Skip to main content

Section 1.5 Simple Logic

A mathematical statement is a sentence that is either true or false, but not both at the same time. For example, consider the following sentences:

  1. The Supreme Court’s ruling on Roe v. Wade on January 22, 1973 gave people the right to access abortion legally all across the country

  2. The Supreme Court ruled to overturn Roe v. Wade in 2022 making abortion a federal crime within the United States.

  3. Abortion should be illegal.

The first sentence is a sentence which is true, making the sentence a mathematical statement. The second sentence is false which also means that it is a mathematical statement. Lastly, the third sentence is an opinion which is neither true nor false. Therefore, the third sentence is not a mathematical statement. Determining if a sentence is a statement can be challenging as this causes us to challenge the difference between our individual belief systems and factual information. We will discuss this in further detail in our last section of this module.

Mathematical statements, or simply statements, are the building blocks to mathematical logic. While mathematicians primarily focus on statements that involve mathematics such as, "The number 2 is greater than 0", the methods used by mathematicians to make sense of their discipline can be applied to other topics which use critical thinking.

There are a series of operations that we can perform on statements to get new statements. This is analogous to taking two numbers and adding, subtracting, or multiplying to get a new number. In algebra, the letters \(x\text{,}\) \(y\text{,}\) and \(z\) are often used to represent numbers. These are called variables. In logic we typically use the variables \(p\text{,}\) \(q\text{,}\) and \(r\) to represent a statement. When we perform an operation on one or more statements, we will call the result a compound statement. It is important to note that the resulting statement is, in fact, a statement. This means that a compound statement must either be true or false, but not both at the same time.

Subsection 1.5.1 Negating and Combining Statements with "and" and "or"

Subsubsection 1.5.1.1 Negations

The negation of a given statement switches the truth value of the original statement. For example, the negation of "Abortion is a federal crime" is "Abortion is not a federal crime". Many times, a statement can be negated by inserting the word "not", or an equivalent word into the original statement. There are many ways to symbolically represent the negation of a statement. For this module, if we represent a statement by the variable \(p\text{,}\) we represent the negation of \(p\) by \(\neg p\text{.}\) We can use a logic calculator to see how the value of \(p\) and \(\neg p\) are related.

In the calculation above, the left most column represents the values for \(p\) and the column under "value" represents the corresponding values for \(\neg p\text{.}\)

Subsubsection 1.5.1.2 And & Or

As we can see, when \(p\) is true, \(~p\) is false and vice versa.The calculator used above generated a truth table. A truth table is a table that will produce the truth values for compound statements based off of the statement variable which are typically placed to the left of the table and the truth values for the compound statement which are typically to the right in the table. We will use this truth table generator for the remainder of this module.

Two more common ways to make a compound statement is by taking two simple statements and combining them with either the word "or" or "and". For example, consider the two statements " In August 2022, abortion is legal with not gestational limit in the state of New Jersey"and "In August 2022, abortion was banned with no exception for rape or incest in the state of Alabama"[1.12.2]. We can combine these statements in the following two ways:

  1. In August 2022, abortion is legal with no gestational limit in the state of New Jersey or in August 2022, abortion was banned with no exception for rape or incest in the state of Alabama.

  2. In August 2022, abortion is legal with no gestational limit in the state of New Jersey and in August 2022, abortion was banned with no exception for rape or incest in the state of Alabama.

These statements are a bit awkward and can be rewritten without losing their meaning as simply:

  1. In August 2022, abortion is legal with not gestational limit in the state of New Jersey or was banned with no exception for rape or incest in the state of Alabama.

  2. In August 2022, abortion is legal with not gestational limit in the state of New Jersey and was banned with no exception for rape or incest in the state of Alabama.

Both of these compound statements are true statements because the original simple statements are both true. The truth table for an "or" statement, more formally known as a disjunction is given below:

The truth table for an "and" statement, more formally known as a conjunction is given next:
The columns on the far left represent all possible truth values of the simple statements \(p\) and \(q\) used in the compound statement. The entries on the far left of each row represent the different truth value of the simple statements used in the compound statement. The final column on the far right represents the truth value of the compound statement for the given simple statement truth values. For example the second row in the truth table above reads "When \(p\) is true and \(q\) is false, \(p\) and \(q\) is false".

Subsection 1.5.2 Equivalent Statements

Given a statement \(p\text{,}\) we say that a statement \(q\) is equivalent to \(p\) if they have the same truth value. We denote equivalent statements, \(p\) and \(q\text{,}\) by

\begin{equation*} p \equiv q\text{.} \end{equation*}

For example, \(\neg(p\vee q)\equiv \neg p\wedge \neg q\) because for corresponding values of \(p\) and \(q\text{,}\) these compound statements have the same truth results. This can be seen by examining the truth tables below:

Because equivalent statements have the same truth value, we can interchange one for the other without losing the meaning of a statement. Let's try the following example. You may find the following website helpful for entering in the symbols used in these calculators Propositional Calculus 1 .

Determine which of the following statements is equivalent to \(\neg(p\wedge q)\text{:}\)

  1. \(\displaystyle \neg(p\vee q)\)

  2. \(\displaystyle \neg p\vee \neg q\)

  3. \(\displaystyle \neg p\wedge \neg q\)

Solution.

  1. Not equivalent

  2. Equivalent

  3. not equivalent

The examples above are called DeMorgan’s Laws. Summarizing DeMorgan’s Laws:

  1. \(\displaystyle \neg(p\vee q)\equiv \neg p\wedge \neg q\)

  2. \(\displaystyle \neg(p\wedge q)\equiv \neg p\vee \neg q\)

There are many other equivalent statements, as we will see in the next section.

Subsection 1.5.3 Implications (If-Then Statements)

An implication is a statement that has a premise and a conclusion such as, "If I take birth control, then I can’t get pregnant". In this implication, the premise is the statement "I take birth control" and the conclusion is the statement "I can’t get pregnant". With an implication, the truth value of the premise and conclusion determine the truth value of the actual implication. Given two statements \(p\) and \(q\text{,}\) the statement: if \(p\text{,}\) then \(q\) can be represented by \(p\rightarrow q\) and read as "if \(p\text{,}\) then \(q\)", "\(p\) implies \(q\)", "\(p\) only if \(q\)", and "\(q\text{,}\) if \(p\)". Below is the truth table for the implication "if \(p\text{,}\) then \(q\)":

The truth value of implications can be counterintuitive at times. This occurs primarily when the premise is false because the implication will always be true. For example the implications below are true since the implications in all are false:

  1. If abortions are federally protected, then people will use abortions to terminate unwanted pregnancies.

  2. If abortions are federally protected, then people will not use abortions to terminate unwanted pregnancies.

  3. If abortions are a federal crime, then more women will be jailed.

  4. If abortions are a federal crime, then more women will be jailed.

In these statements, it seems as if only one of the first two statements should be true and one of the last two statements should be true as the conclusions are negations of one another. However, all four statements are true.

It turns out that the only way for an implication to be false is when the premise is true and the conclusion is false. An example would be, "If abortions are not a federally protected right, then no-one will get an abortion".

Another counterintuitive concept that arises with implications is their negation. The negation, \(\sim (p \rightarrow q)\) is equivalent to the statement \(p\wedge \sim q\text{.}\) This is verified by examining their truth tables.

Verify that the following statements are not equivalent to \(\sim (p \rightarrow q)\) using the truth table generator below:

  1. \(\displaystyle \sim p\rightarrow q\)

  2. \(\displaystyle \sim p\rightarrow \sim q\)

  3. \(\displaystyle p\rightarrow \sim q\)

  4. \(\displaystyle q\rightarrow p\)

Solution.

  1. Not equivalent

  2. Not equivalent

  3. Equivalent

  4. not equivalent

It is important to notice that the negation of an implication is not an implication. For example, the negation of "If abortions are federally protected, then people will not use abortions to terminate unwanted pregnancies" is "Abortions are federally protected and people use abortions to terminate unwanted pregnancies" which is a false statement.

doc.sagemath.org/html/en/reference/logic/sage/logic/propcalc.html