Ya, well, we would disagree there.

Ya, well, we would disagree there. — Sam26

Looks like it. Anyway, kudos to you for starting this thread. It's a good place to brush up on my logic. I hope others feel the same way. Good luck. :smile:

Thanks.

Hey! Does anyone have time to help me solve a conditional proof with assumption???

Logic Post 12

Continuing with symbolization...

What is a statement-form? A statement-form is a proposition that consists of logical symbols (∨ · ⊃≡ ~) and statement variables (p, q, r, etc.). For example,

p · (q ∨ r), which can be derived from the statement "If William is a liar, then either he is stupid or he is crazy."

Any substitution instance of a statement-form is any statement with that form. For example, the substitution instance of ~ p would stand in for any statement with the form "It is not the case that George Washington was our 4th president." Statement forms (~ p) are not intrinsically true or false, only substitution instances are true or false. In other words, only where ~ p represents a particular statement, is it said to be true or false.

Truth-Tables

Truth-tables allow us to determine the truth-values of a particular statement given certain input values. For example, if we allow the letter p to serve as a marker for a given statement, and let the letter T stand for its truth-value true, and the letter F for the truth-value false, we can then show that any simple statement has two possible truth-values.

The following is an example:

p
---
T
F

Any compound statement p and q has only four possible sets of truth-values. The following is an example:

p-----q
_____
T-----T
F-----T
T-----F
F-----F

If we used truth-functors that involved three different statements, then we would need three statement-variables (p, q, and r). In the above example using two statement-variables it required four lines. If we used three variables it would involve eight lines. The following is another example:

p-----q-----r
_________
T-----T-----T
T-----T-----F
T-----F-----T
T-----F-----F
F-----T-----T
F-----T-----F
F-----F-----T
F-----F-----F

As you can see this can be quite cumbersome, because a table with one variable has 2^1= 2 lines; a table with two variables has 2^2 = 4 lines; a table with three variables has 2^3 = 8 lines; and so on.

I'm skipping over categorical deductive logic, which was originated by Aristotle in the Organon.

Logic Post 13

Continuing with truth-tables

Negation

The · symbol that is used as a sentence connective is a kind of operator, that is, it can operate on two separate sentences to produce a third compound sentence A · B. For instance, the operator "It is well known that" operates on the following sentence "Abraham Lincoln was the sixteenth president of the United States" to construct the compound sentence "It is well known that Abraham Lincoln was the sixteenth president of the United States".

The tilde symbol ~ that is used to symbolize negations is an operator of this kind, i.e., it can generate a new sentence out of just one starting sentence. The negation operator is the only one that does this in standard sentential logic. All (or almost all) others including "It is well known that" are non-truth-functional operators.

Negation is one of the easiest truth-functional operators to learn, because it only operates on individual sentences. The operation of the negation symbol is straight forward, because if you negate a true sentence, you get a false sentence, and if you negate a false sentence, you get a true one.

We negate sentences in English in a variety of ways. Examples are given in the following sentences:

1) "Eleven is not even."
2) "Eleven is uneven."
3) "It is not the case that eleven is even."
4) "It is false that eleven is even."
5) "Eleven is odd."

The above five examples were taken from Kegley and Kegley's Introduction to Logic p. 226.

Finally, let us use the following truth-table to define the truth-functor negation:

The statement "The earth has one moon" has two possible truth-values, and the statement "It is not the case that the earth has one moon" has two possible truth-values. Hence, letting p stand in for each of the aforementioned statements we get the following truth-table:

p------- ~p
_______
T---------F
F---------T

There can also be statements that involve more than one negation. Consider - "It is false that it is not the case that Abraham Lincoln is not tall." Let p represent "Abraham Lincoln is tall" and we will construct the following truth-table.

p--------- ~p---------- ~~p---------- ~~~p
____________________________
T-----------F--------------T----------------F
F-----------T--------------F----------------T

It is best when using the negation symbol to express it by the statement, "It is not the case that," which reverses the truth-value of the statement. While it is true that the ~ symbol is equivalent to most English uses of the word not, it doesn't always convey the correct meaning. In logic, ~ always means contradictory. However, there are uses of the word not in English that do not convey a contradiction. For example, "Some males do not smoke pot" does not contradict the statement that some males do smoke pot. In other words, both statements can be true, so the not in this case doesn't involve a contradiction.

Logic Post 14

Conjunction

When two sentences are joined together by the truth-functor and, they are called conjuncts; and compound sentences formed by the truth-functor and are called conjunctions. An example of a simple conjunction is "Wittgenstein was a philosopher, and he was also an engineer"; and since we are using the truth-functor symbol · this sentence would be symbolized in the following way:

Using p to refer to "Wittgenstein was a philosopher", and q to refer to "He was also an engineer" - the sentence would be symbolized p · q.


There are a number of ways to express a conjunction in English.

1) "Mathew stays but Jane leaves."
2) "Mathew stays, however Jane leaves."
3) "Mathew stays, moreover Jane leaves."
4) "Mathew stays although Jane leaves."
5) "Mathew stays yet Jane leaves."
6) "Mathew stays even though Jane leaves."

The previous six examples taken from Kegley and Kegley's book, Introduction to Logic, p. 228.

In order for someone to commit himself to the truth of a conjunctive statement, that person would have to accept that both p and q are true. Otherwise, the conjunctive statement is false. This is clearly seen in the following truth-table:

p--------q--------p · q
_______________
T--------T----------T
F--------T----------F
T--------F----------F
F--------F----------F

As can be seen in this truth-table a conjunctive statement is only true if both of its component statements are true. So, what this means is that if we are committed to the truth of p, and committed to the truth of q, then we are committed to the truth of p · q. The argument p · q is therefore valid for the conjunctive form.

p
q
__
Therefore, p · q.

Later you will come to know this as one of the rules of inference known as conjunction.

Those of you who remember Ephilosopher (the philosophy forum), most of this comes from a thread I did back then on logic. I'm revising parts of it as I go along, but much of the work is already done. This is why I'm able to post so fast. I'll also be adding to the original thread, so that part will take more work.

Logic Post 15

Disjunction

We will now consider a statement joined by the truth-functor symbol v. When two statements are connected by the truth-functor or it is called a disjunction. Each component statement is called a disjunct. An instance of a disjunction is: "Either Plato was a philosopher, or he was a physician." This disjunctive sentence is symbolized as:

p v q

When we state a disjunct we are putting forth two possibilities. These two possibilities have two senses - one is called inclusive, and the other is called exclusive. The inclusive sense is when we are admitting the possibility that each of the disjuncts may be true. The inclusive sense is represented by the following truth-table:

p--------q--------p v q
_______________
T--------T----------T
T--------F----------T
F--------T----------T
F--------F----------F

As we can observe by this truth-table the only case in which the inclusive disjunction is false is when both disjuncts are false.

As for the exclusive sense of the disjunction, this is a case where we may want to rule out both disjuncts as true. As in the following example:

George is in France, or George is in Italy.

This is clearly an exclusive sense, because George cannot be in both places at once. The exclusive sense says either p is true or q is true, but not both. However, this sense is not used in truth-functional logic, although, the two senses share a common trait, that is, that at least one of the disjuncts must be true (Kegley and Kegley, Introduction to Logic, p. 232).

Given the definition of a disjunctive statement, that it is false only when both disjuncts are false, and true in each of the other three alternatives, we get the following two valid disjunctive argument forms:

p v q
~p
____
Therefore, q.

p v q
~q
____
Therefore, p.

These are called disjunctive argument forms, and they are another one of the rules of inference that you will learn later.

There is an interesting case of the disjunction involving a negative, which can be written in two ways. The first is ~(p v q), and the second is (~p · ~q) - these are equivalent forms, and can be seen as such in the following truth-tables.

p--------q--------p v q-------- ~(p v q)
_________________________
T--------T----------T----------------F
F--------T----------T----------------F
T--------F----------T----------------F
F--------F----------F----------------T


p--------q-------- ~p-------- ~q-------- ~p · ~q
________________________________
T--------T----------F-----------F-------------F
T--------T----------T-----------F-------------F
T--------F----------F-----------T-------------F
F--------F----------T-----------T-------------T

Logic Post 16

Conditional Statements and Material Implications

A conditional statement is a statement that is composed of two component statements joined by the truth-functor if...then (⊃). These statements can also be referred to as hypothetical statements or material implications. Speaking of implication keep in mind that the English word implies more than one meaning, and these meanings can be conveyed using the connective if...then. Statements using the aforementioned connective can imply a logical, causal, definitional, or decisional relationship.

The component statements that make up the conditional are called the antecedent and the consequent. The antecedent precedes then, and the consequent follows then in the conditional. A simple conditional statement that implies a logical relationship is the following:

If John is a philosopher, then John is a thinker.

p = John is a philosopher.
q = John is a thinker.

p ⊃ q

The connective if...then is defined using truth-tables in the following way: If a substitution instance of p or any other variable is true, and a substitution instance of q or any other variable is false, then the substitution instance of p ⊃ q is false. For all other substitution instances of p and q, the statement form p ⊃ q is true. This can be clearly seen in the example that follows:

p--------q------------p ⊃ q
___________________
T--------T---------------T
T--------F---------------F
F--------T---------------T
F--------F---------------T

Therefore, the preceding truth-table says that it is never the case that p can be true and q false, which is to say, that it denies the conjunction of its antecedent with the negation of its consequent.

Conditional statements are unlike conjunction and disjunction in that the order of the statements in a conditional make a difference when constructing truth-tables. This can be clearly seen in the following instance:

If the car runs out of gas, then the car will stop running.
If the car stops running, then the car runs out of gas.

What falsifies each of these statements is different, i.e., in the first statement p ⊃ q, when p is true and q is false we get a false statement, but if we reverse the p and q, as in q ⊃ p, we get a false statement when p is false and q is true. Hence, the truth-tables will look like the following:

p ------- q ------------ p ⊃ q ------------ q ⊃ p
_________________________________
T---------T ---------------- T ----------------T
T -------- F ---------------- F ---------------T
F -------- T ---------------- T ----------------F
F -------- F ---------------- T ----------------T

If you look at the main connectives in each of the conditionals you can see that what falsifies one does not falsify the other.

Remember that in a conditional statement the antecedent implies the consequent, which means that if the antecedent is true, then the consequent is true. However, this is hypothetical, and as such, the conditional statement does not tell us anything about the truth of its component statements; it is only saying that IF the antecedent is true, the consequent is true.

There are more complicated issues when it comes to conditional statements. For instance, "If John or Mary go to the movies, then Mary or John will go to the movies", is a logical relationship. However, in the statement "If John goes to the movies, then Mary will go to the movies," is a factual relationship.

The most important way in which conditional statements differ, has to do with their truth-functionality, that is, most conditionals uttered in out daily lives are not as straightforward. Consider the following:

"If the Red Sox beat Pittsburgh, then the Red Sox will win the World Series" - symbolized R ⊃ W. Now let us suppose that someone places a bet that this conditional is true. We know that if the Red Sox beat Pittsburgh, and yet the Red Sox fail to win the World Series, then obviously R is true and W is false, and the person fails to win the bet. On the other hand, if the Red Sox beat Pittsburgh, and they also win the World Series, then the statement is true and the person wins the bet. Now let us look at this in a standard truth-table where the two results are represented.

R ------- W -------------R ⊃ W
_____________________
T -------- T -----------------T
T -------- F -----------------F
F -------- T
F -------- F

A problem arises if the Red Sox fail to beat Pittsburgh, because then it is not clear what to say about this conditional as it relates to the last two results. However, we cannot simply leave the last two results blank even though it would be odd to call the statement true, it would definitely not be false. Hence, when faced with a choice (for our purposes) we will construct our truth-tables so that all statements with false antecedents are true.

A conditional statement, as currently defined, provides us with a minimal common meaning for uses of the "if... then" statement, which once again means that the consequent cannot be false if the antecedent is true. So the point is, that since this is a minimal condition for the meaning of a conditional statement, it only partially satisfies the uses of the "if...then" statement in English, just as the disjunction only partially fulfills all the meanings of or in English.

Finally, it should be noted that there are other more powerful forms of logic that are better equipped to handle these kinds of problems. However, it is a good idea to get a good handle on sentential logic first before going on to master other forms of logic (like quantification theory).

I have not completely analyzed conditional statements and material implications. If you want a more complete analysis, you'll have to do some research. This thread is a guide, nothing more.

Logic Post 17

There is much that can be accomplished using the ⊃ symbol. For instance, using the definition of the ⊃ symbol we can get the valid argument form known as Modus Ponens.

Modus Ponens

p ⊃ q
p
_____
∴(Therefore), q

How do we know that this necessarily follows? Because using our truth-tables we know that any instance where the antecedent is true, the consequent is true. Hence, using Modus Ponens, we have constructed a valid argument form. Keep in mind the differences between validity and soundness, which we discussed earlier.

Another valid argument form that follows from the definition of the symbol ⊃ is called Modus Tollens, i.e., if we deny the consequent, we can conclude the denial of the antecedent.

Modus Tollens
p ⊃ q
~q
_____
∴ ~p

There are two corresponding fallacies that are derived from the Modus Ponens and Modus Tollens. First, the valid form...

Modus Ponens
____________
p ⊃ q
p
_____
∴ q

"If we have desegregation we will have some busing.
We have desegregation.
______________________________________
Therefore, we will have some busing(Kegley and Kegley, p. 240)."

The invalid form is the following:

p ⊃ q
q
_____
∴ p

If we have desegregation we will have some busing.
We have some busing
__________________________________
Therefore, we have desegregation.

The above invalid form commits the fallacy of affirming the consequent. We know this because the definition of a conditional for our purposes, states that we cannot have false consequent when the antecedent is true. We can see this in line two the following truth-table.

_______________________________________________________

The fallacy that corresponds to Modus Tollens is the fallacy of denying the antecedent. Let us first look at the valid form...

Modus Tollens
____________

p ⊃ q
~q
______
∴ ~p

"If the paper burns, there is sufficient oxygen present.
There is not sufficient oxygen present.
__________________________________________
Therefore, the paper does not burn (Kegley and Kegley, p. 240)."

This is obviously a valid form, however, if we deny the antecedent, then we commit the fallacy of denying the antecedent.

Fallacious Form
_____________
p ⊃ q
~p
______
∴ ~q

If the paper burns, there is sufficient oxygen present.
The paper does not burn.
__________________________________________
Therefore, there is not sufficient oxygen.

We can obviously see that this could be false, because it is certainly possible that oxygen is present and the paper still will not burn. Maybe the paper is wet, or maybe there is not enough oxygen.

Logic Post 18

Biconditionals

Two statements are materially equivalent if they have the same truth-values. The symbol ≡ is the symbol we are using in this thread to stand for material equivalence. Thus, if we say that "Two times two equals four, if and only if, four times one equals four," then the two statements are materially equivalent, since both have the same truth-value.

Material Equivalence

p ≡ q

Truth-table
Given the definition of material equivalence - that two statements are materially equivalent if and only if they have the same truth-values - we can see this is so by looking at line one and line four.

These compound statements are more commonly referred to as material biconditionals or just biconditionals, because they are equivalent to material conditionals. For example, the material biconditional "Two times two equals four, if and only if, four times one equals four" in symbolic form looks like this, p ≡ q; and it is equivalent to the two-directional conditional "If two times two equals four, then four times one equals four, and if four times one equals four, then two times two equals four," which is symbolized as, (p ≡ q) · (q ≡ p).

Remember that the material biconditional only captures the minimal truth-functionality of the English biconditional. There is no connection implied by the component statements. It only states that they both have the same truth-value.

Logic Post 19

Tautologies, Contradictions, and Contingent Sentences

Before going on we will examine the differences between tautologies, contradictions, and contingent sentences.

First, a tautology is a statement form that is true under all possible interpretations of its variables; or another way of saying it is that a sentence is tautological if and only if there is no interpretation of the sentence which produces a false truth-value. Keep in mind that it is under the main connective that one looks to find the appropriate truth-value. Here are some examples:

1)
2) 3)
In the above three examples we are using brackets around the main connective to ONLY illustrate our point. And the point is, that in each of these truth-tables under the main connective we have all Ts, that is to say, all substitution instances are true. Therefore, given our definition of a tautology each of the above examples are tautological.

There are a couple of feature that we need to be aware of when thinking about tautologies. First, tautologies tell us nothing about the world, i.e., they are noninformative. For instance, if I say "Either it is snowing, or it is not snowing"- this statement is necessarily true, but it tells us nothing about the whether.

Second, we can determine the truth-value of a tautology a priori, which simply means, that we can know the truth of the statement quite apart from the evidence.

A statement whose truth is logically impossible is called contradictory; or a statement is a contradiction if and only if there is no line of the truth table that shows a truth-value of true. The following statement forms are contradictory, and can be seen as such by their truth-tables:

Note the line of truth-values under the bracketed main connective.



Any statement such as "Triangles have three sides and triangles do not have three sides" is contradictory in virtue of its form, p · ~p.

We have discussed statement forms that have all true truth-values (tautologies), and statement forms that have all false truth-values (contradictions). We will now complete this section with a definition of contingent statement forms.

The first two statement forms had either all true truth-values, or all false truth-values. And as you would expect the final statement form has a mixture of both true and false truth-values. It is considered a contingent statement because its truth-values are not dependent upon logic alone, but are contingent upon some state-of-affairs. For instance, the statement "The glass is sitting on my desk, or it is not sitting on my desk" is contingent upon things other than the form of the statement. The following is an example of a contingent statement form, and its corresponding truth-table:

Note that under the bracketed main connective there is a mixture of true and false truth-values, which means that the statement form is contingent.

Logic Post 20

Rules of Inference

While it is true that truth-tables are very good at testing the validity of truth-functional arguments, they tend to be a bit cumbersome; especially if you have four or more variables (remember that truth columns grow exponentially at a rate of 2^n).

It is much easier to deduce validity using deduction, i.e., you use deductive argument forms that have already been shown to be valid to perform a sequence of elementary arguments which will then confirm the main conclusion. “The elementary arguments, in essence, are a set of rules, called transformation rules, for they specify which truth-functional statement forms may be inferred from which others. The transformation rules are then subdivided into inference rules and substitution rules. Systems made up such sets of rules are called natural deduction systems. The selection of the rules in these systems is relatively arbitrary; any set will do so long as it is complete (Kegley and Kegley, Introduction to Logic, p. 271).”


Rules of Inference

1) Modus Ponens (MP):

p ⊃ q
p
_____
∴ q

2) Modus Tollens (MT):

p ⊃ q
~q
______
∴ ~p

3) Disjunctive Syllogism (DS):

p v q
~p
____
∴ q

p v q
~q
____
∴ p

4) Hypothetical Syllogism (HS):

p ⊃ q
q ⊃ r
______
∴ p ⊃ r

5) Simplification (Simp):

p · q
_____
∴ p

p · q
_____
∴ q

6) Conjunction (Conj):

p
q
__
∴ p · q

7) Addition (Add):

p
__
∴ p v r

8) Constructive Dilemma (CD):

(p ⊃ q) · (r ⊃ s)

p ∨ r
______
∴ q v s

You can use these inference rules on entire lines only. Never use them on parts of a line. For example, do not use simplification in the following way: (p · q) ⊃ r to get p ⊃ r

Also these inference rules are used in one direction only. For example, you can work your way from p · q to p using simplification, but not from p to p · q.

You want to memorize the rules of inference.

@Sam26 Sweet.

Hopefully there aren't too many errors. :gasp:

ALignment in the tables is an obvious thing. They are a bit hard to read.

Ya, they are hard to read. I'll try to line them up.

It looks a bit better now I think.

Using [ code ] means you can write it in a monospace text editor and paste it straight in:

Ah, I see what you mean. Thanks.

_______________________________________________

:up:

You should start your own thread on why questions should be considered propositions. The question is settled for me, I'm not going to debate it. — Sam26

If it were settled for you,, you'd be able to answer these questions. True wisdom comes in questioning everything - in never being settled until all posdible questions have been asked and answered. It seems to me that you are just covering your ears and closing your eyes and screaming,"lalalalala, I can't hear you!"

It seems that lately, a lot of threads on this forum are started as a means to proselytize, not to engage people in any meaningful debate or to learn from others. What a shame.

True wisdom comes in questioning everything - in never being settled until all posdible questions have been asked and answered. — Harry Hindu

I don't know where you get the idea that "true wisdom comes in questioning everything," I don't agree with that either. As I said earlier, this thread is just a guide for people. If you think it's an important point, then start a thread and debate the issue with those who want to debate. I'm not going to debate the issue.

I don't know where you get the idea that "true wisdom comes in questioning everything," I don't agree with that either. — Sam26

"The only true wisdom is in knowing that you know nothing." - Socrates

Logic Post 21

Deductive Methods

When analyzing arguments you want to look for forms that correspond to valid rules of inference. For example, consider the following argument form:

Premise 1. [p v (~q ⊃ r)] ⊃ [~s v (r · t)]

Premise 2. ~ [~s v (r · t)]

Conclusion: ~ [p v (~q ⊃ r)]

First, just because the argument has a large number of variables don’t let that intimidate you. Second, you want to keep the conclusion in mind, since this is where you are heading. Next you want to take note of the major connective in the first premise, which is ⊃, it has the form p ⊃ q. Now notice that the second premise is a negation, and it has the form ~q. At this point if you have memorized the eight rules of inference you should be able to see where this is leading.

So let’s break premise one down so that we can see how it corresponds with p ⊃ q.

Premise 1. [p v (~q ⊃ r)] ⊃ [~s v (r · t)]

p = [p v (~q ⊃ r)]

then comes the major connective ⊃

q = [~s v (r · t)]

So premise one has the form p ⊃ q.

Now let’s look at premise two.

~ [~s v (r · t)]

Premise two is the denial of q in the argument form p ⊃ q, so it has the form ~ q.

We now have

p ⊃ q

~q

You should now be able to see that the example above matches the rule of inference called Modus Tollens. Premise two denies the consequent. If you kept your eye on the conclusion Modus Tollens is the obvious choice.

The conclusion is ~[p v (~q ⊃ r)], which is the denial of p.

Therefore, the argument form

Premise 1. [p v (~q ⊃ r)] --> [~s v (r · t)]

Premise 2. ~ [~s v (r · t)]

Conclusion: ~ [p v (~q ⊃ r)]

is the same as


Modus Tollens

p ⊃ q
~q
______
∴ ~p

We have now figured out a simple proof using one of the rules of inference.

Logic Post 22

Rules of Replacement

You need to memorize these rules of replacement along with the rules of inference.


1) Absorption

(p ⊃ q) ≡ p ⊃ (p · q)

2) Double Negation

p ≡ ~~p

3) De Morgan’s Theorems

~(p v q) ≡ (~p · ~q)
~(p · q) ≡ (~p v ~q)

4) Commutation

(p v q) ≡ (q v p)
(p · q) ≡ (q · p)

5) Association

[(p v q) v r] ≡ [p v (q v r)]
[p · q · r] ≡ [p · (q · r)]

6) Distribution

[p v (q · r)] ≡ [(p v q) · (p v r)]
[p · (q v r)] ≡ [(p · q) v (p · r)]

7) Transposition or Contraposition

(p ⊃ q) ≡ (~q ⊃ ~p)

8) Material Implication

(p ⊃ q) ≡ (~p v q)

9) Material Equivalence

(p ≡ q) ≡ [(p ⊃ q) · (q ⊃ p)]
(p ≡ q) ≡ [(p · q) v (~p · ~q)]

10) Exportation

[(p · q) ⊃ r] ≡ [p ⊃ (q ⊃ r)]

11) Tautology

p ≡ (p v p)
p ≡ (p · p)


“It should be noted that the eight Inference Rules and the eleven Rules of Replacement constitute a complete system of truth-functional logic in the sense that the construction of a formal proof of validity for any valid truth-functional argument is possible. However, some of the rules are redundant. Thus, for example, Modus Tollens is redundant because every instance in which Modus Tollens is used, the Principle of Transposition and Modus Ponens can function equally well. Disjunctive Syllogism could also be replaced. But these two argument forms are easy to grasp and the use of all nineteen rules makes proofs considerably easier (Kegley and Kegley, Introduction to Logic, p. 280 and 281).”

Logic Post 23

There is an important difference between the Rules of Replacement and the Rules of Inference. The Rules of Inference can only be used on entire lines of a proof. So, in a proof, X can be inferred from X · Y, if X · Y make up the entire line. You cannot infer X from W ⊃ (X · Y) using Simplification. When using the Rules of Replacement this is not the case, because logically equivalent expressions can replace other logical equivalent expressions even if they do not constitute a whole line in a proof.

You're going to need more information than what I've given you here to learn to use these correctly. You should find a book with exercises, and one that explains the Rules of Replacement more thoroughly. Hopefully, this will give you somewhat of a guide to know what to study. I also would recommend studying the categorical syllogism. There are videos on Youtube that will explain much of this in detail.

Logic Post 24

Enthymemes

Enthymemes are arguments in which a premise or premises are left out. Sometimes even the conclusion is left out - it is supposedly understood.

Enthymemes are quite ubiquitous in discourse, so it is important to familiarize yourself with them. They are used because the premise or conclusion is understood, and stating them would be to state the obvious. However, sometimes people will leave out part of an argument, because to state the premise or conclusion would obviously make the argument false. So to avoid criticism sometimes people will purposely leave a premise or a conclusion unstated. I know it is hard to believe that people actually do this.

You need practice to get good at solving enthymemes. Understanding these concepts is one thing, but actually solving the problems is quite another. Do not assume that because you understand what I am writing that you automatically can solve the problems. Logic is like math you need practice. Without it you will not be able to reason well.