Inconsistency In Sentence Logic and in Predicate Logic

[Bacle]

Does anyone know/understand the different definitions for Inconsistency
in Sentence Logic and in Predicate Logic?

I know in Sentence Logic, that a sentence ( a Wff, actually) S is
contradictory, if from S we can derive (using theorems of
truth-functional logic ) a sentence of the type A&~A , where '&'
is 'and' and '~' stands for negation, i.e., we assume S, and, using
theorems, we can conclude, using MP, that S->(A&~A).

How do we define contradiction in Predicate Logic, tho? Is it
defined both syntactically and semantically, i.e., do we say
S|- (B&~B) and S|=(B&~B), i.e., we can both derive syntactically
(i.e., have a proof of) B&~B from S, and have a model for S in which
B&~B is true?

Thanks.

 

[Stephen Tashi]

The terms you are using ("sentence logic", "predicate logic") may not be standardized precisely enough for your question to have a definite answer. When I think of "sentence logic" I think of symbolic logic that uses letters to represent entire sentences, like T = Tom is happy. When I think of "predicate logic", I think of symbolic logic that breaks sentences down into finer parts like properties and variables, such as h(x)= x is happy and H(x) = the set of x such that h(x).

My simplistic thought about this is that there are textbooks that treat predicate logic but never discuss model theory, so when those texts talk about "contradiction" they apparently aren't referring to semantics based on a model.

This brings up an interesting point. if we define a system of logic that deals with things called "statements" and we assert that a statement cannot be both true and false, then we must admit that the result of a "law" (like modus ponens etc.) is not always a statement since proofs by contradiction must produce results like "B and not-B". I wonder how well this technicality is observed in textbooks on elementary logic.

 

[Bacle]

I found some info., in case someone else is interested. From B&B's Dictionary of Math:
(paraphrase; words with _ . _ are also defined therein.)
Predicate calculus: the system of symbolic logic concerned not only to represent
the logical relations between sentences or propositions as wholes, but also to
consider their internal structure, in terms of subject and _predicate_. The primitive terms are individual names, predicates and variables... if quantification is restricted to
individuals, it is 1st order, then it is _consistent_, _complete_, but not _decidable_.


Consistent:
adj.
1) (Of a set of statements): when all statements can be true under the same _interpretation_

2)(Of a formal system): not allowing the deduction of a contradiction from the
axioms; more generally, not having an atomic sentence as a theorem.

Completeness:(of a logical theory)
having the property that every semantically valid formula can be proved syntactically from the axioms.

Valid. ( of a sentence in a formal language): true in every interpretation.; satisfied by every assignment of values to the variables in the interpretation, so that every interpretation is a model for the statement.

 

[sigurdW]

This brings up an interesting point. if we define a system of logic that deals with things called "statements" and we assert that a statement cannot be both true and false, then we must admit that the result of a "law" (like modus ponens etc.) is not always a statement since proofs by contradiction must produce results like "B and not-B". I wonder how well this technicality is observed in textbooks on elementary logic.

How do you mean: is not "B and not-B" = "B and B"? since "-B" = "not B"?

 

[Bacle2]

"This brings up an interesting point. if we define a system of logic that deals with things called "statements" and we assert that a statement cannot be both true and false, then we must admit that the result of a "law" (like modus ponens etc.) is not always a statement since proofs by contradiction must produce results like "B and not-B". I wonder how well this technicality is observed in textbooks on elementary logic. "

Sorry, I missed your reply from a while back. I think this issue can be resolved by

making a distinction between sentences (which are the atoms, and whose internal structure,

as you rightly pointed out, does not matter to the effects of truth-functionality) and wff's,

i.e., well-formed-formulas in the truth-functional calculus. Basically we form wff's by

using sentences as building blocks, and then using logical connectives and, or, etc. to

form wff's. Then B&~B is a wff, and the standard format for a contradiction.

 

[mbs]

This brings up an interesting point. if we define a system of logic that deals with things called "statements" and we assert that a statement cannot be both true and false, then we must admit that the result of a "law" (like modus ponens etc.) is not always a statement since proofs by contradiction must produce results like "B and not-B". I wonder how well this technicality is observed in textbooks on elementary logic.

Proof by contradiction in the natural deduction system does not involve a direct derivation of $B\wedge\neg B$. The rule is...

|-$\underline{A\Rightarrow(B\wedge\neg B)}$
|-$\neg A$

The "contradiction" in the premise is hypothetical since it is to the right of an implication connective. Therefore the premise as a whole is not contradictory.

If $A$ is "the moon is made of blue cheese" and $B$ is "1=0" then...

|-$A\Rightarrow B$

is not necessarily a contradictory statement. Only if I can prove "the moon is made of blue cheese" can I prove a contradiction (through modus ponens) and show that my axioms must be inconsistent.

 

[mbs]

The thing is in informal proofs it is customary to not carry the hypothetical A through multiple lines of reasoning as that would be tedious to write. Instead you just say something like...

"Assume $A$"...

proceed to derive both $B$ and $\neg B$...

and finally conclude $\neg A$.

The informal proof can all be converted into a tree of explicitly formal deduction rules, but it can be a rather tedious process.