introduction to Logical Forms

Formal logic is the branch of mathematics that deals with the principles of valid reasoning and argumentation. It provides a systematic framework for analyzing statements and their relationships, allowing us to determine the truth or falsity of an argument based on its structure. In formal logic, we aren’t really concerned with the actual content of statements, but rather with its form and structure.

Basic Concepts

It is helpful to start with some basic concepts in formal logic. Of course, it is not possible to cover all aspects of formal logic in a single introduction, but we will touch on some of the fundamental ideas. It is also difficult to start from absolute zero, so we will take some definitions for granted.

Propositions and Variables

Let’s begin with the building blocks of an argument.

Variables

A variable is a placeholder typically represented by some symbol that can take on different values and are used when talking about something with an unknown value or to represent a general case rather than a specific instance.

Propositions

A proposition (sometimes called a Statement) is a sentence that is strictly either true or false. There can be no ambiguity: the truth value must be ascertained without any additional context. A sentence such as “Black is a color.” and $2 + 5 = 7$ are examples of a proposition. A statement such as $x + 2 = 5$ is not a proposition, because its truth value depends on the value of $x$ and is therefore ambigious.

There are a number of common propostions that are used in formal logic:

• A Universal Propostion states that a certain property holds for all elements in a given set (which we define as a collection of objects). They tend to contain language such as “for all…”
• An Existential Propostion states that there is at least one element in a given set for which a certain property holds. They tend to contain language such as “there exists…”
• A Conditional Propostion states that if one statement (the antecedent) is true, then another statement (the consequent) is also true. We will discuss these more later in this section. COnditional Propositions are typically expressed in the form “If … then ..”

Logical Forms

In formal logic, an argument is not a verbal fight between two people, but rather a sequence of propositions (called premises) that aim to establish the truth of an assertion (the conclusion). Before we start analyzing arguments, we need to understand how to represent them in logical form.

As mentioned earlier, In formal logic, we are more concerned with the logical form rather than the specific content. The logical form of an argument is ascertained from its content. The form is essentially an abstract representation of the argument that focuses on the relationships between its premises, thus allowing us to evaluate the validity of an argument just from its form. To create this abstract representation, we often use symbols to represent the premises and utilize logical connectives.

A propositional variable is a symbol that represents a proposition, and are typically denoted with symbols like $p, q, r$. We can chain together multiple propositional variables using logical connectives to form more complex propositions.

Logical Connectives

Notice how we used logical connectives to build more complex propositions from simpler ones. Some common logical connectives include:

• Negation ($\neg$): Represents “not p”, $\neg p$ means “not $p$.”
  • If $p$ represents “It is raining”, then $\neg p$ represents “It is not raining”.
  • Convention may vary across different disciplines (because we like making life difficult)
  • Symbols like $\sim p$, $\~{p}$ or $!p$ are also used to denote the negation of $p$.
  • If $p$ is true, then $\neg p$ is false; if $p$ is false, then $\neg p$ is true.
• Conjunction ($\land$): Represents ”$p$ and $q$”. $p \land q$ means ”$p$ and $q$.”
  • If $p$ represents “It is raining” and $q$ represents “It is soggy”, then $p \land q$ represents “It is raining and it is soggy.”
  • $p$ but $q$ means the same thing as $p \land q$.
  • $p \land q$ is true only if both $p$ and $q$ are true; otherwise, it is false.
• Disjunction ($\lor$): Represents ”$p$ or $q$”. $p \lor q$ means ”$p$ or $q$.”
  • If $p$ represents “It is raining” and $q$ represents “It is sunny”, then $p \lor q$ represents “It is raining or it is sunny.”
  • Neither $p$ nor $q$ means the same thing as $\neg p \land \neg q$.
  • We generally interpret “or” in the inclusive sense, meaning that is at least one of $p$ or $q$ is true, then $p \lor q$ is true. (both can be true as well).
• Implication ($\to$): Represents “If $p$, then $q$”. $p \to q$ means “If $p$, then $q$.”
  • We call statements of this form conditional statements or implications, as the value of $q$ is conditional on the value of $p$.
  • If $p$ represents “It is raining” and $q$ represents “The ground is wet”, then $p \to q$ represents “If it is raining, then the ground is wet.”
  • We say $p$ implies $q$.
  • $p$ only if $q$ means the same thing as $p \to q$.
  • $p \to q$ is false only if $p$ is true and $q$ is false; otherwise, it is true.
  • This may seem counterintuitive at first, but remember that if $p$ is false, then the implication $p \to q$ is considered true regardless of the truth value of $q$. Such statements are consided to be vacuously true. Thus, $p \to q$ is only false when a true premise leads to a false conclusion.

We will discuss more logical connectives later on, but these are the most common ones used in propositional logic.

Using these logical connectives, we can construct complex arguments and ascertain their truth value. We can now analyze the logical structure of arguments and determine their validity based on the truth values of their components.

Tautologies and Contradictions

A tautology is a propositional statement that is always true, regardless of the truth values of its components. An example of a tautology is the statement $p \lor \neg p$, which states that either $p$ is true or $p$ is not true. This statement is always true because one of the two possibilities must hold.

A contradiction is a propositional statement that is always false, regardless of the truth values of its components. An example of a contradiction is the statement $p \land \neg p$, which states that $p$ is both true and not true at the same time. This statement is always false because it is impossible for both conditions to hold simultaneously.

Some philosophers argue that mathematics is just a collection of tautologies. Of course, this is incorrect as mathematics is just a Government PsyOp. I jest, but this topic is quite interesting. Imre Lakatos, a mathematical philosopher, wrote a famous book about it. You can read more here .

Evaluating Propostional Statements

To evaluate the truth value of a logical form, we can use tools such as truth tables. We can systematically list all possible truth values for the propositions involved and determine the truth value of the overall statement based on the logical connectives used.

We can write the truth tables for $p \land q, p \lor q, \neg p$, and $p \to q$ as follows:

These truth tables help us understand how the truth values of complex propositions are determined based on the truth values of their components. By using these tools, we can analyze and evaluate arguments in formal logic effectively.

We can chain together multiple propositions using these logical connectives to form more complex statements. For example, we can create a statement like:

Operations are evaluated based on their precedence. Negation has the highest precedence. Parentheses can be used to explicitly specify the order of operations. Statements such as $p \lor q \land r$ are ambigious, so it is better to use parentheses to clarify the intended meaning, like so

The truth table for complex statements can be constructed by evaluating the truth values of their components step by step. Start from the innermost expressions and work your way outwards. The truth table for the above statement $(p \land q) \to (\neg r)$ would look like this:

Two statements are logically equivalent if they have the same truth value in every possible scenario. Logical equivalences can be used to simplify complex statements or to prove the validity of arguments.

Try to create some some propositions statements and plug them into truth tables to see how they work! Ensure your statements are well-defined and unambiguous.

Logical Equivalences

Logical equivalences are statements that are always true and can be used to simplify complex logical expressions. Suppose that $p,q, r$ are propositional variables and $t, c$ are a tautology and contradiction respectively. We have the following logical equivalences:

Commutativity

Associativity

Distrubutivity

Identity Laws

Negation

Double Negation

Idempotent Laws

Universal Bound Laws

Absorption Laws

De Morgan’s Laws

De Morgan’s laws are essentailly a distrubution rule for negation over conjunctions and disjunctions. They state the following equivalences:

• The negation of a conjunction is equivalent to the disjunction of the negations: $$\neg (p \land q) \equiv (\neg p) \lor (\neg q)$$
• The negation of a disjunction is equivalent to the conjunction of the negations: $$\neg (p \lor q) \equiv (\neg p) \land (\neg q)$$

Proof of De Morgan’s Laws

We can prove De Morgan’s laws using truth tables. Let’s start with the first law:

Observe that the columns for $\neg (p \land q)$ and $(\neg p) \lor (\neg q)$ are identical, confirming the first of De Morgan’s Laws. Now, let’s prove the second law:

Observe that the columns for $\neg (p \lor q)$ and $(\neg p) \land (\neg q)$ are identical, confirming the second of De Morgan’s Laws.

More on Conditional Statements and Implication

Logical Equivalences of Conditional Statements

Conditional Statements, also known as implications, are a fundamental concept in formal logic. A conditional statement is typically expressed in the form “If $p$, then $q$,” denoted as $p \to q$. Here, $p$ is called the antecedent (or premise), and $q$ is called the consequent (or conclusion). Arguments involving conditional statements are used to establish relationships between propositions and to derive conclusions based on given premises. Logical inferencing often relies on understanding and manipulating these conditional statements. We can express conditional statements using other logical connectives.

Recall, the truth table for implication ($p \to q$) is given by

We have $p \ to q$ is logicallly equivalent to $\neg p \lor q$.

Addtionally, we have the contrapositive of a conditional statement. If we have $p \to q$, the contrapositive is the statement $\neg q \to \neg p$. The contrapositive is logically equivalent to the original conditional statement. This means that if $p \to q$ is true, then $\neg q \to \neg p$ is also true, and vice versa. Sometimes, the contrapositive might be easier to prove or work with than the original implication. This usage is a fundamental concept in logic and is often used in mathematical proofs and reasoning.

Proof of Logical Equivalence of Conditional Statements

To prove that $p \to q$ is logically equivalent to $\neg p \lor q$, we can construct a truth table for both expressions and compare their truth values.

Observe that the columns for $p \to q$ and $\neg p \lor q$ are identical, confirming that $p \to q$ is logically equivalent to $\neg p \lor q$.

To prove that $p \to q$ is logically equivalent to its contrapositive $\neg q \to \neg p$, we can construct a truth table for both expressions and compare their truth values.

Observe that the columns for $p \to q$ and $\neg q \to \neg p$ are identical, confirming that $p \to q$ is logically equivalent to its contrapositive $\neg q \to \neg p$.

Converses and Inverses

We can define the converse and inverse of a conditional statement $p \to q$ as follows:

• The converse is the statement $q \to p$. The converse is not logically equivalent to the original conditional statement.
• The inverse is the statement $\neg p \to \neg q$. The inverse is also not logically equivalent to the original conditional statement

Conditions

$p$ only if q means “if not p then not q”. In logical form, this is expressed as $\neg p \to \neg q$.

A biconditional statement is a statement that combines two conditional statements: “if $p$ then $q$” and “if $q$ then $p$”. It is denoted as $p \leftrightarrow q$ and is read as ”$p$ if and only if $q$”. A biconditional statement is true when both $p$ and $q$ have the same truth value (both true or both false) and false otherwise. In logical form, a biconditional statement can be expressed as the conjunction of two implications: $(p \to q) \land (q \to p)$.

Proof of Logical Equivalence of Biconditional Statements

To prove that $p \leftrightarrow q$ is logically equivalent to $(p \to q) \land (q \to p)$, we can construct a truth table for both expressions and compare their truth values.

Observe that the columns for $p \leftrightarrow q$ and $(p \to q) \land (q \to p)$ are identical, confirming that $p \leftrightarrow q$ is logically equivalent to $(p \to q) \land (q \to p)$.

Suppose that $r,s$ are propositional variables.

We say that $r$ is a sufficient condition for $s$ if we have $r \to s$.

We say that $r$ is a necessary condition for $s$ if we have $s \to r$. We can write this with the contrapositive as if not $r$ then not $s$ as $\neg r \to \neg s$.

The biconditional statement $r \leftrightarrow s$ indicates that $r$ is both a necessary and sufficient condition for $s$. In other words, $r$ is true if and only if $s$ is true.