#logicalreasoning

i took 1 full lsat exam today, which took 2 hours-ish! I also met with my lsat tutor for the 4th time and went over the logical reasoning sections i did (particularly in the conclusions and disputes category!)

I am a homosexual and a (at least in public) Orthodox jew. I would love if you could hear what I have to say.

It seems to be that the anti-homosexual sentiment in Judaism is no longer halachic. It never was. The torah prohibits a man lying with a man as one does with a woman. Most people, including the rabbinical interpretation (Which is generally considered full authority.) have said that anal male-male intercourse is what is forbidden, along with the separate prohibition of spilling seed. I have asked quite a few people whom I’m “supposed” to talk with this about on their opinions and they all get avoidant, and a few of them had never shied away from any sort of sensitive topics. I have asked with sincerity on how to have a loving relationship with another man without breaking these two prohibitions, but still, I am tossed to the side by these people. Also note that the people I have confided in are much more open minded than the average orthodox religious (Frum) jew. I thought the prohibition of homosexuality was simply halachic, possibly kabbalistic as well, but no, even when halachic concerns are no longer a problem, we still are shunned and not allowed to live a life without loneliness. This is akin to any unfair bias, and it causes pain. No harm is done to the parties who participate in a halachically permissible relationship, so why should it not be allowed, or at least tolerated as long as it’s kept private? I’m not a proponent of forcing my opinions on others, but think for a second, where is this even coming from?

Smack about Crypto.

In reasoning , you come across this topic known as syllogism. In it , can not is used in indefinite conclusions. So if someone said that you can not do it ( anything ) , just know that it's nothing more than a possibility and possibility can be reversed in your favor with the help of frequency.

Imagine waking up in a society where questioning authority is a crime. Your personal choices are being closely monitored. Are we already there? So, is true freedom and autonomy possible when extremists rule society? Let's examine this question in more detail. Governments can have a positive or negative effect on their citizens in several ways. Democracies promote policies that reflect positive values, benefiting everyone. We celebrate positive changes in the outcomes of fulfilled lives. A democracy can turn into a dictatorship or oligarchy. When this happens, people lose basic freedoms. Let's start by defining the terms we use.

What is True Freedom and Autonomy?

This is the point where posts get longer. I've done my best to simplify and shorten where I can without leaving out unnecessary information. To make navigation easier I'll be using some key formatting tools. Definitions are boldened, examples are numbered and indented, and sections have headers.

Finding Simple Sentences

Sentential logic's foundation is rooted in taking large blocks of information (commonly text), and converting it into something we call simple sentences. What are simple sentences? They are statements without any logical connectives. Unfortunately this definition creates a sort of paradox because to understand simple sentences you need to know what logical connectives are, but in order to use logical connectives you need to know what simple sentences are. So the goal of the first part of this post will be to provide enough examples of both on their own so that we can begin using them together in the latter parts.

Logical connectives come in one of the following forms:

And

Or

If ..., Then...

Not

Statements with one of the above logical connectives in it are not simple sentences. That sentence can actually be broken down further until we get to the simple sentence. Another key feature of simple sentences is that it must either be true or false. You can't have partially true statements in this logical environment. This is an area where errors in logic could form. If something is subjective, within the context of logic, you need to qualify it in some way as to determine whether it is true or false. For example, if you want to prove that someone is happy, you may need to create a more precise definition of what happiness looks like for them given the context.

Example 1; "If the pizza has pepperoni on it, then the pizza is good". Sounds like a solid argument. How many simple sentences are there? The answer is two. Notice the logical connectives we mentioned earlier, "if" and "then". Removing those connectives we're left with two simple sentences. 1. The pizza has pepperoni on it. 2. The pizza is good.

Example 2; "The pepperoni is crispy or the pizza is bad". Clearly an objective statement. How many this time? Two again. The logical connector "or" connects two statements. 1. The pepperoni is crispy. 2. The pizza is bad.

Example 3; "The cheese is not gooey". How many? The answer is one but don't be fooled, there is still a logical connective that needs to be removed. So the simple sentence looks like this: 1. The cheese is gooey.

Example 4; "If Clare, Kat, and Rachel eat pizza, then it is Friday". How many simple sentences and what are they? If you guessed four you would be correct. Because of the list of subjects the "if" is applied to each person. 1. Clare eats pizza. 2. Kat eats pizza. 3. Rachel eats pizza. 4. It is Friday.

Representing Simple Sentences

So we understand the differences in simple sentences and logical connectives. We'll go deeper into each of the connectors, but first we need a better way to represent our statements. Why should we spend time learning how to shorten out simple sentences? Lets take a look at a few examples and it should become apparent.

Example 5; If the moon is full, then houses transform into warehouses. 1. The moon is full. 2. Houses transform into warehouses.

Example 6; If the light is on, then I am not alone in my home. 1. The light is on. 2. I am not alone in my home.

Example 7; If the McRib is back, then traffic is worse than the previous month. 1. The McRib is back. 2. Traffic is worse than the previous month.

Each of these arguments follow the same structure. They are logically identical arguments in every way other than the subjects. Then why should we consider them individually when we could look at them all at once? We shouldn't. It's more efficient to represent them all with the same notation. We do the same thing in algebra by substituting logically identical equations with variables. If something applies to one equation, it applies to the others.

In the case of logic, we use letters such as P, Q, and R or F, G, and H. There's lots of letters and we could have many, many simple sentences we need to represent. So for example;

Instead of

Example 8; If guitars are red, then they sound better. 1. Guitars are red. 2. They sound better.

We use

Example 8; If R then S 1. R: Guitars are red. 2. S: They sound better.

The change may seem small but when we begin doing truth tables with many simple sentences this will save a lot of time that didn't need to be wasted.

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Representing Logical Operators Part I

Negation ~

Negation is found with "NOT" To represent the negation of a premise we use the ~ (Tilda). Why represent the negation of a premise with a logical operator instead of a new variable? Short answer, it saves a lot of time and space when you have a large number of premises. Also it may help to think of representing logic in terms of algebra.

Example 9; Africa is not a country. 1. A: Africa is a country. 2. ~A

Disjunction v

Disjunction is found with "OR". Say we have two simple sentences, H: "I am hungry." and T: "I am thirsty." and we add a v (wedge) between them, we get an Inclusive Or statement. I am hungry or I am thirsty or I am both hungry and thirsty. This is in contrast to an Exclusive Or statement wherein both parts cannot be true at the same time. Be on the lookout for when an or statement is inclusive or exclusive.

Inclusive OR

Example 10; I am hungry or I am thirsty. 1. H: I am hungry. 2. T: I am thirsty. 3. HvT

Exclusive OR

Example 11; I am asleep or I am awake. 1. S: I am asleep. 2. W: I am awake. 3. SvW

Compound Sentences

Quick pause from demonstrating how to represent logical connectives. We're going to take the last two operations (negation and disjunction) to talk about compound sentences. Compound sentences are sentences featuring multiple logical operators.

Example 12; I will eat pizza or I will not brush my teeth. 1. P v ~T

Logical operators can manipulate compound sentences in the same way they can manipulate simple sentences. One operator can only link two things together. The next example will illustrate this.

Example 13; They will compete in rugby, tennis, or golf. 1. (RvT)vG 2. Rv(TvG) 3. BUT NOT RvTvG

Its also crucial to pay attention to the meaning of a sentence.

Example 14; It is not the case that I running or I am female. Is this: ~RvF [Female/not running/female who is not running] Or is it: ~(RvF) [I am not female and I am not running] Technically the statement could mean either of these and they do not mean the same thing. (Converting these from logical notation back to English will be expanded upon when we cover DeMorgan's Law in the Rules of Replacement post.)

You need to be careful with how you write down sentences so you can avoid situations like in Example 14. There is only one way to interpret a logical statement in English. But there could be many ways to interpret English statements into logic.

Representing Logical Operators Part II

Conjunction ^

The AND operator connects two simple sentences such that they are both simultaneously true. We notate the and operator with the ^ (carrot).

Example 15; He is tall and has brown hair. 1. T: He is tall. 2. H: He has brown hair. 3. T^H

Its important to note that there are occasions where a conjunction will exist without an and; such as:

Example 16; I am happy you are learning logic but I am upset I have very little free time to write my posts. 1. L: I am happy you are learning logic. 2. F: I am upset I have very little free time to write my posts. 3. L^F The "but" divides the statement and acts as a logical connective, conjoining the simple sentences. In this case, but is an and. Often in real speech but is just another way to say and. This also applies for "as well as" and "in addition to"

Conditional Statements

Conditional statements take two simple sentences and connect them in an "if/then" situation. We show this with an => (arrow). Conditionals come in the following forms: If-Then, Only If, Required, Necessary, and Sufficient.

Example 17; If you are President of the United States, you are both Older then 35 and an American Citizen. 1. P=> (O^C) If President, then older and citizen.

Example 18; You can eat Desert only if you have eaten your Vegetables. (it may help to rewrite the statement in a simpler form. Such as "If you can eat your desert, you have eaten your vegetables.") 1. D=>V If D then V

Example 19; Submitting your Paper is required to having your paper Graded. 1. G=>P If G then P. If having a graded paper is true, then you must have submitted the paper.

Example 20; To not Lose track of logic, it is necessary to Practice each step multiple times. 1. ~L=>P If one does not lose track of logic, then one practices each step.

Example 21; Wearing Sunscreen is sufficient to not getting Burnt. S=>~B If one wears sunscreen, then they will not get burnt.

Important Notes:

If the antecedent is false, the conditional statement is (vacuously) true. We'll cover vacuously true in more detail when we go over truth tables.

P=>Q is the same as ~PvQ. This may seem counter intuitive but we'll see why this works when we cover replacement rules.

Biconditional

The biconditional operator is the logical equivalent to the equal sign. This means that whatever is on one side must be identical to what is on the other side regardless of superficial differences. We can locate a biconditional statement with "if and only if" and we notate them with the <=> (two way arrow).

Example 22; I am in a long line at Taco Bell if and only if it is after 11:00pm. 1. L: I am in a long line at Taco Bell. 2. A: It is after 11:00pm. We can think of the above as the following two conditional statements put together 3. L=>A; If I am in a long line at Taco Bell, it is after 11:00pm. 4. A=>L; If it is after 11:00pm, I am in a long line at Taco Bell.

You may, if you were very observant, realize the biconditional is just two conditional statements put together. In fact, the biconditional is redundant. The following two statements are identical:

L<=>A

(L=>A)^(A=>L)

So we don't actually need a unique operator for biconditional, right? Though it is redundant and can be avoided by rewriting the statement, we're still going to be using biconditionals. On one hand, it's all over logic and you need know how to communicate with others who use logic. On the other hand the biconditional is very intuitive. How could be participate in logic without an equality identity? Its an efficient redundancy.

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In the next post we'll be going over truth tables. We'll take everything we learned here and apply it to a structural matrix that allows us to simplify the process of applying operators to multiple premises simultaneously. If you have any questions please feel free to send me an ask and I'll put all the asks together into one response post. If I don't reply to your particular question that likely means I'm going to cover it in an upcoming post.