Logic: The Laws of Truth

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Next, in the third column, I list the values of based on the values of P. I use the truth table for negation: When P is true is false, and when P is false, is true. In the fourth column, I list the values for. The fifth column gives the values for my compound expression.

It is an "and" of the third column and the fourth column. An "and" is true only if both parts of the "and" are true; otherwise, it is false. A tautology is a formula which is "always true" that is, it is true for every assignment of truth values to its simple components. You can think of a tautology as a rule of logic. The opposite of a tautology is a contradiction , a formula which is "always false". In other words, a contradiction is false for every assignment of truth values to its simple components.

Propositional Logic − Logical Equivalences

Show that is a tautology. I construct the truth table for and show that the formula is always true. The last column contains only T's. Therefore, the formula is a tautology. Construct a truth table for. You can see that constructing truth tables for statements with lots of connectives or lots of simple statements is pretty tedious and error-prone. While there might be some applications of this e.

The point here is to understand how the truth value of a complex statement depends on the truth values of its simple statements and its logical connectives. In most work, mathematicians don't normally use statements which are very complicated from a logical point of view.

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I want to determine the truth value of. Therefore, the statement is true. You can't tell whether the statement "Ichabod Xerxes eats chocolate cupcakes" is true or false but it doesn't matter.

Nicholas J. J. Smith, Logic: The Laws of Truth

If the "if" part of an "if-then" statement is false, then the "if-then" statement is true. Check the truth table for if you're not sure about this! So the given statement must be true.

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Two statements X and Y are logically equivalent if is a tautology. Another way to say this is: For each assignment of truth values to the simple statements which make up X and Y, the statements X and Y have identical truth values. From a practical point of view, you can replace a statement in a proof by any logically equivalent statement. To test whether X and Y are logically equivalent, you could set up a truth table to test whether is a tautology that is, whether "has all T's in its column".

However, it's easier to set up a table containing X and Y and then check whether the columns for X and for Y are the same. Show that and are logically equivalent. Since the columns for and are identical, the two statements are logically equivalent. This tautology is called Conditional Disjunction. You can use this equivalence to replace a conditional by a disjunction.

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There are an infinite number of tautologies and logical equivalences; I've listed a few below; a more extensive list is given at the end of this section. When a tautology has the form of a biconditional, the two statements which make up the biconditional are logically equivalent.

Hence, you can replace one side with the other without changing the logical meaning. You will often need to negate a mathematical statement. To see how to do this, we'll begin by showing how to negate symbolic statements.


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Write down the negation of the following statements, simplifying so that only simple statements are negated. I've given the names of the logical equivalences on the right so you can see which ones I used. I showed that and are logically equivalent in an earlier example.

In the following examples, we'll negate statements written in words. This is more typical of what you'll need to do in mathematics. The idea is to convert the word-statement to a symbolic statement, then use logical equivalences as we did in the last example. Use DeMorgan's Law to write the negation of the following statement, simplifying so that only simple statements are negated:. Let C be the statement "Calvin is home" and let B be the statement "Bonzo is at the moves". The given statement is. I'm supposed to negate the statement, then simplify:.

The result is "Calvin is home and Bonzo is not at the movies".

Truth Tables, Tautologies, and Logical Equivalences

Let P be the statement "Phoebe buys a pizza" and let C be the statement "Calvin buys popcorn". To simplify the negation, I'll use the Conditional Disjunction tautology which says. That is, I can replace with or vice versa. The result is "Phoebe buys the pizza and Calvin doesn't buy popcorn". Next, we'll apply our work on truth tables and negating statements to problems involving constructing the converse, inverse, and contrapositive of an "if-then" statement.

Logic : The Laws of Truth

By the contrapositive equivalence, this statement is the same as "If is not rational, then it is not the case that both x and y are rational". This answer is correct as it stands, but we can express it in a slightly better way which removes some of the explicit negations. Most people find a positive statement easier to comprehend than a negative statement. By definition, a real number is irrational if it is not rational. So I could replace the "if" part of the contrapositive with " is irrational".

The "then" part of the contrapositive is the negation of an "and" statement. As a teacher of logic, I see real benefits in Smith's approach.

I predict that this one will be widely adopted throughout the English-speaking world. One of its unique strengths is that it broaches important philosophical issues that naturally arise in connection with symbolic logic. The book thus serves both as an introduction to logic itself and to the philosophy of logic. Review quote "[I]f you are a teacher in the market for a new logic text, or a student looking for very helpful reading, this could indeed be the book for you. It has the expected virtues of clarity, precision and accessibility The book deserves to be used widely, both as a text for courses and for self-study.

Smith succeeds in introducing his readers to the complexities of symbolic logic in a very gentle manner--there is not a formula or a definition in the book that is not thoroughly explained or illustrated with an abundance of examples. Paoli, History and Philosophy of Logic "An excellent textbook for an undergraduate course on this topic. Evans, Philosophy in Review show more.

About Nicholas J. Smith Nicholas J. Smith is senior lecturer in philosophy at the University of Sydney in Australia. He is the author of Vagueness and Degrees of Truth. Rating details. Book ratings by Goodreads. Goodreads is the world's largest site for readers with over 50 million reviews. We're featuring millions of their reader ratings on our book pages to help you find your new favourite book.

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