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3.3. Truth Tables & Logical Equivalence

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3.3. TRUTH TABLES & LOGICAL EQUIVALENCE What is a Truth Table? A truth table is an essential tool in propositional logic. Truth tables have been widely used since the 1920’s when the American logician Emile Post (1897 – 1954) and the Austrian philosopher Ludwig Wittgenstein (1889 – 1951) popularized them in their work. Nowadays, they are ubiquitous in the study of logic as they provide an efficient method for determining the truth value of a compound statement. Consider the following compound statement: “The table was delivered on Thursday and the couch was delivered on Friday.” The truth value of this statement depends entirely on whether its components are true or false. In fact, the following conclusions must follow from the sentence. If the dining room table was delivered on any day other than Thursday (say, Wednesday), then the statement cannot be true. In fact, in this case it is false regardless of whether the couch was delivered on Friday or not. If the couch failed to be delivered on Friday, then the statement cannot be true. It is again false no matter what happened with the table delivery. The only time the statement is true is when both deliveries occurred according to plan. This type of situation, which requires looking at all logical possibilities, is neatly handled by a table summarizing all possible truth value outcomes that result from whether the table or couch deliveries occurred as planned. This kind of table is called a truth table. In general, a truth table generates the truth values of any compound statement based on all the possible combinations of truth values of its components. Since any basic component can be either true or false, this means that the truth table for a statement with 푛 components has 2푛 rows (4 rows with 2 components, 8 rows with 3 components, 16 rows with 4 components, etc.) Note that this mirrors precisely the number of regions in a general Venn diagram with 푛 sets. In practice, truth tables with more than 8 rows are typically handled by computers using Boolean algebra – where 1 and 0 are the actual values given to components that are, respectively, true or false. Basic Truth Tables NEGATION For a component 푝, the statement “Not 푝” is called the negation of 푝 and is written symbolically as ~푝. Negations always change the truth value of the statement. Hence, the negation of a true statement is false and the negation of a false statement is true. For example, negating the true statement “Men are mortal” results in the false statement “Men are immortal” and the negation of the false statement “More than four U.S. states border Mexico” is the true statement “At most four U.S. states border Mexico.” (In fact, there are exactly four states that border Mexico: California, Arizona, New Mexico, and Texas.) This rule is summarized in Table 3 below. 풑 ~풑 T F F T Table 3: Basic Truth Table for the Negation “Not 푝” Double Negations The negation of the negation ~푝 is the original statement 푝. This is why double negations in a statement logically cancel each other out. For instance, the statement “The chef is not displeased with his new lobster dish” is logically identical to the statement “The chef is pleased with his new lobster dish.” Here the prefix “dis-” denotes a negation. Similarly, the statement “A prime number cannot be even” can be replaced with the statement “A prime number can be odd” as “odd” is the same as “not even” and vice- versa. While negations are straightforward from a logical perspective, their use in everyday language is more complicated and can be, at times, outright illogical! It turns out that some double negations fail to follow the logical rule that “Not not 푝” is “푝”. Consider the famous song by the Rolling Stones “I can’t get no satisfaction.” Shouldn’t Mick Jagger technically sing “I get no satisfaction?” This logical aberration is apparent in many other classic songs like Pink Floyd’s “We don’t need no education” and Marvin Gaye’s “Ain’t no mountain high enough.” More insidiously, some languages have a built- in grammar that allows double negations to act as negations. In Spanish, for instance, “No hay ningun problema” is translated as the sentence “There is no problem.” However, this Spanish sentence literally states that “It is not the case that there is no problem.” Similarly, in Italian the sentence “Non vedo niente,” which translates to “I see nothing,” actually reads as “I do not see nothing.” This departure from the rule of negation laid out in Table 3 is particularly commonplace in informal speech, dialects (such as African-American vernacular English or Afrikaans), Slavic languages (such as Russian or Polish), and Romance languages (such as Portuguese or Italian). It is also prevalent in some old English literature. For instance, Chaucer writes in his Canterbury Tales “There wasn’t no man nowhere as virtuous […]” Negations in Set Theory A negation can be viewed as the complement of a set. Consider a component statement 푝. This can always be written as “푥 ∈ 푃,” where 푃 is the predicate and 푥 is the subject of the statement. The negation ~푝 is then given by “푥 ∉ 푃” or “푥 ∈ 푃̅,” where 푃̅ is the complement of 푃. For instance, the statement “Peewee is not a scary animal” can be depicted by the following Venn diagram: Make the universal set a box that represents the set of all animals. Now place inside this box a circle representing the set of all scary animals (i.e. a subset of the universal set). Peewee would then be placed inside the box (since he’s an animal), but outside the circle (since he’s not scary). [Insert Venn Diagram with Peewee] Negations With Quantifiers 풑 ~풑 “All … are ….” “Some … are not …” “Some … are …” “No … is …” Table 4: Forms of Negations for Basic Statements with Quantifiers CONJUNCTION For two components 푝 and 푞, the statement “푝 and 푞” is called a conjunction and is written symbolically as 푝 ∧ 푞. Conjunctions are only true when both components 풑 and 풒 are true. Otherwise they are false. Stated differently, conjunctions are false if at least one of its component 푝 or 푞 is false. The basic truth table for this connective is shown in Table 4 below. 풑 풒 풑 ∧ 풒 T T T T F F F T F F F F Table 5: Basic Truth Table for the Conjunction “푝 and 푞” DISJUNCTION For two basic statements 푝 and 푞, the statement “푝 or 푞” is called a disjunction and is written symbolically as 푝 ∨ 푞. Disjunctions are only false when both components 풑 and 풒 are false. Otherwise they are true. Stated differently, disjunctions are true if at least one of its component 푝 or 푞 is true. The basic truth table for this connective is shown in Table 5 below. 풑 풒 풑 ∨ 풒 T T T T F T F T T F F F Table 6: Basic Truth Table for the Disjunction “푝 or 푞” Truth Tables for Compound Statements Tautologies and Contradictions Logical Equivalence De Morgan’s Laws .
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