: Soundness of Prop. Fitch Lecture 6
Now that we’ve defined the propositional subset of Fitch,
, we consider some important properties of a proof
system:
Soundness: If a statement can be proved from a set of
statements , then is a tautological consequence of
. (If , then
Completeness: If is a tautological consequence of ,
then can be proved from . (If
, then .)
Compactness: If every finite subset of can be satisified
by some assignment, then there exists an assignment
satisfying all of .
The intuition behind soundness is clear: each of the rules corresponds correctly to the meaning of the symbols, so none of them should be able to prove false things from true premises. Our proof will follow this intuition.
Let’s restate compactness in a form that’s easier to prove by induction:
Theorem 6.4 Let be a statement in a proof and let
^ be the premises in force when occurs. Then
any truth assignment that makes each
true also makes
true.
Proof: [BE] says “if the conclusion ever fails for any step in any proof, consider the first step on which it fails”. I would be more inclined to view proofs as being induc- tively constructed by adding steps. In either case, it suf- fices to prove the conclusion in the case in which every
statement of before is a tautological consequence of
its premises.
For to occur in the proof, it must have been produced
by some rule. We thus are reduced to twelve cases, one for each of the rules. We’ll do a few of these, and do a few more in the exercises.
# Intro:
Here is of the form # and there is a subproof,
within the scope of , of from . The premises in
force during that subproof were a subset of the premises
in force for .
So in any truth assignment that makes all of ’s premises
true, the premises for the subproof are also true. Since the inductive hypothesis holds for , any assignment that
also makes true must make true.
Thus any such assignment either makes false or makes
true, and by the definition of truth for # either of these
makes # , and this statement is .
Elim:
Here is an arbitrary sentence, and the statement is in
its scope. Since the inductive hypothesis holds for , we
know that any assignment making the premises of true
makes itself true. There can thus be no such assign-
ment. But this means no assignment could make all the
premises of true, since the premises of are a subset
of these. The conclusion for is thus vacuously satisfied.
Remaining Steps: There are eight of these, with proofs similar to the above four cases. You’ll be assigned to write out some of these on HW#2.
This proof is emotionally unsatisfying in the way that many proofs in mathematical logic are unsatisfying. We seem to be proving that a step is valid if and only if it’s valid, and not really saying anything at all.
The key point are that truth and provability are two differ- ent properties, even though for sound and complete proof systems they hold for exactly the same sentences. The reason that this system is sound and complete is that the proof steps correspond properly to the definitions of truth for each operator, and vice versa. We shouldn’t be confi- dent that this correspondence holds until we’ve checked it in each case.
Our proof will require one more definition. A set of sen-
tences is called formally complete if for every sen-
tence , either or . (Note that we are
not saying that “formal completeness” has anything to do with “completeness” – one is a property of sets of statements and the other a property of the whole proof system.)
We will show:
Every formally consistent, formally complete set of
sentences is tt-satisfiable.
Every formally consistent set of sentences can be ex-
tended to a formally complete set while remaining formally consistent.
Since a truth assignment satisfying an extension of also
satisfies itself, these two statements imply that any for-
mally consistent set is tt-satisfiable, and thus imply the completeness of propositional Fitch.
Let’s first show that a formally consistent, formally com-
plete set of sentences is satisfiable. Suppose that is
such a set. Let
be any of the variables of the sys- tem. Since “
” is a sentence, we know that either
or
. Could both be true? No, because then by
writing both these proofs and doing one -Intro step we
could prove from .
This tells us how to define our truth assignment! We set each
to be true if
and false otherwise, giving us
a mapping from the set of variables to the set of truth
values. No truth assignment other than could make
all of true, but we still have to show that itself does
makes true.
So we have our result for formally complete and formally
consistent sets. Now we must show that if is formally
consistent we can extend it (by adding sentences to it) to make it formally complete while keeping it formally consistent.
We consider the atomic variables
in order, a potentially
infinite process. At stage of our process we look at
and
and ask whether either is provable from as
extended so far. If one of them is already provable we do nothing. But if neither is provable, we add
(an
arbitrary choice) to and go on.
At the end of this process, has become formally com-
plete, because we just proved that a system that proves each variable or its negation proves each sentence or its negation. Could it fail to be formally consistent?
The only way we could have destroyed the formal con- sistency property would be when we added a variable
to . But we only did this if the former could not prove
. If there were a proof of from and
, we could adapt this proof to prove
from alone by -Elim –
just take
as the premise in the presence of and derive
We are done! The “infinite process” might worry you a bit, and perhaps it should. We can’t carry out this pro- cess with any kind of finite algorithm. But all we want
to argue is that the final exists , and it does – for any
individual
, it is well-defined whether it is in the final
or not.
To review one more time:
If is formally consistent we extend it to be formally
complete, and
we then know that it settles all sentences consistently
with the assignment it places on the variables,
so the original is tt-satisfiable, by this assignment.
Note that compactness is a semantic rather than a proof- theoretic property. Its definition for propositional logic depended on tt-satisfiability, which depends on the notion of a truth assignment being a model for a set of sentences.
More generally, a semantics for a logical system is a way of defining models and whether a given model satisfies a given set of sentences. The general compactness prop- erty of such a system can be stated: “If every finite subset of has a model, then has a model.”
Our proof of compactness for propositional logic used no specific properties of propositional logic, only the fact that it has a proof system that is sound and complete, and in which every proof is finite.
CMPSCI 601: Other Proof Systems Lecture 6
Propositional Fitch is not the only sound and complete proof system for propositional logic. [BE] introduces an- other system called resolution in section 17.4, and there are many others.
The relative power of such proof systems is an active area of study in computer science and mathematical logic. Qualitatively , of course, any two sound and complete systems have the same power, because they can each prove the true tautological consequences and no others.
But quantitatively there are differences among the sys- tems, some proved and some conjectured. For example, there are families of tautologies that have polynomial- size proofs in propositional Fitch but require exponential- size resolution proofs. We won’t prove this in this course, but we’ll later talk about how such results relate to com- putational complexity.
