Propositional logic is the study of the five logical connectives not, or, and, if ... then ... & ... if and only if ... . In this chapter we describe a language for propositional logic, assign truth functions to the five logical connectives, and define the fundamental concept tautological consequence. In Chapter 3 we will introduce a formal system whose intended interpretation is propositional logic. Thus, one may regard Chapter 2 as a discussion of the language of the formal system that we propose to study in Chapter 3; alternatively, Chapter 2 is a semantic approach to propositional logic.
The only time that we can be confident that the conclusion of an argument is true, is when the argument is valid and all hypotheses of the argument are true. This is the basis for the definition of a valid argument - more about this later.
Symbols: The symbols of the language are
The letters $p_1, p_2, p_3, \dots$ are called propositional variables and are interpreted as simple propositions. The list is infinite to ensure that we never run out of propositional variables when constructing formulas. The two symbols ( and ) are used for punctuation. Finally, the remaining symbols are called logical connectives. Their names and interpretations are as follows:
| Symbol | Name | Interpretation |
|---|---|---|
| $\neg$ | Negation | not |
| $\lor$ | Disjunction | or |
| $\land$ | Conjunction | and |
| $\rightarrow$ | Conditional | if ... then ... |
| $\leftrightarrow$ | Biconditional | ... if and only if ... |
The symbols $(, \space ), \space \neg, \space \lor, \space \land, \rightarrow$ and $\leftrightarrow$ are often called logical symbols. In translating English sentences into the language of propositional logic, these symbols are always interpreted in the same way
Formulas: The formulas are defined inductively by these three rules:
In other words, the formulas of propositional logic either are propositional variables or are built up from simpler formulas using the operators $\neg$, $\lor$, $\land$, $\rightarrow$ and $\leftrightarrow$. The formula $\neg A$ is called the negation of $A$; the formula $(A \lor B)$ is called the disjunction with disjuncts $A$ and $B$; the formula $(A \land B)$ is called the conjunction with disjuncts $A$ and $B$; the formula $(A \rightarrow B)$ is called the conditional (or implication) with antecedent $A$ and consequent $B$; the formula $(A \leftrightarrow B)$ is called the biconditional of $A$ and $B$.
This is an example
This is an example
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A set $C$ is convex if for all $x,y \in C$ and for all $\alpha \in [0,1]$ the point $\alpha x + (1-\alpha) y \in C$.
This is a theorem
This is an example
This is a lemma
This is a proof. The proof starts with a simple assumption and then uses logical reasoning to arrive at the conclusion. Different techniques of reasoning may be employed when trying to prove a specific statement. Rules of inference include Modus Ponens, Modus, Tollens, etc.
This is a corollary
This is a beamer theorem
This is a count example
This is another count example
This is another count example