I can see immediately that there is really only one row that I need to work. See if you can figure out which one it is. I only need to focus on rows where I know the premises might be true or the conclusion might be false.
So, I can safely ignore rows 3 and 4, because the second premise is false on those rows. When I look at rows 1 and 2, I see that the conclusion is true on line 1. So, the only row that has a chance of showing this argument to be invalid is row 2. After working it, I see that one of the premises turned out to be false. On which rows do you think we should focus? Notice that the conclusion is false only on rows 3 and 4. On row 4, though, the second premise is false.
So, the only row that could make this invalid is row 3. Since a conditional with a false antecedent is true, the first premise if true on line 3. The second premise is also true, but the conclusion is false. So, this argument is invalid. There is no line with all true premises and a false conclusion, so the argument is valid. This argument type is called by the Latin name, Modus Tollens.
The third line has all true premises and a false conclusion, so this argument is invalid. On the first line, since A and B are both true, the first premise is true.
So, the second premise is also true. The third premise is false, since it is a negation of a true conjunction. Finally, the conclusion is true. Truth tables can be used to determine the validity of any argument in propositional logic. The only drawback is that they get very large, very quickly. A truth table for an argument with six simple sentences in it has 64 rows—not something that most of us would look forward to doing.
It would be nice if there was a way that we could go straight to the row that showed an argument to be invalid, if there was one. Fortunately, there is, although it can be tricky at times. If it does not lead to a contradiction, then there will be a line like that, and the argument will be invalid.
We can see from the last cell of the row that the conclusion is also true under such an assignment. So this argument has been shown to be valid. In general, to determine validity, go through every row of the truth-table to find a row where ALL the premises are true AND the conclusion is false.
Can you find such a row? If not, the argument is valid. If there is one or more rows, then the argument is not valid. Note that in the table above the conclusion is false in the second and the forth row.
Why don't they show that the argument is invalid? Look at the truth-table, and determine which line is supposed to show that? To show that a sequent is invalid, we find one or more assignment where all the premises are true and the conclusion is false.
Such an assignment is known as an invalidating assignment a counterexample for the sequent. To help us calculate the truth-values of the WFFs under each assignment, we use the full truth-table method to write down the truth-values of the sentence letters first, and then work out the truth-values of the whole WFFs step by step.
It turns out that this complex expression is only true in one case: if A is true, B is false, and C is false. When we discussed conditions earlier, we discussed the type where we take an action based on the value of the condition. We are now going to talk about a more general version of a conditional, sometimes called an implication.
Implications are logical conditional sentences stating that a statement p , called the antecedent, implies a consequence q. Implications are a logical statement that suggest that the consequence must logically follow if the antecedent is true. Notice that the statement tells us nothing of what to expect if it is not raining. If the antecedent is false, then the implication becomes irrelevant. There is only one possible case where your friend was lying—the first option where you upload the picture and keep your job.
In traditional logic, an implication is considered valid true as long as there are no cases in which the antecedent is true and the consequence is false. It is important to keep in mind that symbolic logic cannot capture all the intricacies of the English language. For any implication, there are three related statements, the converse, the inverse, and the contrapositive.
Looking at truth tables, we can see that the original conditional and the contrapositive are logically equivalent, and that the converse and inverse are logically equivalent.
A logical argument is a claim that a set of premises support a conclusion. There are two general types of arguments: inductive and deductive arguments. An inductive argument uses a collection of specific examples as its premises and uses them to propose a general conclusion. A deductive argument uses a collection of general statements as its premises and uses them to propose a specific situation as the conclusion. Notice that the premises are specific situations, while the conclusion is a general statement.
In this case, this is a fairly weak argument, since it is based on only two instances. An inductive argument is never able to prove the conclusion true, but it can provide either weak or strong evidence to suggest it may be true. 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". 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. 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. You could restate it as "It's not the case that both x is rational and y is rational". The word "both" ensures that the negation applies to the whole "and" statement, not just to "x is rational".
By DeMorgan's Law, this is equivalent to: "x is not rational or y is not rational". Alternatively, I could say: "x is irrational or y is irrational". Putting everything together, I could express the contrapositive as: "If is irrational, then either x is irrational or y is irrational". As usual, I added the word "either" to make it clear that the "then" part is the whole "or" statement.
Show that the inverse and the converse of a conditional are logically equivalent. Let be the conditional. The inverse is. The converse is. I could show that the inverse and converse are equivalent by constructing a truth table for. I'll use some known tautologies instead. Start with :. Remember that I can replace a statement with one that is logically equivalent.
For example, in the last step I replaced with Q, because the two statements are equivalent by Double negation.
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