Is rational logic a proper tool of philosophy? Why? How?

[grannymh]

One of many links which can be used. thefreedictionary.com/logic

There was a time when some universities required philosophy as a minor for a bachelor’s degree. A logic course was a key component. In my humble observation, it appears that the popular three statement syllogism that “all humans are mortal” is inadequate when it comes to obvious basic Catholic doctrines surrounding human origin and human nature.

Sidebar:
“All humans are mortal” etc. appears in the definition of syllogism, American Heritage College Dictionary, fourth edition. The definition of logic preferred for this thread is - The study of the principles of reasoning, especially of the structure of propositions as distinguished from their content and of method and validity in deductive reasoning. thefreedictionary.com/logic In other words, the content for this thread is based on structure, method, validity and deductive reasoning. Examples from the thread “Adam & Logic, Third Edition, Original Relationship between Humanity and Divinity” can be used.

The intent for this thread is to examine the Why? How? of logic.
Those familiar with journalism 101 can throw in Who? What? When? Where?

 

[BartholomewB]

Grannymh

A syllogism is a form of argument that draws a conclusion from two premises, a major (general) premise and a minor (specific) premise. For example, if you have a canary called Jim, you could argue like this:

All canaries are birds.
Jim is a canary.
Therefore Jim is a bird.

You’re not attempting to prove that all canaries are birds, right? It’s not about ornithology, it’s about logical method. All you’re saying is that ***if ***it’s true that all canaries are birds and if it’s also true that Jim is a canary, then it must be true that Jim is a bird.

The standard example used in the textbooks since the time of Aristotle or thereabouts is this:

All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.

In other words, if it’s true that all men are mortal, and so on.

The logic can be correctly done even if the premises are false. Here’s an example:

All canaries are crocodiles.
Jim is a canary.
Therefore Jim is a crocodile.

Is this case, one of the two premises (the first one) is false. As a result, you haven’t proved that Jim is a crocodile, even though the logic of the syllogism is flawless.

Hope this helps.

 

[Thomas_White]

In logic, the antecedent of a proposition is called the hypothesis. In the proposition, “If P, then Q”, P is the antecedent or hypothesis. P is an assumption.

It is perhaps in the scientific method, rather than in formal logic, that your concern would be an issue: An hypothesis is an assumption. If the attempt is to prove the hypothesis by probability and statistical methods, there typically remains a level of uncertainty about the validity of the assumption.

For instance, in his “The Grand Design”, Stephen Hawking extrapolates probabilities from quantum mechanics and uses those uncertainties to argue that there need not be a God behind the creation of the universe. Hawking presents his conclusions as facts, but they are not facts. They are at best probabilities.

 

[unstoppable_II]

Because it is common to all mankind.

There have been other methods developed to acquire truth, among them polylogism.

Polylogism is the belief that different people or groups have different forms of logic. Since logic is the art of non-contradiction, polylogism can have only two possible meanings. Either reality is different for each group, or logic is a loose term for method of acquiring knowledge. The latter, though, is not how it is used. Those speaking of polylogism state that the conclusions from the different logics are different. This means that although they both claim to be methods of acquiring knowledge, the truth of the knowledge is different for different groups. This can mean only that reality is different for the different groups. …

The rest is here:
“Polylogism”
solohq . com/Objectivism101/Irrational_Polylogism . shtml
See also:
“Steven Yates on Polylogism”

He who addresses fellow men, who wants to inform and convince them, who asks questions and answers other people’s questions, can proceed in this way only because he can appeal to something common to all men – namely, the logical structure of human reason. The idea that A could at the same time be non-A or that to prefer A to B could at the same time be to prefer B to A is simply inconceivable and absurd to a human mind.
…
Polylogism is not a philosophy or an epistemological theory. It is an attitude of narrow-minded fanatics, who cannot imagine that anybody could be more reasonable or more clever than they themselves. Nor is polylogism scientific. It is rather the replacement of reasoning and science by superstitions.
archive . lewrockwell . com/yates/yates62 . html

 

[laircy]

Grannymh

A syllogism is a form of argument that draws a conclusion from two premises, a major (general) premise and a minor (specific) premise. For example, if you have a canary called Jim, you could argue like this:

All canaries are birds.
Jim is a canary.
Therefore Jim is a bird.

You’re not attempting to prove that all canaries are birds, right? It’s not about ornithology, it’s about logical method. All you’re saying is that ***if ***it’s true that all canaries are birds and if it’s also true that Jim is a canary, then it must be true that Jim is a bird.

The standard example used in the textbooks since the time of Aristotle or thereabouts is this:

All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.

In other words, if it’s true that all men are mortal, and so on.

The logic can be correctly done even if the premises are false. Here’s an example:

All canaries are crocodiles.
Jim is a canary.
Therefore Jim is a crocodile.

Is this case, one of the two premises (the first one) is false. As a result, you haven’t proved that Jim is a crocodile, even though the logic of the syllogism is flawless.

Hope this helps.

There does not have to be a general and a specific premise. There can be two general premises:

All animals are mortal
All men are animals
Therefore, all men are mortal.

Though yes, you are correct that if the premises of a syllogism are true, the conclusion must be true as well. Syllogisms do work even when they make no sense or are false. That is the difference between the Prior Analytics and the Posterior Analytics. The first deals with the syllogism simply, while the second deals with what makes syllogisms true and productive of “scientific knowledge,” where the middle term is the cause, the conclusion is universal and affirmative, etc.

---

Re: the thread

Most of the time when we reason, we do not speak in syllogisms, but rather those neat syllogisms are the basic, stripped down form of the argument that we do use. For example, a book I read recently would spend a whole chapter dedicated to an argument, which was then boiled down to syllogism in the appendix.

 

[grannymh]

There does not have to be a general and a specific premise. There can be two general premises:

All animals are mortal
All men are animals
Therefore, all men are mortal.

Though yes, you are correct that if the premises of a syllogism are true, the conclusion must be true as well. Syllogisms do work even when they make no sense or are false. That is the difference between the Prior Analytics and the Posterior Analytics. The first deals with the syllogism simply, while the second deals with what makes syllogisms true and productive of “scientific knowledge,” where the middle term is the cause, the conclusion is universal and affirmative, etc.

---

Re: the thread

Most of the time when we reason, we do not speak in syllogisms, but rather those neat syllogisms are the basic, stripped down form of the argument that we do use. For example, a book I read recently would spend a whole chapter dedicated to an argument, which was then boiled down to syllogism in the appendix.

I am using paragraphs in the book, *Catechism of the Catholic Church, Second Edition *as a source of truth. How do I acknowledge my source for a syllogism?

 

[grannymh]

because it is common to all mankind.

There have been other methods developed to acquire truth, among them polylogism.

Polylogism is the belief that different people or groups have different forms of logic. Since logic is the art of non-contradiction, polylogism can have only two possible meanings. Either reality is different for each group, or logic is a loose term for method of acquiring knowledge. The latter, though, is not how it is used. Those speaking of polylogism state that the conclusions from the different logics are different. This means that although they both claim to be methods of acquiring knowledge, the truth of the knowledge is different for different groups. This can mean only that reality is different for the different groups. …

​

The rest is here:
“Polylogism”
solohq . Com/Objectivism101/Irrational_Polylogism . shtml
See also:
“Steven Yates on polylogism”

He who addresses fellow men, who wants to inform and convince them, who asks questions and answers other people’s questions, can proceed in this way only because he can appeal to something common to all men – namely, the logical structure of human reason. The idea that a could at the same time be non-A or that to prefer A to B could at the same time be to prefer b to a is simply inconceivable and absurd to a human mind.
…
polylogism is not a philosophy or an epistemological theory. It is an attitude of narrow-minded fanatics, who cannot imagine that anybody could be more reasonable or more clever than they themselves. Nor is polylogism scientific. it is rather the replacement of reasoning and science by superstitions.
archive . lewrockwell . com/yates/yates62 . html
​

The principle of non-contradiction is very important.

 

[grannymh]

Grannymh

A syllogism is a form of argument that draws a conclusion from two premises, a major (general) premise and a minor (specific) premise. For example, if you have a canary called Jim, you could argue like this:

All canaries are birds.
Jim is a canary.
Therefore Jim is a bird.

You’re not attempting to prove that all canaries are birds, right? It’s not about ornithology, it’s about logical method. All you’re saying is that ***if ***it’s true that all canaries are birds and if it’s also true that Jim is a canary, then it must be true that Jim is a bird.

The standard example used in the textbooks since the time of Aristotle or thereabouts is this:

All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.

In other words, if it’s true that all men are mortal, and so on.

The logic can be correctly done even if the premises are false. Here’s an example:

All canaries are crocodiles.
Jim is a canary.
Therefore Jim is a crocodile.

Is this case, one of the two premises (the first one) is false. As a result, you haven’t proved that Jim is a crocodile, even though the logic of the syllogism is flawless.

Hope this helps.

Thank you.

Re: “You’re not attempting to prove that all canaries are birds, right?”

As a journalist before the birth of Google, I had to have in my back pocket evidence or a demonstration that all canaries are birds.

Currently – it is hard for an old bird to change feathers.

 

[grannymh]

In logic, the antecedent of a proposition is called the hypothesis. In the proposition, “If P, then Q”, P is the antecedent or hypothesis. P is an assumption.

It is perhaps in the scientific method, rather than in formal logic, that your concern would be an issue: An hypothesis is an assumption. If the attempt is to prove the hypothesis by probability and statistical methods, there typically remains a level of uncertainty about the validity of the assumption.

For instance, in his “The Grand Design”, Stephen Hawking extrapolates probabilities from quantum mechanics and uses those uncertainties to argue that there need not be a God behind the creation of the universe. Hawking presents his conclusions as facts, but they are not facts. They are at best probabilities.

It is the conditional “if” that I find troubling. Most likely that is because I would consider the “hypothesis” as an axiom or a declared truth. But, a hypothesis is not necessarily an actual undeniable truth. This older than dirt brain needs help.

In my opening post, I referred to obvious basic Catholic doctrines surrounding human origin and human nature. In a New York minute, I can see doctrines as declared truth being questioned. Starting with the questionable existence of God. My answer would be that, like in theatre, there needs to be a “willing suspension of disbelief.” On my part, I need to present a nearly impeccable logical method.

It seems to me that I have to fit my faith in Adam into the tool of understandable proper logic. This is the basic reason for this thread. I want to know how logic per se works so I can use it properly. As for why, I agree with the comment of Unstoppable II, post 4, that logic is common to all mankind.

Regarding the scientific method. That is based on the inductive method of reasoning. It is possible to gather together all the reasons why God decided that the best plan for humanity in general would be to start the human species with a single founder biblically known as Adam. Still, I would be faced with the question – But is that logical?

 

[laircy]

I am using paragraphs in the book, *Catechism of the Catholic Church, Second Edition *as a source of truth. How do I acknowledge my source for a syllogism?

Is the Catechism making rational arguments or stating doctrine (faith)?

 

[Rhubarb]

In all the logic I’ve studied, the Aristotelian syllogism isn’t the focus. The use of the syllogism was wrapped up in Aristotle’s epistemology and metaphysics. Aristotelian logic has been supplanted by the descriptive power of more modern logics. This isn’t to say that Aristotelian logic is bad, per se, but that most arguments require a more nuanced approach in regimenting the arguments. Modern logics have more benefits too, which I’ll talk about further down.

Logic is a language, at least modern logic is. It is a formal language, unlike English, or Latin, or Urdu, etc. which are natural languages. The formal language of logic lets us take the information that is given in natural languages and regiment them out in such a way that lets us see what the relationship between concepts and ideas are. I describe it like algebra. Sentences are broken up into symbols - like I did in your prior thread - and then we can see how the sentences relate to one another.

This lets us do a few things: First, it lets us regiment our arguments and test them for validity and truth. I think the usefulness of this is clear.

The other thing it lets us do is see what we’re committed to accepting based on what we accept already. This is a bit more complicated so I’ll use a very simple example. Let’s say that together we agree to the sentence “It is raining”, that is, we both take that sentence to be true. Also, we both accept the sentence “If it is raining, then the sidewalk is wet”. Now, if we take both of those sentences as true, then our logic tells us we MUST accept “The sidewalk is wet” by itself based on what we have already admitted. The relationships between the first two sentences give us the third. With long arguments, this benefit is important because sometimes unintended consequences creep into arguments.

 

[unstoppable_II]

… Let’s say that together we agree to the sentence “It is raining”, that is, we both take that sentence to be true. Also, we both accept the sentence “If it is raining, then the sidewalk is wet”. …

I can see you’ve never been president of a homeowners association.

 

[grannymh]

Is the Catechism making rational arguments or stating doctrine (faith)?

The universal Catechism of the Catholic Church, Second Edition, does both and a lot more.

There are* both* rational arguments and statements of Faith. Then there are weird looking paragraphs in small print which contain supplementary material, some of which appears to be on the poetic side. There are cross-references in the margins which give additional perspectives. For those interested in historical data, the footnotes are extensive. However, in my opinion, “the Index of Citations” is primary when one wants to check Scripture used in basic Catholic teachings. Granted that some explanations can go on and on. The “Glossary” is a great short cut. And there are those times, when one simply wants to cut to the chase. At the end of a section, there is a summary titled “IN BRIEF” which lives up to its name.

Caution – Be sure to read CCC 20-21 first. It explains the use of small print in the text.

 

[grannymh]

In all the logic I’ve studied, the Aristotelian syllogism isn’t the focus. The use of the syllogism was wrapped up in Aristotle’s epistemology and metaphysics. Aristotelian logic has been supplanted by the descriptive power of more modern logics. This isn’t to say that Aristotelian logic is bad, per se, but that most arguments require a more nuanced approach in regimenting the arguments. Modern logics have more benefits too, which I’ll talk about further down.

Logic is a language, at least modern logic is. It is a formal language, unlike English, or Latin, or Urdu, etc. which are natural languages. The formal language of logic lets us take the information that is given in natural languages and regiment them out in such a way that lets us see what the relationship between concepts and ideas are. I describe it like algebra. Sentences are broken up into symbols - like I did in your prior thread - and then we can see how the sentences relate to one another.

**This lets us do a few things: First, it lets us regiment our arguments and test them for validity and truth. I think the usefulness of this is clear. **

The other thing it lets us do is see what we’re committed to accepting based on what we accept already. This is a bit more complicated so I’ll use a very simple example. Let’s say that together we agree to the sentence “It is raining”, that is, we both take that sentence to be true. Also, we both accept the sentence “If it is raining, then the sidewalk is wet”. Now, if we take both of those sentences as true, then our logic tells us we MUST accept “The sidewalk is wet” by itself based on what we have already admitted. The relationships between the first two sentences give us the third. With long arguments, this benefit is important because sometimes unintended consequences creep into arguments.

I put in bold, **The other thing it lets us do is see what we’re committed **to accepting based on what we accept already. because it is what I think logic does.

I would start with three statements which, from a Catholic position, would be considered undeniable truth.

We would have to accept the following.

1. God as Creator exists.
2. God as Creator interacts personally with each individual human.
3. Every individual human has the inherent capacity to interact with God as Creator.

What kind of statements would fulfill the following?
**“This lets us do a few things: First, it lets us regiment our arguments and test them for validity and truth. I think the usefulness of this is clear.” **

Would 2. & 3. have to be accepted because they would be characteristics of the Creator in 1.? Or does each statement have to be validated?

 

[Rhubarb]

I put in bold, **The other thing it lets us do is see what we’re committed **to accepting based on what we accept already. because it is what I think logic does.

I would start with three statements which, from a Catholic position, would be considered undeniable truth.

We would have to accept the following.

1. God as Creator exists.
2. God as Creator interacts personally with each individual human.
3. Every individual human has the inherent capacity to interact with God as Creator.

What kind of statements would fulfill the following?
**“This lets us do a few things: First, it lets us regiment our arguments and test them for validity and truth. I think the usefulness of this is clear.” **

Would 2. & 3. have to be accepted because they would be characteristics of the Creator in 1.? Or does each statement have to be validated?

1, 2 and 3 are propositions - that is, they are sentences that can be true or false. Logic deals with propositions. There are lots of sentences in natural languages, like imperatives such as “Shut the door!” or questions “Why did you leave the door open?” These sentences can have no truth value. Validity isn’t a property of propositions, at least, in the logical sense. Validity is a property of arguments.

A valid argument is an argument where, should the propositions that make up the premises be true, then the proposition that is the conclusion must necessarily be true. There are ways to test for validity - such as truth trees, and natural deductions. (Which only work when the argument is valid) When we’re talking about logic, that’s all validity means. That the truth of the conclusion is guaranteed by the truth of the premises. It’s a technical sense of the word ‘valid’, and in philosophy, precise use of jargon is important. (At least in analytical philosophy)

To the question at hand, 2 and 3 does not follow from 1 as they are written, logically. Something that would follow from 1, “God as Creator exists” could be “something exists” (in the logical use of the word ‘something’, which doesn’t imply that God is a thing) or “a creator exists.” These are simple existential sentences were are bound to accept by rule of logic if we accept 1 - that God as Creator exists. We certainly can accept 2 and 3, but logic doesn’t demand that we do.

 

[wmw]

1, 2 and 3 are propositions - Validity isn’t a property of propositions, at least, in the logical sense. Validity is a property of arguments.

A valid argument is an argument where, should the propositions that make up the premises be true, then the proposition that is the conclusion must necessarily be true. There are ways to test for validity - such as truth trees, and natural deductions. (Which only work when the argument is valid) When we’re talking about logic, that’s all validity means.

I’m sorry I’m a bit confused about truth trees, I thought they were a method of checking proper logical form, but your talking about the underlying validity of the premises. This is different than what I thought. They only seem to be the very formal mathematics like symbolism for the logical deductions.

Also, “natural deductions” they are, from what I’ve looked up on Wikipedia, that arose as an alternate logical language that uses “judgments” as propositions rather than axioms and was developed with it’s own symbols. As time has gone on, is it true that this has all been combined into one standard system? But still this all appears to be checking on “validity” in terms of the logical construction, not the initial truth of the premises.

To the question at hand, 2 and 3 does not follow from 1 as they are written, logically. [2 & 3 could be] Something that would follow from 1, … but logic doesn’t demand that we do.

Yes, I think grannymh and I are similar in thinking that there are logical reasoning’s that would connect 1, 2, & 3, but you are right that as stated they are stand alone premises. This is a bit confusing in that we also have a reasoning behind thinking that God would be so kind as to be this way referring to 2 & 3. This “there’s a little more to it” backdrop does buzz in one’s mind that there are logical deductions between 1, 2, & 3. At the same time we didn’t want to expand the argument into that argument which I think we are in agreement would be better made separately.

 

[Rhubarb]

I’m sorry I’m a bit confused about truth trees, I thought they were a method of checking proper logical form, but your talking about the underlying validity of the premises. This is different than what I thought. They only seem to be the very formal mathematics like symbolism for the logical deductions.

Also, “natural deductions” they are, from what I’ve looked up on Wikipedia, that arose as an alternate logical language that uses “judgments” as propositions rather than axioms and was developed with it’s own symbols. As time has gone on, is it true that this has all been combined into one standard system? But still this all appears to be checking on “validity” in terms of the logical construction, not the initial truth of the premises.

Yes, I think grannymh and I are similar in thinking that there are logical reasoning’s that would connect 1, 2, & 3, but you are right that as stated they are stand alone premises. This is a bit confusing in that we also have a reasoning behind thinking that God would be so kind as to be this way referring to 2 & 3. This “there’s a little more to it” backdrop does buzz in one’s mind that there are logical deductions between 1, 2, & 3. At the same time we didn’t want to expand the argument into that argument which I think we are in agreement would be better made separately.

Truth trees can be used to test sentences and entire arguments in logic for various things, depending on how you apply them. I don’t know what you mean by ‘proper logical form’, but I assume you mean ‘validity’ by it. (I would consider an argument that is valid to have a proper logical form, at least) And this is correct, you can use a truth tree to see if an argument is valid or invalid. I could give a list of other ‘logical properties’ that it can be used to find, but it’s all technical jargon (quantifactional truth and falsity, for instance) that probably won’t be helpful, though I’ll explain it if you’d like.

I should’ve specified that I meant Fitch-style natural deduction. These are the classic proofs, where you outline your premises at the top of a table, and your conclusion at the bottom, and go through the motions to ‘prove’ the argument. What it does is give you a sure-fire demonstration that an argument is valid. These are similar to the geometric and trigonometric proofs from math class. Mathematics is given in first-order logic, even the math we use as kids. They just leave out all the unnecessary parts for what we do with it so we don’t get confused.

And you’re right. 1, 2, and 3 certainly can be connected in a logical manner. That’s why I said they are not ‘as written.’ 2 and 3 could be made to follow from 1, for instance, if the words were different. Remember - logic is about information, not about the world around us. When we make a valid argument, we’re learning about parts of language - propositions. The wording of the language matters, to the logic.

 

[wmw]

Will it be profitable to know the meaning of these symbols?
Symbols :
Keyboard ~ & | $ % ^

or would you expand the set to:

Symbols :
Keyboard ~ & | $ % ^ @ # ≠ ∈ ∉ ⊆ ≈ ≡ α β φ ψ θ

This seems a rather daunting task to learn this Fitch natural deduction language.

 

[Rhubarb]

Will it be profitable to know the meaning of these symbols?
Symbols :
Keyboard ~ & | $ % ^

or would you expand the set to:

Symbols :
Keyboard ~ & | $ % ^ @ # ≠ ∈ ∉ ⊆ ≈ ≡ α β φ ψ θ

This seems a rather daunting task to learn this Fitch natural deduction language.

Well I completed two of the three undergraduate logic courses offered at my university. The first course dealt with sentential logic. The second class dealt with the more complex first-order logic. The symbols we used were the five basic connectives, and then two quantifiers. There are additional connectives and operators. For instance, “∈” is used in set-theory. If I recall, it means ‘is an element of [a set]’ and we didn’t use it. I’ll start from the top, hopefully that’ll make things clear. Natural deductions can be done with ALL the connectives and operators, so a complete explanation would include all of them. But, alas, I never did set theory or the crazy graduate logic courses. Yet. This might get long - and please remember I am not a teacher nor into grad studies, nor was logic my strongest suit. Any logicians out there, if I left something out or made a mistake, please correct me.

So logic is about sentences called propositions. These are sentences with truth values. There are what are called ‘atomic sentences’ and ‘compound sentences’. Atomic sentences that do not use connectives, and compound sentences use connectives. For instance, ‘It is raining right now’ is an atomic sentence. But ‘it is raining right now and I’m thirsty right now’ is a compound sentence. In the compound sentence, ‘and’ is the connective. When we regiment the sentences into logic, we can symbolize the former as just A. The latter requires the symbolization ‘A and B.’ See how the information given in each atomic sentence of the compound sentence is different? So they need different variables.

So, the five basic connectives are: negation (~ or ¬), conjunction (& or ^), disjunction (v), if-then, conditional, or implication (→ or ⊃), and biconditional (≡). These were all introduced in my (so I assume most) intro to logic course. First-order logic introduces the quantifiers: the existential quantifier (∃) and the universal quantifier (∀)

Negation is simple. It is just ‘not’ or ‘it is not the case that…’ So, if ‘It is raining right now’ is A, ~A reverses back to ‘it is not raining right now’. It makes true statements false, and false statements true. So, if A is false, ~A is true. If ~A is false, A is true.

Conjunction is what we use when see ‘and’ or anything that joins two sentences together. ‘It is raining right now AND I am thirsty right now’ or ‘The sun is up right now BUT I don’t like onions’ both use conjunction as their main connective. They can both be symbolized as A&B. (But not in the same argument) A sentence that uses conjunction as the main connective is true only when both conjuncts are true. So, A and B both have to be true for A&B to be true. If either is false, then the conjoined sentence is false.

Disjunction is what we use whenever we use ‘or’ or anything that works in a similar way. This is what is called the ‘inclusive or’ because it isn’t a ‘one or the other’ type of ors. “I will go to bed OR I’ll stay up talking about logic” is a disjunctive sentence, symbolized as ‘AvB.’ Disjunctive sentences are true so long as one of the disjuncts are true. (So A or B could be true, it doesn’t matter which. So long as one is true, the whole sentence AvB is true) For instance, “I live in California or the moon is made of green cheese” is a true sentence, at this moment.

Conditionals are the if-then statement. If it is raining then the sidewalk is wet. This is symbolized as A⊃B. This can be tricky. If A= It is raining and B= The sidewalk is wet, the sentence ‘the sidewalk is wet when it is raining’ can also be symbolized as A⊃B because it expresses the same information as the first sentence I gave. Conditional sentences are false only when the first part (the antecedent) is true and the second part (the consequent) is false. For the sample sentence, if ‘it is raining’ is true, and ‘the sidewalk is wet’ is false, then ’ if it is raining then the sidewalk is wet’ is false.

Biconditionals are the if-and-only-if statement. ‘The sidewalk is wet if and only if it is raining’ would be symbolized as A≡B. Biconditionals are true when both atomic sentences connected are true, or both false. When one is true and the other is false, then the whole compound sentence is false.

The quantifiers are a little more tricky as they aren’t used in the simpler sentential logic. They require the use of predicate-variable symbolization. I’ll give you the briefest of outlines, but really I think you can gloss over this. Basically, the existential quantifier, ∃, states that something exist. So, ∃xPx could mean ‘at least one thing is a peach’ or ‘peaches exist’ or ‘there exists some X, and that X is a peach.’ The truth of the sentence depends on what is called the ‘universe, or domain, of discourse’, which is just a fancy way of saying ‘the things we’re talking about’. For instance, if our domain is ‘furniture’ then the sentence ‘a chair exist’ is true because I happen to be sitting in one.

The universal quantifier, ∀, attributes a certain predicate to all things. (that’s a very fuzzy way to describe it, but I tried for like ten minutes to give a better description. The example might help) So, ∀xMx could say ‘everything is made of matter’ or ‘for all Xs, X is made of matter.’ Again, the truth of these sentences depend on our domain. If our domain is furniture, I think it’s true - all pieces of furniture are made of matter. But, perhaps,if the domain is ‘everything’, the sentence might be false. A soul isn’t made of matter, right? (Assuming Catholic teaching is true for the sake of argument)

 

[Rhubarb]

PART 2

So I dropped a lot of information in your lap and I don’t know if it’s all so helpful. But the take-away is that with the connectives and quantifiers we can translate a LOT of propositions into logic so we can analyze them for logical properties. Even VERY complex sentences can be laid out and their structure, which is sometimes obscure or ambiguous in natural language, can be made explicit.

Fitch-style deductions are easy to learn, but TERRIBLE to master. They were easily the hardest part of my logic courses, mostly because I’m very dyslexic and patters tend to not pop out at me. Basically, each connective has two rules and so do the quantifiers. One to introduce them, and one to eliminate them. (For the moment, let’s stick to just the connectives - sentential logic is way easier) One you know the rules, you can have at them and start deriving. Our professors always let us have a cheat sheet with the rules on them. Even logicians that have been doing it for years can be stumped on Fitch deductions. It just takes time cranking away at it.

There are more operators, too, called modal operators. But, let’s not get into them. I didn’t take modal logic, and my understanding of how they work is barely conversational.