Statements, Logical Connectives, and Logical Relations

Statements

In logic we often talk about the logical properties of statements and how one statement is related to another. So what is a statement?

There are three main sentence types in English:

• Declarative sentences are used for assertions, e.g. "He is here."
• Interrogative sentences are used to ask questions, e.g. "Is he here?"
• Imperative sentences are used for making requests or issuing commands, e.g. "Come here!"

For present purposes, we shall take a statement to be any declarative sentence. A declarative sentence is a complete and grammatical sentence that makes a claim.

So here are some examples of statements in English:

• Snow is white.
• The moon is made of green cheese.
• Everyone is here.
• Whatever will be, will be.
• The data and information provided on this web page is for informational purposes only, and is not intended for trading or commercial purposes, unless written prior permission is obtained by the user from the author, though the author will not be liable for any errors or delays in the content, or for any actions taken in reliance thereon.

As you can see, statements can be true or false, and they can be simple or complex. But they must be grammatical and complete sentences. So these are not statements :

• The United Nations [ A proper name, but not a sentence ]
• Bridge over troubled waters. [ Not a complete sentence ]
• Come here right now! [ A command that is not a complete sentence making a claim ]
• Will you be available on Tuesday or Wednesday? [ A question ]
• HJGAS&*^@#JHGKJAS*&^*!@GJHGAA*&S [ Ungrammatical ]

There is an easy test to decide whether something is a statement in English. Suppose you have a sentence φ and you add "it is true that" to the front. If the resulting expression is grammatical, then φ is a statement. Otherwise it is not.

So for example, φ might be "bridge over troubled waters". We append "it is true that" to the front, and end up with "it is true that bridge over troubled waters." But this expression is not grammatical. So "bridge over troubled waters" is not a statement. However, "I am like a bridge over troubled waters" is a statement, because "it is true that I am like a bridge over troubled waters" is grammatical.

Exercise #1

Logical connectives

Here are a few basic concepts in logic that you ought to be familar with, whether you are studying symbolic logic or not.

§1. Negation

The negation of a statement α is a statement whose truth-value is necessarily opposite to that of α. So for example, for any English sentence α, you can form its negation by appending "it is not the case that" to α to form the longer statement "it is not the case that α".

In formal logic, the negation of α can be written as "not-α", "~α" or "¬α".

Here are some concrete examples:

There are two points about negation which should be obvious to you:

• A statement and its negation can never be true together. They are logically inconsistent with each other.
• A statement and its negation exhaust all logical possibilities - in any situation, one and only one of them must be true.

Exercise #1

§2. Disjunction

A disjunction is a kind of complex sentence typically expressed in English by the word "or", such as:

Either we meet tonight, or we do not meet at all.

The sentence has the structure of "either P or Q", where P and Q are statements

In logic, we often make a distinction between exclusive disjunction and inclusive disjunction.

According to the exclusive interpretation, "P or Q" is true when P is true, or when Q is true, false when P and Q are both true, and also false when P and Q are false. Many people take the exclusive interpretation to be what is intended in for example "You can have tea or you can have coffee", where it is supposed to be implied that you can only have one or the other but not both.

On the inclusive interpretation, "P or Q" is false when P and Q are both false, and it is true in all other situations, including when both P and Q are true.

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:

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