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• Logical Statements
• Negation, Conjunction, and Disjunction
• Equivalent Statements
• Tautologies and Contradictions
• Variable Statements

Symbolic logic is the study of logic and logical arguments by writing everything in terms of symbols. Logical statements are given symbols such as \(P\), \(Q\), and \(R\), and can be assigned a truth value of (\(\text{T}\))rue or (\(\text{F}\))alse.

Compound statements are statements that are built from simpler statements using logical conjunctives. Logical conjunctives are logical operations such as negation (logical not), conjunction (logical and), and disjunction (logical or).

The truth value of a compound statement is determined by the truth value of its base statements and the rules of the conjunctives used. A truth table is a table which gives the truth value of a compound statement for all possible truth value combinations of the base statements.

Logical Statements

A mathematical structure is a collection of definitions and logical statements (axioms and theorems) about that structure. As such it is reasonable to start by defining what a logical statement is.

• A logical statement (the zyBook calls them propositions) is any statement that can be classified as either true or false, but not both at the same time.

• True or false are just a binary classification of a statement within a particular structure. Try not to think of these as universal truths, but only a classification with in a particular structure. Different structures may have different classifications for the same statement.

• Statements must be a sentence that can be classified as true or false. For example commands and questions are not statements, since it is not possible to assign them a truth value.

  • Do the math.
  • What time is class?

• Statements cannot be both true and false at the same time. Such sentences are called paradoxes. The liar’s paradox is the statement “I am a liar¹”.

  • If this statement was true, then I would be a liar. But a liar would not state they were a liar.
  • If this statement was false, then I would not be a liar. But a truth teller would not say they were a liar.

  In either case the statement cannot be true or false without leading to a contradiction, which makes it a paradox.

• The zyBook states that opinions can be statements, such as

  • The movie was funny.

  Here the truth value is a matter of opinion and depends on which individual is assigning this a true or false value. Since any individual (in theory) could assign a truth value, it is considered a statement.

Negation, Conjunction, and Disjunction

The three primary logical conjunctives are the negation (not), conjunction (and), and disjunction (or). These conjunctives are defined as follows:

1. Negation: Negation is the logical not which has the opposite truth value of the original statement.

   • In symbolic logic the negation symbol is \(\neg\).

   • If \(P\) is a statement then \(\neg P\) (thought of as not \(P\)) is the negation of the statement.

   • The truth table for negation is

     \(P\)	\(\neg P\)
     T	F
     F	T

2. Conjunction: Conjunction is the logical and which is only true if both the base statements are true.

   • In symbolic logic the conjunction symbol is \(\land\).

   • If \(P\) and \(Q\) are statements, then \(P\land Q\) (thought of as \(P\) and \(Q\)) is the conjunction of the two statements.

   • The truth table for conjunction is

     \(P\)	\(Q\)	\(P \land Q\)
     T	T	T
     T	F	F
     F	T	F
     F	F	F

3. Disjunction: Disjunction is the logical or which is true if at least one of the base statements are true.

   • In symbolic logic the disjunction symbol is \(\lor\).

   • If \(P\) and \(Q\) are statements, then \(P \lor Q\) (thought of as \(P\) or \(Q\)) is the disjunction of the two statements.

   • The truth table for disjunction is

     \(P\)	\(Q\)	\(P \land Q\)
     T	T	T
     T	F	T
     F	T	T
     F	F	F

   • Disjunction, the logical or, is the inclusive or, which allows for both statements to be true at the same time. This is not the same as the exclusive or (xor), which assigns a false value to both statements being true.

     In the English language, it can be ambiguous if or means inclusive or exclusive, since sometimes it is used to be the inclusive or and others it is used to be the exclusive or. While in logic, or always means inclusive or.

In addition to the three basic logical conjunctives listed above, there are other ones that are sometimes used. Such as the exclusive or (called xor) mentioned above or not-and (called nand) seen in circuits. In practice the other logical conjunctives can be reduced to using only the three listed above.

Equivalent Statements

Equivalent Statements are two symbolic statements that produce the same truth table. That is they have the same truth value for all possible truth values of the base statements. If two logical statements are equivalent, we state this by using the \(\equiv\) symbol. For example

\[P\lor Q\equiv \neg\Big( \neg P \land \neg Q\Big)\]

To see these two logical statements are equivalent create their truth tables. The truth table of \(P\lor Q\) is above and the truth table of \(\neg\Big( \neg P \land \neg Q\Big)\) is:

\(P\)	\(Q\)	\(\neg P \land \neg Q\)	\(\neg\Big( \neg P \land \neg Q \Big)\)
T	T	F	T
T	F	F	T
F	T	F	T
F	F	T	F

Since this truth table is the same as the truth table for disjunction, it follows that the two statements are equivalent.

De Morgan’s Laws give equivalent statements and show how negation, conjunction, and disjunction are linked. The two equivalent statements are

\[\begin{align*} \neg\Big( P\land Q\Big) &\equiv \neg P \lor \neg Q \\ \neg\Big( P\lor Q\Big) &\equiv \neg P \land \neg Q \end{align*}\]

Creating a truth table for all four statements above will prove the above two equivalencies.

Tautologies and Contradictions

• A compound statement which is always true is called a tautology. For example, the statement \(P\vee \neg P\) has the following truth table:

  \(P\)	\(\neg P\)	\(P\vee \neg P\)
  T	F	T
  F	T	T

  Since this statement is always true, it is a tautology.

• A compound statement which is always false is called a contradiction. For example, the statement \(P\wedge \neg P\) has the following truth table:

  \(P\)	\(\neg P\)	\(P\wedge \neg P\)
  T	F	F
  F	T	F

  Since this statement is always false, it is a contradiction.

Variable Statements

In mathematics, statements often depend on some variable. Some statements are true for all possibilities of the variable, while others are true for only some (or no) possibilities of the variable. As such having a notion of a variable statement is useful.

• A variable statement is a statement which contains a variable that can take on multiple values.

• Some variable statements are true for all possible values of the variables, such as \((a+b)^2 = a^2 + 2ab + b^2\) is true for all real numbers \(a\) and \(b\).

• Some variable statements are only true for some values of the variable and false for others, for example the variable statement

  \[x^2 = 25\]

  is true for some values of \(x\), such as \(x=5\) or \(x=-5\), while this statement is false for other values of \(x\), such as \(x=3\).

• Some variable statements are true for no possible values of the variable, such as \(x^2 = -1\) is false for all real numbers \(x\).

• Statements which are opinions can be thought of as a variable statement, where the variable is the individual in question. For example the statement “the movie was funny” could be thought of as the statement “Person \(X\) thought the movie was funny.”

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