A logical argument in math

In the next section we will use what we have learned about constructing statements to build arguments with logical statements. We will also use more Venn diagrams to evaluate whether an argument is logical, and introduce how to use a truth table to evaluate a logical statement.

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.

Many scientific theories, such as the big bang theory, can never be proven. Instead, they are inductive arguments supported by a wide variety of evidence. Usually in science, an idea is considered a hypothesis until it has been well tested, at which point it graduates to being considered a theory. For gravity, this happened when Einstein proposed the theory of general relativity. A deductive argument is considered valid if all the premises are true, and the conclusion follows logically from those premises.

In other words, the premises are true, and the conclusion follows necessarily from those premises. Both the premises are true. To see that the premises must logically lead to the conclusion, one approach would be use a Venn diagram. From the first premise, we can conclude that the set of cats is a subset of the set of mammals. From the second premise, we are told that a tiger lies within the set of cats. From that, we can see in the Venn diagram that the tiger also lies inside the set of mammals, so the conclusion is valid.

From the first premise, we know that firefighters all lie inside the set of those who know CPR. From the second premise, we know that Jill is a member of that larger set, but we do not have enough information to know if she also is a member of the smaller subset that is firefighters. Since the conclusion does not necessarily follow from the premises, this is an invalid argument, regardless of whether Jill actually is a firefighter.

It is important to note that whether or not Jill is actually a firefighter is not important in evaluating the validity of the argument; we are only concerned with whether the premises are enough to prove the conclusion. From the first premise, we know that the set of people who live in Seattle is inside the set of those who live in Washington. From the second premise, we know that Marcus does not lie in the Seattle set, but we have insufficient information to know whether or not Marcus lives in Washington or not.

This is an invalid argument. This is an invalid argument, since there are, at least in parts of the world, men who are married to other men, so the premise not insufficient to imply the conclusion. While this example is hopefully fairly obviously a valid argument, we can analyze it using a truth table by representing each of the premises symbolically. We can then look at the implication that the premises together imply the conclusion.Mathematicians often develop ways to construct new mathematical objects from existing mathematical objects.

A logical operator or connective on mathematical statements is a word or combination of words that combines one or more mathematical statements to make a new mathematical statement. A compound statement is a statement that contains one or more operators.

Because some operators are used so frequently in logic and mathematics, we give them names and use special symbols to represent them. Some comments about the disjunction. In everyday life, we often use the exclusive or. Some comments about the negation.

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For example, we would usually say or write :. For example, in Question 1we will assume that each statement is true. In each part, determine the truth value of each of the following statements:.

Which of the four statements [ a through d ] are true and which are false in each of the following four situations? In the preview activities for this section, we learned about compound statements and their truth values. This information can be summarized with truth tables as is shown below.

Rather than memorizing the truth tables, for many people it is easier to remember the rules summarized in Table 2. Conditional statements are extremely important in mathematics because almost all mathematical theorems are or can be stated in the form of a conditional statement in the following form:. It is imperative that all students studying mathematics thoroughly understand the meaning of a conditional statement and the truth table for a conditional statement.

Recall that a quadrilateral is a four-sided polygon. Truth tables for compound statements can be constructed by using the truth tables for the basic connectives. To illustrate this, we will construct a truth table for. The first step is to determine the number of rows needed.

The next step is to determine the columns to be used. One way to do this is to work backward from the form of the given statement. Table 2. The step numbers correspond to the order in which the columns were completed.

Progress Check 2. That is, a tautology is necessarily true in all circumstances, and a contradiction is necessarily false in all circumstances.

2.1: Statements and Logical Operators

Do not delete this text first. Constructing Truth Tables Truth tables for compound statements can be constructed by using the truth tables for the basic connectives.

a logical argument in math

For a truth table with two different simple statements, four rows are needed since there are four different combinations of truth values for the two statements. We should be consistent with how we set up the rows.

All truth tables in the text have this scheme. For a truth table with three different simple statements, eight rows are needed since there are eight different combinations of truth values for the three statements. Our standard scheme for this type of truth table is shown in Table 2.

Answer Add texts here. Exercises for Section 2. Support your conclusion.In the next section we will use what we have learned about constructing statements to build arguments with logical statements. We will also use more Venn diagrams to evaluate whether an argument is logical, and introduce how to use a truth table to evaluate a logical statement.

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.

Episode 1.4: Premises and Conclusions

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. Many scientific theories, such as the big bang theory, can never be proven. Instead, they are inductive arguments supported by a wide variety of evidence.

Usually in science, an idea is considered a hypothesis until it has been well tested, at which point it graduates to being considered a theory. For gravity, this happened when Einstein proposed the theory of general relativity.

A deductive argument is considered valid if all the premises are true, and the conclusion follows logically from those premises. In other words, the premises are true, and the conclusion follows necessarily from those premises. Both the premises are true. To see that the premises must logically lead to the conclusion, one approach would be use a Venn diagram.

Logical Argument

From the first premise, we can conclude that the set of cats is a subset of the set of mammals. From the second premise, we are told that a tiger lies within the set of cats. From that, we can see in the Venn diagram that the tiger also lies inside the set of mammals, so the conclusion is valid. From the first premise, we know that firefighters all lie inside the set of those who know CPR.In logical terms, this three-step process involves building a logical argument.

An argument contains a set of premises at the beginning and a conclusion at the end. In many cases, the premises and the conclusion will be linked by a series of intermediate steps. The premises are the facts of the matter: The statements that you know or strongly believe to be true. In many situations, writing down a set of premises is a great first step to problem solving. Everyone is very excited about the project, but you make some phone calls and piece together your facts, or premises.

So far, you only have a set of premises. Sometimes an argument is just a set of premises followed by a conclusion. In many cases, however, an argument also includes intermediate steps that show how the premises lead incrementally to that conclusion. Using the school construction example from the previous section, you may want to spell things out like this:. But, school begins in September. The conclusion is the outcome of your argument. For the school construction example, here it is:.

In some cases, an argument may not be valid. If not, the argument is invalid. The school construction example argument may seem valid, but you also may have a few doubts. For example, if another source of funding became available, the construction company may start earlier and perhaps finish by September. Thus, the argument has a hidden premise called an enthymeme pronounced EN-thi-meemas follows:. Logical arguments about real-world situations in contrast to mathematical or scientific arguments almost always have enthymemes.

So, the clearer you become about the enthymemes hidden in an argument, the better chance you have of making sure your argument is valid. Building Logical Arguments. He has earned his living for many years writing vast quantities of logic puzzles, a hefty chunk of software documentation, and the occasional book or film review. He likes writing best, though.A logical argument consists of one or more premises followed by one or more conclusions.

The conclusion of a logical argument may be valid or invalid. A premise is a statement that helps support a conclusion.

a logical argument in math

Examples of premises are: The sun is yellow. Socrates is a man. Two distinct lines in a plane either intersect or are parallel. A conclusion is a statement that may follow from the premises. If the logic in the argument is used correctly, the conclusion is valid. If the logic in the argument is not used correctly, the conclusion is invalid.

Examples of conclusions are: The sun is a class G star. Socrates is mortal.

a logical argument in math

Two distinct lines in a plane intersect exactly zero or one times. The letters p and q are often used to represent premises when discussing logical argument in general. The statement: if p then q means, "If the statement p is true, then it follows that the statement q is true. If these types of logical arguments are used incorrectly, any conclusions will be invalid.

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If a shape is a square, then it is a rectangle. HIJK is a square. Therefore q must also be true. Therefore HIJK must also be a rectangle. Indirect Argument If p is true then q is true. HIJK is not a rectangle.

Therefore p can not be true. Therefore HIJK can not be a square. Chain Rule if p is true, then q is true. If a shape is a rectangle, then it is a parallelogram.

Therefore, if p is true, then r is true. Therefore, if a shape is a square, then it is a parallelogram. Or Rule Either p is true or q is true. Figure A is a circle or a square.Name: Email: Password: Leave blank: Free Account Login Click here to access your premium account Username or email: Password: Looking for. Humans can make Mars inhabitable by nuking its poles.

Life is or will be a simulation. Last year, Musk argued the speed at which computer simulations have evolved in recent decades suggests the reality we perceive either already is a simulation or will be at some point in the future. Apple Analyst Shrugs Off Demand Concerns: 'Gross Margin Is The Key'View the discussion thread.

Building Logical Arguments

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Only certain amount of people have access to premium bet tips and strategies. Subscribe to our list to stay informed!. The Saints sit in the middle of the Premier League table ahead of the trip to London, whilst the Gunners dropped to fifth position in the standings following a 3-1 loss to Manchester United.

Both sides are eager to return to winning ways in the national championship and a real football fest. The Reds head into the local derby following back-to-back victories over the likes of Stoke City and Brighton and they are likely to stick to their attacking style of play against the Toffees.

New coach David Moyes is eyein. Watford did well to beat Newcastle United at St. The Eagles sit in the relegation zone in the standings ahead of the visit of Bournemouth and, no doubt, they are going to fight tooth and nail against the Cherries.Some of these types of pages do have incredible value. It is also perfectly natural to have some of these types of links in your link profile if you have a reputable site.

Article syndication (for search engine optimisation purposes) is a big no-no for me in 2017. I still think the place for your articles should be on your blog, on the whole, to attract traffic and links, and to build your reputation as an authority. I once wrote an article that had a signature link back to my site, and while testing how well it had penetrated the SERPs and in how many instances, there was one trusted domain with the content republished, and THAT had attracted a link from a then PR 9 page on a very old trusted site.

I also found another couple of sites that were willing to link to that kind of content for future reference. Use sparingly and with GREAT caution.

I avoid article submission sites these days. Google wants the secondary links (from buzz about the news in your press releases) to count toward your ranking, not the actual press release links.

Stay WELL AWAY from anchor text rich article marketing, press releases and advertorials. Google made that clear when they added the following to their guidelines about what not to do:Back To Table Of ContentsNo. Not for me, just yet.

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Not just now, and not in isolation. The idea of people picking the best sites, rather than counting links the traditional way, is an ideal situation, of course. Google certainly has lots of manual quality raters in 2017.

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Eric Ward calls it right for me:the rankings effect can be directed at specific known circles, friends, connections, etc. The one size fits all search result is headed for the museum. Also, I believe the highest caliber most credible link sources will become that much more important as a trust signal for engines.

Much more hit or miss. Would they base their algorithms around a 3rd party metric. It would surprise me if that were the case. To date, that page has not been indexed, despite having quite a few shares (64 according to the OpenGraph). Do we always listen to Matt Cutts.


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