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I am having a little trouble sorting out two definitions from the first chapter in my logic textbook, The Logic Book by Bergmann, Moor and Nelson. I am under the impression that a set in a sentence can be logically consistent without entailing anything (eg. {Sara is right-handed, Mark is right-handed, Jean is right-handed}). Are there no actual arguments within logically consistent and logically entailed sentences? What are the differences and relation between the two?

I know that logical entailment is similar to logical validity. It seems the main difference is that logical entailment is more general than validity and the sentence can be entailed by an empty set (while logical validity must include an argument). Please correct me if I'm wrong. It's difficult for me to conceptualize and would really appreciate if someone could help simplify, thank you!

From the comments:

Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true.

Logical entailment: A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence to be false.

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  • Which textbook are you using? Could you quote the definitions from the textbook? This may help someone clarify the difference. Welcome! – Frank Hubeny Jul 12 at 16:20
  • You should point out that you are specifically learning about Mathematical logic. There are other logic types. Don't assume logic is logic. What works in one logic may not work in another logic. You mention sets and sentences which implies math by your terminology used. Consistency expresses something that will always remain true if you manipulate the original proposition (not a sentence). So the proposition no s is a p is consistent to the proposition all s is non-p. Both have the same truth value. All s is p is NOT consistent with all p is s. One will be true as the second is false. – Logikal Jul 12 at 16:57
  • Thank you Frank. I am using the 6th edition of The Logic Book by Bergmann, Moor and Nelson. Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true. Logical entailment: A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence to be false. It's difficult for me to grasp except that a logically consistent sentence must not contradict itself. Logical entailment is supposed to be similar to validity more than consistency. – Toni Jul 12 at 16:58
  • Thank you Logikal, this is just a very basic introductory text starting with deductive logic. I will try to add that to the description. – Toni Jul 12 at 17:00
  • It is also helpful if you state if the text is a Mathematical text or something else. All subjects have deductive reasoning as part of the topic. I know of no academic discipline where deductive reasoning can't apply. Some of the same terms in a so called logic class can be used in a different context. For example the term contraposition differs in math than logic taught on philosophy. You should not assume the terminology is universal everywhere or even so in reality. So some subjects have a different take on how they use deductive reasoning for the particular topic. – Logikal Jul 12 at 21:21
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They are two different notions that are strictly related :

if a set Γ of sentences and a sentence A are not consistent, then Γ logically implies (or entails) ¬A.

Consistency is a property of a set of sentences [see The Logic Book, page 92].

It can be defined either sintactically (a set Γ of sentences is inconsistent iff we can derive a contradiction from it), or semantically [see page 93] :

a set Γ is consistent iff it is satisfiable, i.e. iff there is at least truth-value assignment on which all the members of the set are true.

Logical consequence (or logical entailment) is a relation defined between a set Γ of sentences and a single sentence A [page 95] : the relation holds when there is no truth-value assignment on which all elements of Γ are true and A is false.

The proof is quite simple : if Γ, A is inconsistent, it is unsatisfiable, i.e. there is no truth-value assignment that satisfies simultaneously all elements of Γ and A.

But this means that, every ruth-value assignment that satisfies Γ will satisfies the negation of A.

Example : a very simple example is the following : { P, P → ¬Q } as Γ and Q as A.

We have that P, P → ¬Q and Q is an inconsistent set of formulas and thus { P, P → ¬Q } entails ¬Q.

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  • Thank you so much! I am going to try to add this to my notes so that I can reference back to it. Just have to reread it a couple times so I can wrap my head around it. Can you provide a brief example of a set Γ of formulas and formula A when it is consistent and when it is not, and then becomes entailed? That would be really helpful, thank you! Very interesting proof as well. – Toni Jul 12 at 17:27
  • Thank you so much Mauro, I am going to write this all down because I think even if I am not yet at the point where I understand this perfectly, I hope to get there! I think this proof itself is very cool and would love to study this more, hope the textbook will get to this later on. – Toni Jul 12 at 21:56
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Logical entailment means that every truth assignment which satisfies statement ɸ also satisfies statement ψ. If we say "All English people drive on the left side of the road," then the statement "someone is English" logically entails "drives on the left." Note that other truth conditions might satisfy "drives on the left" (Scots and Welsh do it as well).

Logical consistency merely means that there exists at least one truth assignment that satisfies all of the statements. The statements "someone is English or drives on the right" and "someone is not English or drives on the left" are logically consistent with each other, because there is at least one truth condition (e.g. someone is English and drives on the left) which satisfies both.

Keep in mind that entailment is directional — the truth assignments that satisfy one statement are a subset of the assignments that satisfy the other — while consistency is a simple relation. You can think of it (roughly) like the difference between causation and correlation, though please don't extend that analogy too far.

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  • Thank you Ted, that is very helpful! I am wondering now the difference between logical consistency and logical equivalency based on your description, which was defined in my textbook just before consistency. It is stated as "sentences p and q are logically equivalent iff it is not possible for one of these sentences to be true while the other sentence is false." I thought of this as restating the statements in two separate sentences, is this right? (e.g. "Jake loves Henry" and "Henry is loved by Jake" are logically equivalent sentences). – Toni Jul 12 at 17:30
  • Also, is it still logically consistent if the second statement "someone is not English or drives on the left" could be contradicted by your previous example that Scots and Welsh do it as well? I am not sure about where truth-values stand except that it must be possible for all members of the set to be true. Is it more based on the relation between them? Sorry if I have confused this. – Toni Jul 12 at 17:32
  • Logical equivalency is merely bi-directional logical entailment: the truth assignments that satisfy one statement are exactly the same as those that satisfy the other. Per the second point, the truth of "someone is not English or drives on the left" is not contradicted by the fact that Welsh people drive on the left. An 'or' statement only needs one term to be true to evaluate to true. That statement would only be false if 'not-English' is false and 'drives on left' is false, i.e. that every English person drives on the right. – Ted Wrigley Jul 12 at 18:09
  • Bi-directional entailment, that is a really good way of looking at it. Thank you very much, for helping with the second point as well. – Toni Jul 12 at 18:41
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Consistency is a relation defined between any two sentences (or statements, propositions, formulas etc.).

Two sentences are consistent if they are not contradictory (to each other). Two sentences are contradictory if any of the two implies (entails etc.) the negation of the other.

This is extended to any set of more than two sentences as follows: A set of sentences is inconsistent if any two sentences of the set are inconsistent with each other.

A set of sentences may be consistent without any of them implying any of the other. You can also have a set of sentences which is consistent and some or all of the sentences imply some or all of the other sentences.

The relation, therefore, between consistency of a set of sentences and implication (entailment) is only that if any sentence of the set implies the negation of any of the other sentences of the set, then the set is inconsistent (and the transposition of that).

Thus, the fact that any two sentences of any set are inconsistent entails that the set itself is inconsistent.

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  • Thanks! I think this is very much in line with the explanations given by the text and seem like a wonderful extension of it. – Toni Jul 12 at 17:50
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Here are the definitions that require clarification:

Logical consistency: A set of sentences is logically consistent if and only if it is possible for all the members of that set to be true.

Logical entailment: A set of sentences logically entails a sentence if and only if it is impossible for the members of the set to be true and that sentence to be false.

Consider logical consistency first using two sentences symbolized as P and P V Q where "V" means "or". If we allow P and Q to range over the values true and false is it possible for some valuation to make both of the sentences in the set true? We can find that out by putting the sentences in a truth table generator joining them with "and" represented by "&":

enter image description here

Note that there are two different valuations of P and Q that will make both of the sentences true. By the definition we only need one valuation so we have more than we need and the set of sentences are consistent.

This set of sentences, by itself, is not an argument. An argument has a set of sentences, called premises, and a sentence called a conclusion. Suppose we make the argument that the set of sentences P and P V Q entail the sentence Q.

Without considering differences between entailment and validity, we could put that in a truth table generator to see what would happen:

enter image description here

Now consider the definition. There are two ways for the set of sentences to be true. We only needed to have one of them for consistency, but we need to consider both for entailment. In both of those cases the conclusion must not be false. However, it is false when P is true and Q is false. So logical entailment fails.

Consider the final question:

I know that logical entailment is similar to logical validity. It seems the main difference is that logical entailment is more general than validity and the sentence can be entailed by an empty set (while logical validity must include an argument).

Without knowing precisely how the textbook distinguishes between logical entailment and logical validity, it is possible to entail a conclusion from no premises at all. Such a conclusion would have to always be true. Since the conclusion is never false, it is impossible to have a set (even an empty set) of true premises and a false conclusion.

An example would be P V ~P. This truth table shows that this sentence is always true:

enter image description here

Such sentences are called tautologies. They are always true.

Michael Rieppel. Truth Table Generator. Retrieved on July 12, 2019 at https://mrieppel.net/prog/truthtable.html

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