FoldUnfold Table of Contents Determining the Validity of an Argument by Rules of Inference Example 1 Determining the Validity of an Argument by Rules of Inference Let $P$, $Q$, and $R$ be statements. So far we have discussed the following rules of inference: Modus Ponens: $(P \wedge (P \rightarrow Q)) \rightarrow Q$. Modus Tollens: $((P \rightarrow Q) \wedge \neg Q) … [Read more...]

## The Law of Syllogism

The Law of Syllogism Definition: Let $P$, $Q$, and $R$ be statements. Then the rule of inference known as The Law of Syllogism is given by the argument $((P \rightarrow Q) \wedge (Q \rightarrow R)) \rightarrow R$. For example, let $P$, $Q$, and $R$ be the following statements: $P$: Bob has studied for the test. $Q$: Bob will pass the test. $R$: … [Read more...]

## Valid Arguments

FoldUnfold Table of Contents Valid Arguments Valid Arguments Definition: Let $P_1$, $P_2$, …, $P_n$ be a collection of statements which we will call Premises and let $Q$ be a statement which we will call a Conclusion. Then an Argument is a formula of the form $(P_1 \wedge P_2 \wedge ... \wedge P_n) \rightarrow Q$. The argument is said to be a Valid Argument if under the … [Read more...]

## The Contrapositive, Converse, and Inverse of an Implication

FoldUnfold Table of Contents The Contrapositive, Converse, and Inverse of an Implication The Contrapositive, Converse, and Inverse of an Implication Definition: Let $P$ and $Q$ be statements and consider the implication $P \rightarrow Q$. The Contrapositive of this implication is the formula $\neg Q \rightarrow \neg P$. The Converse of this implication is the formula $Q … [Read more...]

## Logical Equivalence of Formulas

FoldUnfold Table of Contents Logical Equivalence of Formulas Logical Equivalence of Formulas Definition: Let $\varphi$ and $\psi$ be formulas that are composed of the same component statements. Then $\varphi$ and $\psi$ are said to be Logically Equivalent denoted $\varphi \Leftrightarrow \psi$ if every truth assignment to the component statements cause $\varphi$ and … [Read more...]

## Tautologies and Contradictions

FoldUnfold Table of Contents Tautologies and Contradictions Tautologies and Contradictions Definition: A formula is said to be a Tautology if every truth assignment to its component statements results in the formula being true. A formula is said to be a Contradiction if every truth assignment to its component statements results in the formula being false. It is easy … [Read more...]

## The Semantics of Propositional Logic

FoldUnfold Table of Contents The Semantics of Propositional Logic Negation Conjunction Disjunction Exclusive Disjunction / Exclusive Or Implication Biconditional The Semantics of Propositional Logic So far we have discussed propositional logical connectives and formulas. We noted that formulas derive from statements. We would now like to determine whether a formula is … [Read more...]

## Formulas in Propositional Logic

FoldUnfold Table of Contents Formulas in Propositional Logic Formulas in Propositional Logic We are now ready to properly define a formula in propositional logic. Definition: A Formula in propositional logic must satisfy one of the criteria below: a) A statement is a formula. b) If $\varphi$ is a formula then $\neg \varphi$ is a formula. c) If $\varphi$ and $\psi$ are … [Read more...]

## Propositional Logic Connectives

FoldUnfold Table of Contents Propositional Logic Connectives Propositional Logic Connectives Before we learn about propositional logic we must first understand the meaning of certain symbols which we will call connectives. We will use these connectives with statements. These statements will be denoted with the letters $P$, $Q$, and $R$, and they will state the … [Read more...]

## The AM-GM Inequality

FoldUnfold Table of Contents The AM-GM Inequality The AM-GM Inequality Theorem 1 (The AM-GM Inequality): Let $x_1, x_2, ..., x_n$ be nonnegative real numbers. Then $\displaystyle{\frac{x_1 + x_2 + ... + x_n}{n} \geq \sqrt[n]{x_1 \cdot x_2 \cdot ... \cdot x_n}}$. "AM-GM" stands for "Arithmetic Mean - Geometric Mean". Proof: Consider the function $f(x) = -\ln x$. This … [Read more...]