Fundamentals of Logic - Part 1 (Statements and Symbols)

Summary

This video script delves into the fundamentals of logic, a critical science of correct thinking that has been studied for millennia, starting with Aristotle. It explains the concept of statements and their truth values, distinguishing between declarative sentences and non-statements like questions or commands. The script also introduces symbolic logic for precise argument analysis and touches on paradoxes and fuzzy logic. It further breaks down compound statements, using logical connectives like 'and', 'or', and 'if-then', and concludes with a discussion on logical symbols for formal representation, emphasizing the importance of binary truth values in logic and computer language.

Takeaways

• 📚 Logic is the science of thinking correctly and has been studied for over 2,000 years, with Aristotle being one of the first to study it seriously.
• 🔍 Symbolic logic allows for the breakdown of ordinary language to analyze meaning systematically, enhancing precision in arguments.
• 🗣️ Statements are the foundation of logic, being sentences that can be true or false, but not both.
• 🌰 Examples of statements include 'February has 30 days', '4 plus 2 equals 3', and 'Bill Clinton was US president', with truth values depending on context.
• ❌ Not all sentences are statements; questions and instructions cannot be classified as true or false, hence they lack a truth value.
• 🔄 Paradoxes, such as 'I am lying to you', are complex statements that cannot be given a truth value unambiguously.
• 🤖 Truth values are binary, similar to computer language's bits, existing in states of true or false, which are fundamental to meaning.
• 🌐 Fuzzy logic, a more recent concept, allows for degrees of truth, such as a statement being 30% true and 70% false.
• 🔗 Compound statements are made from simple statements combined with connectives like 'and', 'or', and 'but'.
• 🔑 Connectives are words or symbols that link simple statements to form compound statements, such as the conjunction (∧), disjunction (∨), and negation (¬).
• 🔄 The biconditional (↔) is a two-way relationship between statements, where an if/then clause goes in both directions.

Q & A

What is the fundamental concept of logic discussed in the video?

The fundamental concept of logic discussed in the video is the science of thinking correctly, which has been studied for over 2,000 years and involves the systematic analysis of arguments and statements.

Who is credited with the serious study of logic?

Aristotle is credited with the serious study of logic, although there were a few people looking at it before him.

What is symbolic logic and how does it help in understanding language?

Symbolic logic is a system that allows us to break down ordinary language and examine its meaning in a systematic way, enabling us to be more precise about the arguments we make when discussing any subject.

What is a statement in the context of logic?

In logic, a statement is a sentence that declares something and can be either true or false, but not both.

Why are questions and instructions not considered statements in logic?

Questions and instructions are not considered statements in logic because they do not declare something that can be classified as true or false; they are neither true nor false.

Can you explain what a paradox is in the context of logic?

A paradox in logic is a statement that cannot be given a truth value because it leads to a contradiction or a situation where it can be both true and false, such as the statement 'I am lying to you.'

What is the significance of truth values in logic?

Truth values are significant in logic as they represent the binary states of true or false, which are the foundation of meaning and the only way for anything we say to have meaning.

What is the difference between Boolean logic and fuzzy logic?

Boolean logic, also known as regular logic, states that a statement must be either true or false with no in-between. Fuzzy logic, on the other hand, allows for degrees of truth, where a statement can be partially true and partially false.

What is a compound statement in logic?

A compound statement in logic is a statement made of more than one simple statement or a simple statement and some form of connective, such as 'and', 'or', 'but', etc.

What are the different types of connectives used in logic?

The different types of connectives used in logic include conjunction (and/but), negation (not), disjunction (or), conditional (if-then), and biconditional (if and only if).

How are logical statements represented mathematically or symbolically?

Logical statements are represented using symbols such as ∧ for conjunction, ¬ for negation, ∨ for inclusive disjunction, ⊕ for exclusive disjunction, → for conditional, and ↔ for biconditional.

Outlines

00:00

📚 Introduction to Logic and Its Fundamentals

This paragraph introduces the concept of logic as the science of thinking correctly, which has been studied for over 2000 years. It highlights the importance of Aristotle's contributions and the development of symbolic logic for precise argumentation. The paragraph explains the foundation of logic in statements, which are sentences that can be true or false, and introduces the concept of truth values as binary states. It also distinguishes statements from non-statements such as questions and instructions, which do not have truth values. Paradoxes, such as the liar's paradox, are mentioned as complexities within logic, and the connection between truth values and binary digits in computer language is established, emphasizing the fundamental role of truth values in communication and meaning.

05:02

🔍 Exploring Advanced Logic Concepts: Fuzzy and Compound Logic

The second paragraph delves into advanced logic concepts, starting with fuzzy logic, which allows for degrees of truth rather than the binary true or false of traditional boolean logic. It mentions the relatively recent exploration of fuzzy logic since the 1960s and its application to complex situations. The paragraph then transitions to compound statements, which are made up of multiple simple statements and connectives like 'and', 'or', and 'but'. It explains the types of compound statements, including negations, conjunctions, disjunctions, and introduces the concepts of inclusive and exclusive disjunctions.

10:03

🔄 Understanding Conditional and Biconditional Statements

This paragraph focuses on the conditional and biconditional statements in logic. It explains the 'if-then' structure of conditionals, where the truth value of one statement depends on another. The biconditional is introduced as a two-way relationship between statements, symbolized by a double-headed arrow. The paragraph also discusses the use of mathematical symbols to represent different kinds of logical statements, such as the use of capital letters for simple statements and various symbols for connectives like conjunctions, disjunctions, and negations. It emphasizes the importance of these symbols in formalizing and simplifying the representation of logical arguments.

15:04

📘 Symbolic Representation in Logic

The final paragraph provides a deeper look into the symbolic representation used in logic, including the use of arrows for conditional statements and double-headed arrows for biconditionals. It explains that the biconditional can also be represented as the conjunction of two reversed conditionals. The paragraph reinforces the importance of these symbols for mathematicians to formalize logic and make it easier to write and understand complex logical expressions. It concludes by noting that while there are many symbols, the basic ones mentioned will be the focus of study.

Keywords

💡Logic

Logic is defined as the science of thinking correctly, which is foundational to the video's theme of understanding the basics of logical reasoning. It has been studied for over 2,000 years and is integral to the systematic analysis of arguments in any subject. The script emphasizes that logic allows for precision in discourse by breaking down language and examining meaning.

💡Statements

Statements are sentences that declare something and can be classified as true or false, serving as the building blocks of logic. In the script, examples of statements are given, such as 'February has 30 days' and 'Bill Clinton was US president,' to illustrate how they can be evaluated for their truth value.

💡Truth Value

Truth value refers to the binary state of a statement being either true or false. It is crucial to the video's narrative as it underpins the meaning of any statement made. The script explains that truth values are fundamental to logic and are analogous to binary digits in computer language.

💡Paradoxes

Paradoxes are self-contradictory statements that cannot be assigned a truth value, adding complexity to the study of logic. The script uses the example 'I am lying to you' to demonstrate how paradoxes challenge the binary nature of truth values, as they can neither be definitively true nor false.

💡Boolean Logic

Boolean logic, named after mathematician George Boole, is a form of regular logic where statements are strictly true or false, without any intermediate degrees of truth. The script discusses this as the primary type of logic it will focus on, contrasting it with fuzzy logic.

💡Fuzzy Logic

Fuzzy logic is a form of logic that allows for degrees of truth, rather than the binary true or false of Boolean logic. Mentioned in the script, it is a newer concept that attempts to apply logical principles to complex situations that cannot be easily broken down into Boolean terms.

💡Connectives

Connectives are words or symbols used to combine simple statements into more complex ones. The script provides examples such as 'and,' 'or,' and 'but,' which are essential for constructing compound statements in logic.

💡Compound Statements

Compound statements are formed by combining simple statements using connectives. The script explains that these are analogous to chemical compounds, made up of different elements, and are vital for constructing more complex logical arguments.

💡Negation

Negation is the process of reversing the truth value of a simple statement. In the script, it is demonstrated with phrases like 'not' or 'naught,' showing how a statement like 'tomorrow is Tuesday' can be negated to 'tomorrow is not Tuesday' to create a new statement with an opposite truth value.

💡Conjunction

Conjunction is a type of compound statement formed by connecting two simple statements with 'and' or 'but.' The script uses the phrase 'I am coming but Alice isn't' to illustrate how conjunctions combine statements to convey more complex information.

💡Disjunction

Disjunction is another type of compound statement that connects two simple statements with 'or.' The script differentiates between inclusive disjunction, where both statements can be true, and exclusive disjunction, where only one can be true, using examples like 'I am going to eat pizza or ice cream' and 'I am alive or I am dead.'

💡Conditional

A conditional is a statement with an 'if-then' structure, where the truth of one statement depends on another. The script explains that if one simple statement is true, then the conditional implies that a second statement will also be true, such as 'if it rains, then the ground is wet.'

💡Biconditional

A biconditional is a statement that establishes a two-way relationship between two statements, meaning the 'if-then' clause can go in both directions. The script uses the term 'if and only if' to illustrate this concept, indicating that both statements are true or false simultaneously.

Highlights

Logic is the science of thinking correctly and has been studied for over 2,000 years.

Aristotle was one of the first to seriously study logic, leading to the development of symbolic logic.

Symbolic logic allows for the systematic breakdown and analysis of ordinary language to enhance precision in arguments.

Statements are the foundation of logic, being sentences that declare something and can be true or false.

Paradoxes, such as 'I am lying to you,' cannot be assigned a truth value and are a complexity of logic.

Truth values are binary, existing in states of true or false, similar to the binary nature of computer language.

The concept of truth value is fundamental to meaning, as it is the only way for statements to have meaning.

Fuzzy logic, introduced in the 1960s, allows for degrees of truth, differing from traditional boolean logic.

Boolean logic, named after George Boole, is the focus of the lecture and does not allow for degrees of truth.

Compound statements in logic are made of more than one simple statement and a connective, similar to chemical compounds.

Negation is the reversal of a simple statement's truth value, often indicated by words like 'not' or 'naught'.

Conjunctions combine two simple statements with 'and' or 'but', indicating both must be true for the statement to be true.

Disjunctions use 'or' to connect two simple statements, allowing for one, both, or neither to be true.

Exclusive disjunctions are a type of 'or' statement where only one of the statements can be true, not both.

Conditional statements follow an 'if-then' structure, where the truth of one statement implies the truth of another.

Biconditional statements establish a two-way relationship between two statements, indicated by 'if and only if'.

Mathematical symbols are used to represent different types of statements and their relationships in a formalized way.

The biconditional can also be represented as the conjunction of two reversed conditionals.

Transcripts

hello everyone this is the fundamentals

of logic we're going back to basics

because maths although it can get very

complicated starts from some fairly

basic fundamentals and that's what we're

going to look at here so what is logic

well it's the science of thinking

correctly sounds important doesn't it so

it's been studied for over 2,000 years

it was first studied seriously by

Aristotle although there were a few

people looking at it before him but

since that time it has become much more

ordered and much more well-defined

symbolic logic which we will also be

looking at allows us to break down

ordinary language and look at its

meaning in a systematic way and that

allows us to be much more precise about

the arguments we make when discussing

any subject so the foundation of logic

is statements statements are sentences

which declare something so a statement

can be true when it declares something

that exists or false when it declares

something that doesn't exist but it

cannot be both so here are some examples

of statements February has 30 days four

plus two equals three times two Bill

Clinton was US president and tomorrow is

Tuesday so depending on when you are

watching this either two or three of

those statements are actually true but

you'll notice that there are other

things which are still sentences but

they are not statements they do not

actually classify as either true or

false

have you done it hand me a towel turn

the computer off

and stop the bus why are these not

statements why can they not be true or

false because they are either questions

or they are instructions and they cannot

be given a truth value so there are some

kinds of statements that cannot be given

a truth value and these are called

paradoxes for example I am lying to you

now this is a paradox if I am telling

the truth

when I say online to you then I'm

actually not lying to you and therefore

the statement is false however if I am

lying to you then when I say I'm lying

to you I'm actually telling the truth so

the statement is true so for whatever

reason we cannot give this statement a

truth value it cannot be declared as

true or false

at least not unambiguously so paradoxes

are one of the complexities of logic and

we're not going to go into them in too

much depth

however suffice to say trying to resolve

them and how they work is one of the

ways we create new ways of thinking but

for now let's just keep it simple so I

mentioned this term a truth value and

that is simply the idea of a statement

being true or false so this would seem

kind of obvious but it is impossible to

overestimate how important it is it is

huge

truth values are binary in other words

when they exist they are in one of these

two states true or false

something else that is binary is

computer language and the basic

fundamental unit of computer language is

the bit the binary digit and that is

either a 1

or zero and a bit is the most

fundamental unit of information it is

the smallest amount of information where

it is possible to communicate anything

at all

up or down yes or no in or out 1 or 0 so

truth values are very simply the

foundation of meaning they are the only

way that it is possible for anything we

say to mean anything at all so another

complicated form of logic that we are

not going to go into in detail is fuzzy

logic so I just said that statement must

be either true or false and in regular

logic which is called boolean logic

after the mathematician George Boole

this is true and this is exactly the

kind of logic we will be focusing on but

there is another form of logic where

there can be degrees of truth so for

example a statement might be let's say

30% true and 70% false so this was first

investigated in the 1960s and so it's

still relatively new in mathematics it

hasn't been fully explored but we're

learning more about it all the time in a

way it's attempting to use the

principles of logic on situations which

are too complex to be broken down into

the more straightforward boolean logic

perhaps in fact there is a boolean

output for this fuzzy logic and the

problems it creates but because we do

not have the ability either in our human

minds or even in our great computers to

work

all the variables we have to rely on

this probabilistic form of logic where

there are different degrees of truth but

again we're not going to go into that

but I think it's important to mention

that logic has progressed beyond the

simple form of boolean logic that we

will be investigating so now we're going

to construct more complex statements and

these are called compound statements so

in chemistry a compound is a substance

that is made of two or more different

elements and in logic a compound

statement is a statement that is made of

more than one simple statement or as we

will see a simple statement and some

form of connective what is a connective

well some examples are words like and or

but and this gives us a range of

different types of statement and we'll

look at some of those in the next slide

so what types of statement can we have

well we've already talked about simple

statements a sentence that declares

something true or false and following

that we have a negation which is a

simple statement that has a reversed

truth value how do we do this well we

might use a word like naught or some

other negative word so in the example

above we saw tomorrow is Tuesday and

that can be true or false let's assume

that it is false which will be the case

6 days out of 7 for anyone listening to

this so if tomorrow is Tuesday is a

false statement then if we say tomorrow

is not Tuesday then that

be true so negations are reversed truth

values of simple statements next we have

the conjunction which is two simple

statements connected by and or but so

for example I am coming but Alice isn't

is a conjunction and in fact there's a

negation in there too you might be able

to spot I am coming but Alice is not

coming

so that's actually a conjunction with a

negation thrown in as well or I am going

to play tennis now and then football

later that is also a conjunction using

and so that's the form of conjunction

but we may also have a disjunction and

these are two simple statements

connected by or so there are two types

of disjunction there's one where it can

be either or or possibly both serve for

example I am going to eat pizza or ice

cream tonight there's nothing really

that prevents us from eating both pizza

and ice cream so it could be one or the

other or possibly both at least in the

basic form there on the other hand if I

say I am alive or I am dead

well there's pretty much no way it can

be both of those we're not counting

zombies here so that is an example of a

disjunction which is either/or but not

both and this is called an exclusive

disjunction

there's also the conditional which is a

statement that follows an if-then

structure and if one simple statement

then a second simple statement is the

way it works it's a way of connecting

simple statements together in a form

which means that one of them depends on

the other for its truth value and then

there is the biconditional which is a

statement which establishes a two-way

relationship between statements

so the if/then Clause goes in both

directions in that case so that's just

an introduction to the different types

of statement but we will look at them

all in more detail below so there are

various more mathematical ways to deal

with logic mathematicians like to be

formal and they like to make things

easier to write out so there is a list

of symbols that we can use for

representing different kinds of

statements we often use capital letters

for simple statements and then we

connect them by putting a range of other

symbols in between so for example the

end which is the conjunction function

and this can also of course mean but

which logically is very similar or the

same is represented by this inverted V

or a carry or correct symbol it can also

be represented by the ampersand symbol

and either of those stand for P and Q is

P and Q or possibly P but Q which is

logically the same so that's the

conjunction not or a negation is

represented by the tilde symbol that

little wavy line or alternatively by

that strange

sort of square hook symbol which I don't

know the name for so tilde R or weird

little square hook and R means not R

negation of our next we have the

inclusive or the inclusive disjunction

which is represented by a V or double

line bars and so that means Q inclusive

or R stands for either Q or R or

possibly both so that's the inclusive

disjunction for the exclusive

disjunction we use an underlined V or

that little crosshair symbol and so that

would mean either P or R but certainly

not both P exclusive or r either P or R

but not both and it's generally better

if you're using these symbols to make

your connective symbols slightly smaller

than the capital letters you are using

and obviously stay away from using V or

any other confusing symbols as any of

your simple statements from there we can

move on to the conditional which is

represented by an arrow pointing to the

right so P conditional Q is saying if P

then Q or perhaps P implies that Q so if

P is true then the conditional

relationship means that that implies

that Q will also be true it's important

to note that just because we write a

conditional out it doesn't automatically

follow that that implication is always

correct the by conditional as you may

also have guessed is a double-headed

arrow because as we said it can be a

to weigh truth relationship and there's

also another way to write it and that is

if in other words I double F and this

means R if and only if s so our

biconditional s means r if and only if s

and we'll go into a little bit more

about what that means

shortly so we can also show that the

biconditional is the conjunction of two

reversed conditionals in other words two

conditionals where the first statement

and the second statement are traded

swapped around and those two

conditionals are placed in a conjunction

so as you might expect there are other

symbols that we can use but these are

the basic ones and the ones that we will

stick to in what we study