We should be clear that the word contra positive has Different CONTEXTS.

Aristotle used the word quite differently than mathematicians in the late 1850's.

Aristotelian logic uses semantics to evaluate propositions and not just swapping symbols.

Aristotle proved that semantically some propositions could no be validly and soundly contraposed: you cannot contrapose an E proposition. That is an E proposition is NO S is P.

The No quantifier cannot be contraposed. What you will find some inferences will be true and some will be clearly false. And validity expresses it is impossible to have a false conclusion from TRUE propositions.

Philosophers call this mathematical inference you call contrapositive TRANSPOSITION. All you do is swap positions and negate the variables. The original term contraposition was more involved than TRANSPOSITION. The original contraposition term had three steps you had to complete.

The purpose of mathematical logic is not the same as Aristotelian logic. Mathematical logic simulates human communication, whereas Aristotelian logic had the purpose of evaluation in SOUND arguments to prevent deception. Sound arguments are arguments that must have true premises and also must be formally valid. Valid also can refer to complete arguments and can refer to direct inferences which is seen in mathematical logic. What you call contraposition does NOT refer to valid arguments but valid propositional inferences.

> @ said:

> @ said:

> Treating yo self is an important part of the process leading up to test day as well!

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> So, in othttp://imgur.com/CQhyZOD

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> HAHA! Ice cream > LSAT ;)

It seems to express lsat --> ice cream not the other way around.

I think you are missing the point I am making. I am saying that studying logic should include rules that allow the students to convert those English sentences to arrive at the correct solutions. What you seem to be expressing is the whatever works under the circumstances is what we will use. This is called pragmatism. This is not the same as saying you study logic or you teach logic.

The problem now is what do you mean by conditionally valid.

Conditionally valid usually refers to actual existing things. I am not giving my opinion here.

Aristotelian logic does not assume anything technically. I am aware people use that terminology but if you evaluate it you will find it to be untrue. It can be true in some cases but not 100%.

Deductive reasoning is about methods that work 100% of the time and no less. Scientific reasoning makes up the rest 0 -99%. This is why PURE deductive reasoning is not a science.

At this very moment I am looking a respected logic book authored by Patrick Hurley:

" for example, the AAI-1, is valid from the Aristotelian standpoint If the subject of the conclusion (the minor term) denotes actually existing things."

There is a thread about IF and ONLY IF arguments and how they work. What Hurley is saying is Equivalent to "if the subject of the conclusion (the minor term) denotes actually existing things, THEN the AAI-1 is valid from the Aristotelian standpoint."

I am sorry you disagree. I am not trying to be a trouble maker here. I would like clarity though just in case I made a mistake.

@ said:

Thank you, @, for explaining logic. I think 7Sagers do understand what "only if" means logically since we learn it in the Core Curriculum. As you can see from J.Y.'s comment, we were having a discussion on how ambiguous English can be. :smiley:

Well yes I can agree, but the issue is logical rules are supposed to eliminate that ambiguity. I think what is expressed in this thread is also some students still struggle with ONLY IF and IF claims because they are either not given a universal rule or they were given the rule and they don't understand it. I would think the point of logic is to make life go a bit easier regardless the subject matter and not spots here and there. Most people unfamiliar with logical concepts think everything is already subjective. Subjective techniques which work only tuesdays and thursdays is not as valuable as universal techniques.

I am saying that the form of the argument alone will not always give the correct conclusion. Your conclusion may be true by itself. Your conclusion is not true solely based on the premises.

What logical forms are supposed to do is provide a UNIVERSAL away of getting conclusions no matter what the subject matter is about.

What appears to me in some cases here is that some people are evaluating the truth of the sentences and making rules based on that. So you make sure the conclusion is true based on you making subjective rules.

I have no doubt all people see that my examples have issues and have the SAME form. I used different words in the same position as their words and the truth value changes hence the invalid status.

@ said:

I may be missing something here but the original post says:

[Argument #1]

Premise 1: A ---> C

Premise 2: B ---> C

Conclusion: ~A some ~B

This argument is valid because

Premise 1: Dogs ---> Mammal

Premise 2: Cats ---> Mammal

Conclusion: ~Dogs some ~Cats

Premise 1: All dogs are mammals.

Premise 2: All cats are mammals

Conclusion: Therefore, there must be at least one thing in the world that is neither a dog nor a cat.

I believe that the "Fallacy of the undistributed middle" is:

[Argument #2]

Premise 1: All dogs are mammals.

Premise 2: All cats are mammals

Conclusion: Therefore, all dogs are cats.

[Argument #2] is invalid.

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Notice please when I say an argument is invalid I can state what fallacy it commits. I am not expressing what I think personally. All invalid arguments commit a fallacy. So when anyone says an argument is invalid the person ought to be able to name the fallacy. That is you are not supposed to claim an argument as invalid and walk away. :)

I think people in this Forum understand "validity" is not something personal so no worries :smile:

@ said:

I may be missing something here but the original post says:

[Argument #1]

Premise 1: A ---> C

Premise 2: B ---> C

Conclusion: ~A some ~B

This argument is valid because

Premise 1: Dogs ---> Mammal

Premise 2: Cats ---> Mammal

Conclusion: ~Dogs some ~Cats

Premise 1: All dogs are mammals.

Premise 2: All cats are mammals

Conclusion: Therefore, there must be at least one thing in the world that is neither a dog nor a cat.

I believe that the "Fallacy of the undistributed middle" is:

[Argument #2]

Premise 1: All dogs are mammals.

Premise 2: All cats are mammals

Conclusion: Therefore, all dogs are cats.

[Argument #2] is invalid.

@ said:

Notice please when I say an argument is invalid I can state what fallacy it commits. I am not expressing what I think personally. All invalid arguments commit a fallacy. So when anyone says an argument is invalid the person ought to be able to name the fallacy. That is you are not supposed to claim an argument as invalid and walk away. :)

I think people in this Forum understand "validity" is not something personal so no worries :smile:

Well thank for the response. The issue with what you are doing is STILL a problem. The premises are all claims and you swap it for a SOME because you see the sentence would be true. This is not a legit method because of other issues but I must introduce this here a bit. Consider

All unicorns are animals with a horn.

All rhinos are animals with a horn.

Therefore there must be at least one rhino that is a unicorn.

Controposing this gives FURTHER problems if you understand what contrapositive really means and not the symbol manipulation.

The issue here is called existential import: referring to some non-existent object namely UNICORN in the example. SOME we all agree means AT LEAST One. How can something that does not exit refer to at least one? These quantifies are CONTRADICTORY.

With that said you MUST not CHANGE quantifiers without justification. You start with all you must FINISH with all. You switching terms without justification is what Mathematica's changed logic for. This is a fallacy called the existential fallacy: going from an all to a some in reasoning. All fallacies have true instances where the premises and conclusion are true. The point is the fallacy has cases where it will be FALSE-- that is the form is not always 100% true.

Well the mechanical rules do indeed work: that is you should never be able to get the wrong answer strictly following the rules. All cases you get it wrong is a case you broke at least on or more of the rules. Test this for yourself.

The rule is any words after IF is the subject term in philosophy. Mathematics calls this same thing the hypothesis.

If s then p. Where the s is placed is called the subject or antecedent term. The p is called the consequent. Mathematics uses the term conclusion instead of consequent for the same thing.

For example "you can have dessert if you eat your vegetables" is translated by the rule what follows IF goes first : eat your vegetables then you can have dessert -- not the other way around.

There is a distinction between ONLY IF. "You will make it to heaven only if you believe Jesus Christ is your savior" does not follow the IF rule above. In the case where the words are adjacent the rule is do nothing and put the symbolization in the middle. So let H= you will make it to heaven' let J = you believe Jesus Christ is your savior. The symbolization is H--> S. notice the symbols basically ignore the ONLY IF. Suppose you disagree then what? Well what we want is rules that always work not only tuesdays. Notice what I did there? I used a statement with ONLY. Here I convert my last statement using the RULE: if the rule works then it is Tuesday. So the rule is what follows ONLY is a consequent aka the conclusion of the conditional in mathematics.

So in summary IF indicates the antecedent. The clause ONLY IF means the word order is correct and ignore ONLY IF. Here is another case for you.

"You will be hired only if you meet our requirements" means if I am hired then I must be a person who meets the requirements. H-->R. I have an inkling some of you will say this is not true in that case:nepotism would not exist if that were the case.

Well you could question the truth of the claim then. But the truth value of any conditional with a false first part will be valid.

@ said:

Hi @ :)

@ said:

Akistotle

The form in your post is not the same as the original post and not the same as mine.

The original post said:

@ said:

A ---> C

B ---> C

/A some /B

and I said:

@ said:

@ said:

Can some one clarify how this form is valid???

Are you talking about this form?

A ---> C

B ---> C

/A some /B

The above is a valid argument form.

Your example of an invalid argument:

All men are things that belong to a species.

All men are mortal beings.

All mortal beings are things that belong to a species.

Your example can be represented as

A → B

A → C

C → B

and it is an invalid argument form.

Thank you for the correction. I thought the middle term was placed in the first position in the original.

I still need some clarity on how even this form is valid with the middle term in this position.

First in symbolic logic truth tables are used to establish validity. Inference rules are named after truth tables. There are no inference rules that allow this form. Which inference rules establish this form above.

I am aware there is an inference rule hypothetical syllogism but the terms are not in the correct position to use hypothetical syllogism.

Let me show how this form is invalid:

All dogs are mammals.

All cats are mammals.

All cats are dogs.

Contraposing the conclusion gives:all non-dogs are non-cats.

This is still false with true premises.

In symbolic logic:

If something is a dog then it is a mammal. D --> M

If something is a cat, then it is a mammal. C-->M

If something is a cat then it is a dog. C-->D

Controposing the conclusion gives ~D-->~C

If something is not a dog then it is not a cat. The ~ symbol is called tilde and the official symbol for NOT.

I claim this is a case the premises are true and the conclusion is false even in this form.

The fallacy in this form is undistributed Middle.

What someone has shown is that you can form true premises and a true conclusion with this form BUT it depends what words you use in the form.

The words I chose do not workin this alleged valid form. Is the form valid if I can show a single case where the premises are true and the conclusion false?

Notice please when I say an argument is invalid I can state what fallacy it commits. I am not expressing what I think personally. All invalid arguments commit a fallacy. So when anyone says an argument is invalid the person ought to be able to name the fallacy. That is you are not supposed to claim an argument as invalid and walk away. :)

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None of that is wrong @ , but like you say:

@ said:

Some of what works for the LSAT will not fly In a college class.

We're really only concerned with the LSAT here. Not with a college class. And as far as the LSAT goes, these distinctions don't matter. Sure, a lot of LSAT logic doesn't align with proper logic. For anyone trying to learn Logic, the LSAT is not necessarily a great tool.

Well the distinctions are absolutely important. Why would you say such a thing? I don't see a disclaimer to your students here that these techniques you teach are specialized to get students to reach a goal and that these teachings will not always serve you well outside of this subject.

I would think most people THINK of logic as a universal topic. That is most students come to sites and courses about logic or reasoning thinking they will learn something they can apply outside a class setting. The deception kicks in any place that teaches logic or reasoning (be it the place is a University, College, High school,Website, etc.) when there is no disclaimer stating that what we teach will not always work in every environment in reality.

No one advertises Material logic --aka specialized circumstance logic. The student will likely think "what do I need this for if it works only on SOME days of the week and only in special places?"

Deception is in play when one fails to mention RELEVANT information that could change the outcome.

This is the focus of how original logic was taught in the field of Philosophy. So the term LOGICis being brutalized because people are not clear there are different types, methods and PURPOSES of logic. The majority of students think logic is all the same. All mathematics teaches logic is mathematics which is not the case. The use of the term logic will become useless because different people teach logic one way WITHOUT disclosing there are indeed other ways as well depending on the focus. This deceives the student once he learns he has been duped: what he learned failed in reality and doesn't know why. The teacher forgot to mention this is not universal!

Notice I used the term deductive reasoning. This implies a universal topic and universal concepts at play.

Thank you for understanding my post. What I really wanted to make clear is that the subject you allegedly learn logic from does matter. The common deception is that most people think or believe logic is logic. That all logic is the same. We now see this is not the case. For the LSAT and Sage your logic is different but the original philosophical roots are still present. Rhetoric and law is the pattern of logic that is concerned with PERSUASION. Mathematical logic is the pattern of logic focused on Validity. Science reasoning mostly concerned are psychologist are focused on how humans do reason and how to be practical with it; that is how can we symbolize those practical arguments. Philosophy teaches deductive reasoning which all those other topics use partial knowledge of and they name it logic. The purpose of logic that I learned was defined as the act of correct reasoning which detects deception. Out of all the other subjects that teach their own logic none of those are designed to evaluate arguments for deception. That is deductive reasoning is not to simulate how people do in fact reason. Deductive reasoning is not about persuading an audience such as a jury.

Perhaps you can use detection techniques once in a blue moon but it is not your primary goal for using logic. Mathematical logic focuses on validity even when the premises make no sense: they make a context shift in the term validity. So in reality this will often fail because concepts were removed from mathematical logic.

These forms of logic come from one universal topic: deductive reasoning. That is if you study proper deductive reasoning you will be able to do or follow all the different purposes of logic easier. Nearly all topics use deductive reasoning. This does not make them all the same thing -- aka philosophy. For instance almost all science requires a lot of mathematics but getting a degree in physics does not mean you get a mathematics degree as a bonus. The mathematican will know more about math than the physicist no matter how good at math the physicist is; i.e., no matter how good at algebra, trigonometry and calculus he is.

@ said:

It is logically valid. I think you may have misread the original post, @.

If I did make a mistake can you point it out please?

I referred specifically to the argument form being NOT valid. I agree that a proposition can be contraposed in the conclusion. The problem is that conclusion does not follow from the given premises. That is, there are cases where I can show you I can use that form of argument and find two true premises and a false conclusion. That is what I demonstrated in my first post.

Let me point out again the rhetorical form of logic may define valid differently from deductive reasoning. I can tell you deductive reasoning and mathematical logic express that a valid argument is an argument where the conclusion is impossible to be false with two true premises. That is if the premises are true the conclusion must absolutely be true as well.

What seems to be happening here is you found an true instance of a poor argument form. That is if I substitute different words in the same spot as your form the argument truth value can change. This I demonstrate again:

All men are things that belong to a species.

All men are mortal beings.

All mortal beings are things that belong to a species.

This argument has a middle term named MEN because it repeats in the premises and NOT the conclusion. Notice the argument validity is not based on content value but concepts. Why this is invalid is a concept called distribution.

What the conclusion distributes is NOT distributed in the conclusion. So this a shift in discourse.

Akistotle

The form in your post is not the same as the original post and not the same as mine. Your example is reversed. Your label C is in the wrong spot. Can I use terminology here: the label C is called a middle term because it repeats in the premises and NOT in the conclusion.

So the form is All C are B.

All C are A.

Is distinct from All B are C

All A are C.

So this position of the middle term is important. It can make an argument immediately invalid.

Would it not be easier to learn concepts regarding propositional relations? One must be careful because what I see is a mix of logics here.some people are confusing mathematical logic with classical logic. When you use the symbols it is absolutely not always equivalent to the classical formation. I can provide samples if needed. Let me explain WHY though.

The vocabulary terms change contexts is the simplest answer. For example Your so called contropositive has a different definition in Aristotelian logic. In Aristotelian logic you cannot contrapose a NO quantifier statement. Contropositive had a distinct meaning which is not what you guys use. In math they use contropositive to mean the exact same thing philosopher call TRANSPOSITION which is the more accurate name because you will never confuse the two context that way. Your contrpositve is simply symbol manipulation. I bring this up because logic is associated with philosophy, rhetoric / law, psychology, and mathematics which ALL have different purposes and differ in vocabulary and methods. Some of what works for the LSAT will not fly In a college class. I could say the vice Versa here.

The contradictory is how you negate any statement in mathematical logic. In Aristotelian logic it differs. Some plants are roses can be negated as some plants are non- roses. Before you says it is the same thing as a NOT think twice. Anyone familiar with the square of opposition would have to disagree and say a non is not the same as a NOT. The some s is p proposition is called an I proposition. The some s is not p is called an O proposition and they ARE NOT equivalent.

Can some one clarify how this form is valid??? What I see is an INVALID conclusion from the premises. BUT I also see double word play on the term VALID. The conclusion is inthe contrapositive form of the real conclusion.

So to say the conclusion and it's contrapositive are valid inferences is quite distinct from saying premise one and two MUST bring forth the alleged conclusion.

Here is a clear counter example of why the argument form is INVALID:

All dogs are mammals.

All dogs are canines.

All canines are mammals.

Notice all the claims are true indeed THAT example but it commits a formal fallacy called illicit minor.

Here is another example:

All cobras are snakes.

All cobras are venoumous animals.

All venomous animals are snakes.

Clearly we have true premises and yet the conclusion is FALSE. So we must be clear when we say VALID. Are we talking about transforming a proposition by controposing the proposition or do we mean to refer to THE RELATIONSHIP between the premises and the conclusion.