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.
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:
Here are a few important points about consistency:
Notice that there is a difference between making self-defeating
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:
2. A webpage shown to a user opting out from a mailing list:
Everything we hear is an opinion, not a fact. Everything we see is a perspective, not the truth.
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?
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.
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.
What do these statements entail which they do not entail on their own?
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".