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My textbook says:

A deductive theory is said to be consistent if no two asserted statements of this theory contradict each other, or, in other words, if of any two contradictory sentences at least one cannot be proved.

The "in other words" is what confuses me, how can "no two asserted statements of this theory contradict each other" and "if of any two contradictory sentences at least one cannot be proved" mean the same thing? I've been stuck on this all day can you guys help me please? and please don't give me examples that are too complicated I just started studying logic

Robert
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2 Answers2

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Consider the Peano arithmetic as deductive theory $P$, and the statement of the Goldbach's conjecture:

$A=$ Every even integer greater than $2$ can be expressed as the sum of two primes.

or its negation

$\neg A=$ There is even integer greater than $2$ that cannot be expressed as the sum of two primes.

Today we don't know if one of these two statement can be proved in P. Since P is consistent, than maybe that, in future, someone can prove $A$ , or can prove $\neg A$ but it is not possible to prove all the two.

Maybe also that, in fufutre, someone will prove that no one of the two can be proved in $P$ and is is not contraddictory with the consistency of $P$.

Emilio Novati
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The confusion may be just the same as in your question about completeness:

The i.o.w. part of the definition has as its antecedent all the contradictory sentences that can possibly be built, not the ones that are already provable:

Definition #2:
[...] if of any two grammatical sentences that contradict each other, at least one is not provable

or equivalently.

Definition #2:
[...] if of any two sentences in the language that contradict each other, at least one is not in the theory

Reformulating the first variant of the definition,

Definition #1:
[...] if no two asserted statements of this theory contradict each other
$\Leftrightarrow$ if for any two sentences such that both are provable in the theory, they do not contradict each other

it becomes obvious that the one is just the contaposition of the other:

$P = $ both sentences are provable in the theory,
$\neg P = $ at least one sentence is not provable in the theory
$Q = $ the sentences do not contradict each other,
$\neg Q = $ the sentences contradict each other

Definition #1 = $P \to Q$
Definition #2 = $\neg Q \to \neg P$

Since contrapositive statements are logically equivalent, the two formulations mean the same thing.

But you are probably overthinking matters; the inutition behind consistency is quite simple: It just means that the theory doesn't assert anything contradictory, i.e. it never proves $\phi$ and $\neg \phi$ at the same time.

Natalie Clarius
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  • in "no two asserted statements of this theory contradict each other", "asserted statements of this theory" means all the sentences that one can build in the language or the ones that one "said" already? – Robert Feb 15 '20 at 15:11
  • It means the ones that are provable in the theory. – Natalie Clarius Feb 15 '20 at 15:13
  • so "all the statements in the language which are provable"? – Robert Feb 15 '20 at 15:14
  • Yes. I.o.w., all the statements in the language which are also in the theory. – Natalie Clarius Feb 15 '20 at 15:17
  • Yeah thank you so much I understand now! But how did you understand that "asserted statements of this theory" means "all the statements in the language which are provable"? they sound very differently.. especially because "asserted" means "assert implies stating confidently without need for proof or regard for evidence." (merriam webster) – Robert Feb 15 '20 at 15:31
  • Well, if it says "statements of the theory", then it's about the statements in the theory, not just those in the language. – Natalie Clarius Feb 15 '20 at 17:33