Statements, Logical Connectives, and Logical Relations

Statements are the fundamental units of arguments and proofs in logic. These tutorials explain how to identify statements and introduce some of the basic ways that statements may be related to one another.

Complete the exercises and check your answers.

Logical relations

§1. Consistency

Suppose S is a set that contains one or more statement. S is consistent when it is logically possible for all of the statments in the set to be true at the same time. Otherwise S is inconsistent. Some examples:

  • Consistent: Peter is three years old. Jane is four years old.
  • Consistent: Peter is three years old. Peter is a fat rabbit.
  • Inconsistent: Peter is three years old. Peter is a fat rabbit. Peter is five years old.
  • Inconsistent: Peter is three years old. It is not the case that Peter is three years old.
  • Inconsistent: Peter is a rabbit. All rabbits are three years old. Peter is one year old.
  • Inconsistent: Peter is a completely white rabbit that is completely black.

Here are a few important points about consistency:

  • If you have two statements that are both true, they are certainly consistent with each other.
  • If you have two statements that are both false, they might or might not be consistent with each other. See if you can give your own examples.
  • In the last example above, we have just one single inconsistent statement. An inconsistent statement must be false. But if you have a set of statements { P, Q, R, S }, the whole set is said to be inconsistent even if R and S are both true, and inconsistency is only due to inconsistency between P and Q.
  • Although statements that are inconsistent with each other cannot all be true at the same time, it might be possible for them to be false at the same time.
  • Every statement is inconsistent with its negation.

Inconsistency and self-defeating statements

Notice that there is a difference between making self-defeating statements and inconsistent statements. Suppose a tourist from a non-English speaking country says: "I cannot speak any English." Since what is being spoken is an English sentence, the tourist is obviously saying something false. However, strictly speaking the sentence is not logically inconsistent because it actually describes a logically possible situation. It is quite possible for the speaker not to be able to speak any English. What is impossible is to say the sentence truly. In these situations, it is more appropriate to say that the utterance is self-defeating rather than inconsistent.

Here are some funny actual examples of self-defeating / inconsistent statements:

1. An error message when installing Microsoft Wireless Optical Desktop for Bluetooth:

Keyboard Error or No Keyboard Present
Press F1 to continue, DEL to enter setup

2. A webpage shown to a user opting out from a mailing list:

Exercise #1

§2. Truth

Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth.
Marcus Aurelius

It is not uncommon for people to make very grand and general claims about truth, only for these claims to turn out to be inconsistbut or self-defeating.

For example, some people might say that nothing is true and it is all a matter of opinion. But if that is the case, then the claim is also not true. In other words, it is not true that nothing is true! So why should we believe it?

Or consider the relativist claim that everything is relative and there is no objective truth. Is the claim itself relative or not? If not, then the claim is false since there is something that is not relative. But if the claim is indeed relative, then why should we accept it as opposed to the opposite claim that not everything is relative?

§3. Entailment

A sentence X entails Y if Y follows logically from X. In other words, if X is true then Y must also be true, e.g. "30 people have died in the riots" entails "more than 20 people died in the riots", but not vice-versa.

  1. If X entails Y and we find out that Y is false, then we should conclude that X is also false. But of course, if X entails Y and we find out that X is false, it does not follow that Y is also false.
  2. If X entails Y but Y does not entail X, then we say that X is a stronger claim than Y (or "Y is weaker than X"). For example, "all birds can fly" is stronger than "most birds can fly", which is still stronger than "some birds can fly".

    A stronger claim is of course more likely to be wrong. To use a typical example, suppose we want to praise X but are not sure whether X is the best or not, we might use the weaker claim "X is one of the best" rather than the stronger "X is the best". So we need not be accused of speaking falsely even if it turns out that X is not the best.

Exercise #2

What do these statements entail which they do not entail on their own?

§4. Logical Equivalence

If we have two statements that entail each other then they are logically equivalent. For example, "everyone is happy" is equivalent to "nobody is not happy", and "the glass is half full" is equivalent to "the glass is half empty".

  • If two statements are logically equivalent, then they must always have the same truth value.

Exercise #3