https://maximummathematics.com/books/foundational-mathematics/logic/index.htmlIn most mathematics courses a student has taken so far, they have usually been presented with a collection of results — facts they can use whenever warranted. Polynomials, for instance, come with a wide variety of such facts: how to divide one polynomial by another, how to find the roots of a quadratic equation, and so on. Geometry offers just as many. The Angle Bisector Theorem tells us that the bisector of an angle in a triangle divides the opposite side into two segments proportional to the triangle’s other two sides. A wide variety of theorems describe the chords of a circle. And the famous Pythagorean Theorem relates the three sides of a right triangle.Hugoen-UShttps://maximummathematics.com/books/foundational-mathematics/logic/propositions/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/propositions/index.htmlIn Mathematics, we deal with statements like these: \[ \begin{align*} &\text{Squares have four equal sides.} \\ \\ &\text{Quadratic equations have at most two} \\ &\text{distinct roots.} \end{align*} \]In life, we deal with statements like these: \[ \begin{align*} &\text{If you don't pay your parking tickets, then} \\ &\text{you will go to jail.} \\ \\ &\text{The rent was paid on the first of the month, and the} \\ &\text{air conditioner stopped working.} \end{align*} \]In Math, as in life, some statements are easy to determine if they’re true or false. However, when we encounter more complicated expressions, we typically need a way to carefully evaluate that statement’s truth.https://maximummathematics.com/books/foundational-mathematics/logic/modeling-logic-with-truth-tables/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/modeling-logic-with-truth-tables/index.htmlWe are often working with three or more propositions at a time, usually combined into large numbers of expressions made using the logical connectives discussed previously. It can be cumbersome to work with them individually. Here, we will learn a technique for handling multiple expressions efficiently. A Convenient Shorthand To make our upcoming work easier, we adopt a common shorthand for truth values:https://maximummathematics.com/books/foundational-mathematics/logic/logical-order-of-operations/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/logical-order-of-operations/index.htmlJust like in the arithmetic of real numbers, there is an order we should follow when evaluating logical expressions. For instance, in the arithmetic of real numbers, we abide by the following rules: Parentheses $()$ Exponents Multiplication and division (left to right) Addition and subtraction (left to right) Using this scheme for evaluating arithmetic expressions, we do the following: \[ \begin{align*} 2 - 3^2 \cdot (4 + 6 \div 2) &= 2 - 3^2 \cdot (4 + 3) \\ &= 2 - 3^2 \cdot 7 \\ &= 2 - 9 \cdot 7 \\ &= 2 - 63 \\ &= -61 \end{align*} \]Here we introduce a scheme to help us determine which operations should be done in order to achieve the correct final result.https://maximummathematics.com/books/foundational-mathematics/logic/satisfiability/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/satisfiability/index.htmlWe’ve seen plenty of primitive propositions whose truth values are fixed: \[ \begin{array}{ll} \text{Calvin Coolidge was the 30th President of the United States of America.} & \text{(true)} \\ \\ \text{Mitochondria convert ADP into ATP via cellular respiration.} & \text{(true)} \\ \\ \text{Leonardo da Vinci painted the famous ceiling fresco in the Sistine Chapel.} & \text{(false)} \end{array} \]We’ve also seen compound propositions whose truth values depend on the truth values of its atomic propositions.https://maximummathematics.com/books/foundational-mathematics/logic/laws-of-logic/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/laws-of-logic/index.htmlAt this point, we’re familiar with the fundamental unit of logic — the proposition. We’ve seen how to combine them into compound propositions, and how to use truth tables to identify propositions that are always true — tautologies. With these tools, we are ready to start discussing the heart of logical deduction and our unique ability to reason — the Laws of Logic! A Simple Example Before we dive into the deep end, let’s wade in a shallow example where we examine a few propositions that involve the biconditional connective.https://maximummathematics.com/books/foundational-mathematics/logic/simplifying-logical-expressions/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/simplifying-logical-expressions/index.htmlIn the previous section we saw an example where we used tautologically true biconditionals to “simplify” complicated propositional expressions into simpler propositional expressions. Whatever we could say about the simpler expressions could also be said about their more complicated, logically equivalent counterparts (except perhaps our preference for working with the simpler expressions, of course). In this section we do more work with logical equivalencies, similar to what we saw in the examples seen previously. Our work here will bear a striking resemblance to our experience in dealing with the arithmetic and algebra of real numbers. In fact, the upcoming work we are about to engage with has been dubbed the “algebra of propositions.”https://maximummathematics.com/books/foundational-mathematics/logic/switching-networks/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/switching-networks/index.htmlWhile the content we’ve seen so far seems ethereal, with little application outside of simplifying logical propositions, logic dictates almost every avenue of study. In our everyday lives, we like to think our actions are reasonable and make sense. Certainly, we can use logic to analyze a situation so we can maximize our profit from it, whether our profit is in the form of friendship, promotions at work, happiness, health, or money.https://maximummathematics.com/books/foundational-mathematics/logic/open-propositions/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/open-propositions/index.htmlAll of the propositions we’ve dealt with so far have had definite truth values. For example, $$\text{Thomas Jefferson was the second president of the United States.}$$is a proposition that is known to be false. A statement such as $$\text{2 + 2 = 4, or 2 + 2 = 5.}$$is a compound proposition that is true. However, a statement such as $$\text{$n$ is 1 more than a multiple of 3.}$$is not a proposition, because we don’t know whether it’s true or false. We would need to know the value of $n$ in order to reach such a conclusion. For example, the statement is false when $n = 5$, but true when $n = 16$. Here, we’ll deal with sentences involving variables like this one.https://maximummathematics.com/books/foundational-mathematics/logic/quantifiers/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/quantifiers/index.htmlConsider the following propositional functions, both defined on the universe $\mathcal{U}$ of all integers: \[ \begin{array}{rl} s(n)\text{: } &n^2 \text{ is even.} \\ t(m, n)\text{: } &m^2 - n^2 \text{ is even.} \end{array} \]We can find values of $n$ that make $s(n)$ true, such as $n = -4$. We can also find values of $n$ that make $s(n)$ false, such as $n = 13$. The same is true of $t(m, n)$: the values $m = 3$ and $n = -7$ make $t(m, n)$ true, while $m = 4$ and $n = -1$ make $t(m, n)$ false.https://maximummathematics.com/books/foundational-mathematics/logic/quantified-laws-of-logic/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/quantified-laws-of-logic/index.htmlWhat can we deduce from knowing that a universally quantified open statement is true? What can we deduce from knowing that an existentially quantified statement is true? How are universally quantified statements and existentially quantified statements related to each other? In this section, we dig deeper into quantifiers, and explore propositional logic involving them. A Simple Definition Just like with ordinary statements, we can ask whether two open statements are logically equivalent.https://maximummathematics.com/books/foundational-mathematics/logic/multiple-quantifiers/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/multiple-quantifiers/index.htmlIn an earlier section, we examined propositional functions with two and three variables. For each variable in an open statement, we needed to substitute some value from the respective universe in order to determine the statement’s truth value. Here, we look at quantifying statements with two or more variables. Bound and Free Variables BOUND VARIABLE, FREE VARIABLE For some universe $\mathcal{U}$, consider a propositional function $p(x, y)$ where $x$ and $y$ are both constrained by $\mathcal{U}$, and the quantified statementshttps://maximummathematics.com/books/foundational-mathematics/logic/modeling-logic-puzzles/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/modeling-logic-puzzles/index.htmlLogic puzzles are almost everywhere, from supermarket puzzle books to viral posts on social media. Some are well known, like the Zebra Puzzle and Sudoku. Others are lesser known, including the Knights and Knaves puzzle. Regardless of which puzzle is being tackled, the logical tools we’ve explored throughout this chapter can be used to work out a solution. In this section, we’ll explore a few different kinds of logic puzzles, and use mathematical logic to solve them.