# Inconsistency In Sentence Logic and in Predicate Logic

• Bacle
In summary: B\wedge\neg B)}|-\neg AIf you do that with natural deduction, then you are incorrectly assuming that A is a premise of the argument, rather than some hypothetical premise.In summary, the conversation discusses the different definitions of inconsistency in Sentence Logic and Predicate Logic. In Sentence Logic, a sentence is contradictory if it can be derived using theorems of truth-functional logic. In Predicate Logic, contradiction is defined both syntactically and semantically. The conversation also touches on the concepts of consistency, completeness, and validity in logic. The use of hypothetical premises and proofs by contradiction is also mentioned.
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.

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.

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:
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.

Stephen Tashi said:
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"?

"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.

Stephen Tashi said:
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.

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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.

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## 1. What is inconsistency in sentence logic and in predicate logic?

Inconsistency in sentence logic and in predicate logic refers to a situation where a set of sentences or propositions leads to a contradiction. This means that the set of sentences cannot all be true at the same time.

## 2. What causes inconsistency in sentence logic and in predicate logic?

Inconsistency can occur in sentence logic and predicate logic due to errors in the construction of the sentences or propositions, or due to conflicting information within the set of sentences. It can also be caused by the use of invalid logical rules or principles.

## 3. How is inconsistency detected in sentence logic and in predicate logic?

Inconsistency can be detected through the process of logical inference, where the validity of statements or propositions is evaluated based on the rules and principles of logic. Inconsistency is detected when a contradiction is reached, meaning that the set of sentences cannot all be true at the same time.

## 4. What are the consequences of inconsistency in sentence logic and in predicate logic?

Inconsistency can have significant consequences in logic and reasoning, as it undermines the validity and soundness of arguments. It can also lead to paradoxes, where seemingly true statements lead to contradictory conclusions. Inconsistency also hinders the ability to make reliable and accurate deductions or conclusions from a set of premises.

## 5. How can inconsistency be resolved in sentence logic and in predicate logic?

In order to resolve inconsistency, the conflicting sentences or propositions must be revised or rejected. This can be achieved by identifying the source of the inconsistency and correcting any errors in the sentences or propositions. In some cases, it may also be necessary to revise the logical rules or principles used in constructing the sentences. Inconsistency can also be avoided by carefully constructing sentences and ensuring that they do not lead to contradictions.

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