Introduction to Logical Reasoning

Logical Reasoning and Proofs

Much of theoretical mathematics has to do with writing proofs to connect known truths to new findings. To help you write strong proofs, we’re going to talk about logical reasoning, the foundation on which all valid proofs are built.

One key element of proofs is the premises which are used in them. Premises are statements which are already assumed to be true, and from these true statements, new conclusions may be derived. For example, we already know that the interior angles of a triangle must add up to 180°. This fact may be regarded as a premise.

Proofs as a whole may be referred to as arguments. An argument is a collection of premises followed by a conclusion. If the premises used in an argument are all true, and if the conclusion follows logically from the premises, then the argument is considered valid and the proof is complete.

The individual sentences which compose a proof are called statements and may be either simple statements or compound statements. Statements must have a truth value, meaning they must be able to be proven true or false. “The sky is blue” is a statement because it can be proven true. “There are a lot of clouds in the sky” is not considered a statement because “a lot” is not specifically measurable. I may think there are a lot of clouds in the sky but you may disagree. So, the statement cannot be proven true or false. Opinions are never considered statements in math.

Simple statements are straightforward and are intended to convey one thought. Compound statements, on the other hand, include one or more simple statements along with a logical operator. The five logical operators are negation, conjunction, disjunction, conditional, and biconditional. If that all sounded like gibberish to you, there’s no need to worry. Let’s clear any confusion by going over what each of these mean.

The negation operator is denoted with a tilde, or squiggle, that precedes the simple statement, and its action is to undo, or give the opposite, of that statement. For example, if we wanted to negate the statement, “There are no trucks in the parking lot,” this would be like saying, “It is not true that there are no trucks in the parking lot.” In other words, we could say, “There is at least one truck in the parking lot.” To negate a statement is to claim that it is untrue.

The second type of logical operator is conjunction, which essentially connects two simple statements with the word “and.” The symbol for conjunction is an upside-down “V”. So, if we wanted to say, “There are no trucks in the parking lot and there are no cars in the parking lot,” we could write:

This statement would be true only if the parking lot contained zero cars and zero trucks.

Notice that the conjunction symbol is similar to the symbol we use for the intersections of sets. In fact, a conjunction may be thought of as a type of intersection, because we use the word “and” to show that two statements (like sets) are fulfilled simultaneously.

The third logical operator is disjunction, which is denoted with a V-shaped symbol, and means “or.” If you guessed that disjunction is quite like the union of sets, you would be quite right! Disjunctions are satisfied when at least one of the simple statements given is true. For example, if we say, “There are no trucks in the parking lot or there are no cars in the parking lot,” then we are correct as long as one or both of those are true. We are only wrong if both simple statements are false.

The fourth logical operator is the conditional, which we can think of as an “if, then” statement. For example, the statement, “If it is a holiday, then there is no school” is a conditional and can be denoted by writing a left-to-right arrow between the two simple statements.

This arrow is usually pronounced using the word “implies.” Conditionals are only untrue whenever the first statement is true but the second is false. For example, this conditional is true whenever there is a holiday and school is not being held for the day. The conditional is also true whenever there is no holiday and there is school. And even on days when there is no holiday, but also no school, this statement is still true—it hasn’t become a lie just because there is no holiday. The conditional only becomes false when there is a holiday and there is school.

Conditionals only flow from one way, left to right, but the fifth logical operator, the biconditional, can be read forwards and backwards. Biconditionals usually involve the phrase “if and only if,” and are denoted by a two-sided arrow. For example, the statement, “A shape is a triangle if and only if it is a three-sided polygon,” is a biconditional statement. We would get the same information by reading it backwards: “A shape is a three-sided polygon if and only if it is a triangle.” Biconditional statements are true and satisfied whenever both simple statements are true, or both are false. If one is true and the other is false, then the biconditional is invalid.

A couple final concepts that will help you in logic problems and proofs are existential and universal quantifiers. These names may sound a bit lofty, but in reality they aren’t so bad. These quantifiers are simply tools for expressing how many objects you’re talking about. Earlier, we said that the statement, “There are no trucks in the parking lot” could be negated if there was at least one truck in the parking lot. If we did find a truck, we could say, “There is a truck in the parking lot.” Whether we find one or one-hundred trucks, the initial statement is invalidated. In logic, we use the existential quantifier, a backwards-E symbol, as a shorthand for the words “there is.”

This symbol can also be pronounced with the words “there exists” a truck, or “there is some” truck.

On the other hand, if we searched the parking lot and found zero trucks, we could say, “For all of the vehicles in the parking lot, none of them are trucks.” The words “for all” are used for a universal quantifier, and are written in shorthand by mathematicians as an upside-down “A.”

“For all” can also be pronounced as “for every” or “for each.”

It is even possible to use both of the quantifiers in a sentence, and mathematicians frequently do. For example, we could say, “For all gloves, there is one thumb,” or “There exists some house for which all rooms are painted pink.”

We’ve now discussed that proofs are formed using a sequence of statements, all connected by a flow of logical thought. Many statements are compound in nature, involving one or more simple statements and a logical operator. These operators include negation, conjunction, disjunction, conditional, and biconditional. Additionally, the existential quantifier “there exists” and the universal quantifier “for all” can be very helpful in building successful proofs. With these tools, you’re now ready to try some proof and logic problems on your own!

I hope that this video was helpful. Thanks for watching, and happy studying!