Logics handbook

**Rodion Romanovich** · 02-22-2007, 21:26

A short guide to modern logic, as a simple guide and reference for the philosophical discussions in the backroom. I'll edit this post to add the basics of logic to provide a simple handbook of logic, and hope it could be stickied. This thread will hopefully allow people to refer easily to the basic rules of logic when needed in the discussions here, and will avoid making other threads cluttered with discussions about the basics of logic, which usually isn't relevant to the other discussions. Please add comments in posts below (no off-topic please, just discuss whether some of the rules of logic here are incorrect, and/or you thought a certain section was badly phrased).

INTRODUCTION - what is logic?

Question: what is logic?
Answer: logic is a study of patterns in reasoning. There are certain patterns in reasoning that are general no matter what subject we are reasoning about, such as "either x is true, or x is false". Logic contains, among other things, methods for verifying whether a statement is justified under certain assumption.

Question: what is the difference between classical logic and modern logic?
Answer: in classical logic, most reasoning is done in imprecise natural language such as English, German, French, etc. These languages are problematic because some words have multiple meanings, and it is easy to make fallacies without spotting them. Modern logic is a formal version of classical logic, which allows the user to more easily spot fallacies and evaluate the correctness of proofs. Whereas classical logic recognizes all the fallacies that modern logic recognizes, the usage of classical logic makes it more difficult to spot the fallacious patterns in more complex statements.

Question: has classical logic no advantage over modern logic?
Answer: classical logic is simpler to grasp, and sounds more convincing to laymen, since the statements are made in natural language. Classical logic also allows us to get a point through more easily, since it takes less thought to make an argument in natural language than it takes to carefully formalize an argument to live up to the standards of modern logic. The fact that classical logic is so error-prone in more complex situations, and allows for making incorrect proofs look convincing, is however a good reason to apply modern logic in cases where more complex arguments are given. So one method of taking advantage of the strengths of both classical and modern logic is to first make all arguments in classical logic natural language, and then, if requested by the others in the discussion (because they either doubt its validity, didn't understand it, or thought your presentation was too fuzzy), formalizing the arguments.

CHAPTER 1 - basics of propositional logic

Propositions

The simplest form of logic deals with propositions. Propositions are statements that are either true or false, but not both or neither. Questions and paradoxes are for example not propositions, since a question can't be true or false (the answer to the question that can be true or false though), and a paradox can be both true and false at the same time. Subjective statements aren't propositions, since the taste implied by them belongs to someone, and until that someone has been specified, the statement isn't any proposition. Statements which contain unspecified parts of this type or other types, can be made into propositions by specifying the unspecified part.

Examples of propositions:
- the car is red
- the house is blue
- God exists
- God doesn't exist
- Leprechauns are green (even though Leprechauns don't exist, this proposition, which asserts that something non-existing has a certain property, is a valid proposition)
- John thinks ice cream is tasty (note that even though tasty is subjective, we have specified that we're talking about John's taste here, and that way this statement is specified to be either true or false, i.e. a proposition. "Ice cream is tasty" without tying it to a person, is however not a proposition)

Examples that aren't propositions:
- is the car red?
- does God exist?
- the hairdresser cuts all persons who don't cut themselves (a paradox)
- let S be the set that contains all sets that don't contain themselves (a paradox)
- ice cream is tasty (this is subjective)

Often, propositions are denoted by shorter variables to allow for a more compressed notation. For example:
P1 = the car is red
P2 = Leprechauns are green
Later in reasoning, these variables can be used in shorthand notations like this: "P1 ^ P2 => P3"
Every variable has a truth value. The truth value is either "true" or "false". P1 is said to have the truth value "true" if P1 is true. If P1 is false, it is said that "P1 has truth value false".

The word literal (noun) can denote either a variable such as P1, or a truth values such as true or false.

Dealing with the unknown

A proposition IS either true or false, but sometimes we may not always know the truth value of the proposition - however it still IS either true or false. This is an important distinction to remember! Just because we don't know the truth value of a statement, that doesn't mean it isn't a proposition. To determine if it is a proposition, we must determine whether it IS true or false, not whether we know that it has either of these values.

For example: if we have a statement of the form "the sun is yellow", that statement is either true or false. However, if we have a statement of the form "the person is evil", we can't know whether he/she is evil, until we know which person the statement refers to. There is something unspecified about that statement. More formally, we often write this statement as "x is evil", to make sure that the unspecified part of the proposition is clearly visible. This is often written like this to be even more clear:
P(x) = x is evil
where a comma separated list of unspecified things is placed between the parentheses to the left of the equals sign, and the statement on the right side of the equals sign is the original statement, but with the unspecified things replaced by symbols such as x, y, etc. We can call such unspecified statements logical functions.

We can't determine the truth value of "x is evil" immediately, but we can determine the value of "x is evil" once we have specified what x is. When we for instance write "Sauron is evil", we have a statement that is either true or false, i.e. a proposition, by the definition in the beginning of the chapter. By adding a variable like this we can keep track of which parts of the statement are unspecified and must be specified before we can tell what the truth value of a statement is. In normal discussions, we may say things such as "he is evil", without specifying who "he" is. If it is in a conversation where the context makes clear what "he" we are referring to, we aren't making a fallacy, but to be formally correct (and formal correctness is the best way of checking the validity of an argument, since it uses fool-proof techniques of validation - more on this later), we should in written logical arguments, if we want to be sure to not make implicit assumptions, we could write the statement "he is evil" as either "John Smith is evil" (if it was John Smith we referred to), or "P(John Smith)" and "P(x) = x is evil". To be absolutely strict, there could be many people with the name of John Smith, so it is often difficult in practise to enforce that things are specified so they are completely unambiguous. However, as long as all participants in a logical discussion have had the concepts well enough specified to unambiguously know what is referred to, it is usually a part of formality that can be skipped.

When it comes to unspecified statements of this type, we shall later see some interesting phenomenons. For instance, for some unspecified statements it could be that for all possible specifications of the statement, it will have the same truth value - does it then really differ from a proposition? For instance: "x thinks icecream is tasty, where x is a person" can be specified by providing which person x is. If it would be that all persons like icecream, then this statement is true for all possible specifications of the unspecified parts of it. Before such a thing has been proven, the statement can't be treated as universally true or universally false, but rather as a statement which could possibly become false in some specifications, and true in others. To avoid this problem of sometimes being allowed to use it as a proposition, and sometimes not, we can add quantifiers such as "for all x, where x is a person, x likes icecream". This is a proposition, since it's either true or false. The subject of dealing with unspecified statements of this type (called predicates) really belongs to the next chapter, but I chose to place an introduction on it here, because the principle of partly unspecified statements is a recurring and important theme, which will also be used to some extent in this chapter. To look at all types of partly unspecified statements in a general way can aid in understanding several important principles of logic.

Word definitions

Every proposition contains words, such as "car", "Leprechaun", "red" etc. Since words can have multiple meanings in natural languages, it is important that we make our word definitions more exact before a logical discussion, by clearly stating which definition of the word we are using.

The following is an example from wikipedia of what can happen if we fail to define beforehand what meaning of the word we are using:
A feather is light.
What is light cannot be dark.
Therefore, a feather cannot be dark.

In the beginning, we let the word "light" denote "small weight", whereas in the later half of the argument, we let it denote "highly reflective in all visible radiation frequencies" (or something like that - I will refer to the physics experts to get a better definition of light in the sense of electromagnetic radiation as opposed to light in the meaning of weights...).

FALLACY1: The fallacy of redefining words in the middle of an argument, is called equivocation.

Another fallacy than redefining words during the course of making a logical argument, is that of not defining the words at all - giving them either vague meanings of no meanings at all. The following proposition illustrates the fallacy:
"cars are sdasfsdtgres"

What does "sdasfsdtgres" mean? Until "sdasfsdtgres" has a clear meaning, the proposition can't be either true or false, but can be either, depending on what meaning we choose to assign to the word "sdasfsdtgres" (which from the context, would seem to be an adjective or a noun in plural).

FALLACY2: Failing to define words or giving them a vague meaning, to be able to assign varying meanings to them at will at arbitrary points of an argument. This is basically the same fallacy as equivocation, but many who are know about the equivocation fallacy still make this fallacy. NOTE: we are NOT making a fallacy is we are making a vague definition, but don't use the vague parts of the word definition in our arguments. For instance, if we say "cars are things that have 4 wheels and an engine", and want to prove that all cars have more than 2 wheels, it doesn't matter that our definition is vague about engine performance, speed, type of wheels and many other things, which we would normally need to distinguish a car from say a gokart or toy car, since the proof of the statement that a car has more than 2 wheels only relies on the not at all vague fact that a car has precisely 4 wheels, and 4 is larger than 2. But one has to take care when using vague word definitions, since they make it easy to make fallacies even when not intending to do so.

Combining propositions

Propositions can be combined by operators such as AND, OR and NOT, to form new, more complex propositions. For example: the car is red AND the sky is blue.

There are simple rules for finding out the truth values of propositions formed by combining simpler propositions. These rules are often represented by truth tables. A truth table contains all combinations of truth values for the variables involved in the expression, and what truth value the combined expression has for a particular such combination of component propositions.

For example, the truth table for AND can bewritten like this (note that "f" and "t" below denote "false" and "true", respectively):
A B A AND B
f f f
f t f
t f f
t t t
This truth tables says in clear text that (A AND B) is true only when both A is true, and B is true, but false for all other combinations of truth values for the variables A and B. While a truth table is of little use to describe something as simple as AND (the above table should agree with the intuitive meaning of the operator "AND"), which our language describes just as well, truth tables will later be useful for defining less intuitive operators such as implication (see below), which is often confused with equivalence (see below) in natural language.

More about combining propositions

Here I will present a longer list of methods of combining propositions, and about the most common operators for combining propositions. An operator is (to simplify and not draw in maths into the discussion) something that is applied to one or two propositions, to acquire a new proposition - for example AND, OR, NOT are operators.

Before starting, I will also mentioned that it is possible to form long chains of combinations of propositions, by applying the operators to entire logical expressions. For example, we can write: "the sun is yellow AND leprechauns are green AND it is raining today". When putting expression together in this way, it's important to use parantheses, just like in math, in order to make sure the intended meaning is preserved (Edit: some courses use the notion of truth trees, rather than parentheses, to demonstrate evaluation order). For example, consider the statement:
"it is raining or it is snowing and it is cold"
Let p1 = it is raining, p2 = it is snowing, p3 = it is cold
(side note: these propositions contain unspecified parts, can you see which?)

Spoiler Alert, click show to read:

Let's assume that at the time, these three statements have the following truth values:
p1 =true, p2=false, p3=true
Then consider the following two version of parenthesizing these statements:
(p1 or p2) and p3
p1 or (p2 and p3)
The first version has truth value "true", while the second has truth value false. Obviously, we mean different things when we say things such as:
"it's snowing, or it is raining and it's hot" and "it's snowing or raining, and it's hot".

How to evaluate logical expressions: The way to evalute expressions with parentheses is to first replace variables with their truth values, then first evaluate the truth value of the innermost parenthesis, then replace that parenthesis with the resulting truth value, then repeat this process until only one truth value is left (Edit: some courses use the concept of "truth trees" to describe this). For example:
(p1 or p2) and p3
becomes
(true or false) and true
which becomes
true and true
which becomes
true

Similar methods for simplification can be used with other operators, such as NOT, => (implies) and <=>, for instance. The way to apply them is to look at the literal (literal will here denote either a variable such as A, or a truth value such as "true" or "false") to the left and right of the operator, and see what they are, then look at the row in the truth table which is for that combination of values. I.e. if we have the statement "true OR false", we look at the row where A is true and B is false of the truth table for "A OR B". Then we read the value in the column "A OR B". The value we obtain, is the value of "true OR false". Test this with well-known operators such as AND, OR and NOT to know you're reading from the table in the right way, so you can apply this method of reading correctly to the other operators, which you may not be familiar with beforehand.

Here's a list of more logical operators with their truth tables:
http://en.wikipedia.org/wiki/Truth_table
Logical conjunction is the same as AND.
Logical disjunction is the same as OR.
Logical negation is the same as NOT.

Logical implication needs some further discussion! Logical implication a => b (this is often read as "a implies b", or "if a then b") has the following truth table:
A B A => B
f f t
f t t
t f f
t t t
So the statement A => B is true in all cases, except the case when A is true and B false. Implication is used in logical proofs and is therefore a very important operator to understand well!

Logical equivalence is defined so that A <=> B is true when A=true and B=true, or when A=false and B=false. But when A and B have different truth values, A <=> B is false.

Some controls to see if you got it right. Evaluate the following expressions:
1. true AND false
2. true AND (false OR false)
3. (not true) OR (not false)
4. true => false
5. true <=> false
6. true <=> ((false OR true) AND (true OR true))

Spoiler Alert, click show to read:

Let's not reverse the whole thing. If we know that "A => B" is true, what can we tell about the truth values of A and B? We simply read the truth table backwards:
A B A => B
f f t
f t t
t f f
t t t
we check the rows where A=>B is true (all except the third row), then check the values of A and B we can find on those rows. We see that the following cases exist: "A and B", "(not A) and B", "(not A) and (not B)". This might be a disappointment, if we expected to determine unambiguously what truth values A and B could have - as it is now we know that A can be either true or false, and B can be either true or false, but the combination where A is true and B is false can't exist. However, it is commonplace in logic that we can tell fewer things that we want for sure...

Tautologies

An expression that is always true is called a tautology. Here are examples of tautologies:
- A or (not A)
- A or (A or not A)
- not (A and (not A))
- true
- true AND true
Note that this statements is NOT a tautology:
- A or B
because there is an example of assigning truth values to A and B such that it becomes false: if A = false and B = false, the statement evaluates to false.

I hope it is clear what the distinction is! If you have an expression containing only the literals "true" and "false", you have something that is either true or false, and if it is true it is a tautology. But if you have an expression containing varaibles such as A, P1 or something like that, you have a tautology if for every possible combination of assigning the truth values true and false to all different variables in the expression, the expression will evaluate to true.

Basically, in above sections, we evaluated statements with the literals "true" and "false", but here we will try to evalute statements containing variables, whose truth values we don't know. If the statement evaluates to true for all possible assignments of truth values to the variables, we have found a tautology. If the statement evaluates to false for all possible assignments of truth values to the variables, we have found a false statement. If the statement evaluates to true for some and false for some of the possible assignments of truth values to the variables, we have found neither a tautology nor a false statement, but something that can be true in some cases, and false in other cases (EDIT: according to Sjakihata, these are called contingent statemenets).

Note here we can make a parallell to logical functions - there is something unspecified over a statement such as "P1 AND P2", until we know the truth values of P1 and P2.

Evaluating logical expressions containing variables whose truth values we don't know: How do we evaluate expressions of this form, where we don't know the truth values of all the variables included in the expression? The most general method (and that is also the easiest one to do, but it is quite slow compared to the more complex methods), is to evaluate the expression for all possible combinations of assigning "true" and "false" to all variables in the expression. For example:
- to find the truth value of (A and B), we must evaluate it for the 4 possible ways of assigning "true" and "false" to each variable. The following cases must be tested:
A=true, B=true which gives "true AND true" which evaluates to true
A=true, B=false which gives "true AND false" which evaluates to false
A=false, B=true which gives "false AND true" which evaluates to false
A=false, B=false which gives "false AND false" which evaluates to false
As you can see, there are both cases where A AND B is true, and where it is false, so A AND B isn't a tautology, nor is it a statement that is always false (a so-called contradiction). We will find out whether it is true or false, when we know the truth values of A and B in our particular case.

Here is however an expression that evaluates to true for all combinations:
A OR (NOT A). Let us test all cases:
A=true, B=true which gives "true OR (NOT true)" which evaluates to true
A=true, B=false which gives "true OR (NOT false)" which evaluates to true
A=false, B=true which gives "false OR (NOT true)" which evaluates to true
A=false, B=false which gives "false OR (NOT false)" which evaluates to true
So A OR (NOT A) is a tautology. It was hardly any secret that A OR (NOT A) would be true no matter what truth vale A had... The statement basically said that either something is true, or it is false, and that follows directly from the definition of a proposition...

Tautologies are important in proving things, more on this later.

Form rather than substance: Logic isn't concerned with substance, but with form. Once we've proven that something of the form A or (not A) must be true, we can apply this principle to any statement with this form, no matter what A happens to refer to in this case. Now in a way this also relates to substance of course, since we can hardly find any substance in a statement which tries to contradict a principle such as that A or (not A) is true.

Faster evaluation of logical expressions containing variables: Because A or (not A) is true for all possible choices of A, this form can be applied to all statements containing this form somewhere. This is interesting, because it allows for a much faster method for evaluating expressions, than evaluating the truth value for all possible combinations of assignments of truth values to the variables. By proving several simple theorems such as A and "(not A) <=> false" to be tautologies, we can later use these theorems in expression simplification. An example:
B and (not (B and B))
we immediately recognize the pattern (B and B) to be equivalent to simply "B" alone. By applying this simplification to the innermost parenthesis, we obtain "B and (not B)". We then immediately recognize "B and (not B)" as "false". Recall that equivalence of the form A <=> B means "A is true whenever B is true, A is false whenever B is false". Therefore, logical rules of this form are often written like this: "B and (not B) <=> false". The equivalence means we can, whenever we find the left hand side, replace it with the right hand side if we wish, and whenever we find the right hand side, replace it with the left hand side. Below follows a longer list of logical rules.

Some equivalence rules in propositional logic:
A and true <=> A
A and false <=> false
A or true <=> true
A or false <=> A
A and (not A) <=> false
A or (not A) <=> true
A and (B and C) <=> (A and B) and C
A or (B or C) <=> (A or B) or C
not (A or B) <=> (not A and not B)
not (A and B) <=> (not A or not B)
(A and (B or C)) <=> ((A and B) OR (A and C))
(A => B) <=> not (A and (not B))
not (not A) <=> A

There equivalences are not the only rules we can make in logic. We can also make implication rules. They mean that whenever we find the left hand side, we can replace it with the right hand side, but we can't go the other way (note: in writing, sometimes we consider it practical to write the implication "arrow" pointing to the left, like this: <=, in which case it is the right hand side that can be replaced by the left hand side, and the left hand side can't be replaced by the right hand side).

Some implication rules in propositional logic:
A and (A=>B) => B
this is called "modus ponens" (used in relation to proofs, see below)
(not B) and (A=>B) => not A
this is called "contrapositive" (used in proofs, see below)
B and (A => (not B)) => not A
this is used in proof by contradiction (see below)

Logical arguments

Logical arguments are implications of the form P => C, where C is called conclusion, and P is called premise (P could also be a list of propositions combined by AND, like this: "(pigs can fly AND the sea is blue) AND Leprechauns are green").

Definition: Valid logical argument. A valid logical argument is a logical argument where P => C is a tautology. Recall from the definitions above what this actually means! A tautology is an expression that evaluates to true for all possible combination of inserting truth values to the variables in the expression. In clear text, a valid logical argument is a statement that says: "if the premise P is true, then the conclusion C is true, but if P is false, we have no idea whether C is true or not - if P is false C may be true, but it may also be false".

A proof is a demonstration that a logical argument is valid, i.e. a demonstration that P => C evaluates to true for all possible ways of inserting truth values for the variables in the expression P => C.

Small note: the form of logical arguments might seem unintuitive, since if we know the premise to be false, the argument can be valid while having a false conclusion - by definition of implication, the implication A => B is true is A is false and B is either true. Thus if we consider it a fact that pigs can't fly, the following logical argument is valid:
P1 = Pigs can fly, P2 = Leprechauns exist
P1 => P2
(this is read as: pigs can fly, therefore leprechauns exist)

FALLACY 3 - assuming that a valid logical argument alone proves the conclusion a proof ISN'T an assertion that C is true. It is an assertion that IF the premise P is true, THEN the conclusion C is true. The logical rule called modus ponens is what people usually are searching for in this context. If we want to show that C is true, we need to show that P is true AND that P => C is a tautology (failing to prove either, means that we have failed to prove C, unless we have some other proof of C). The rule for modus ponens says: (A AND (A=>B)) => B. In clear text, this means "if we know both that A is true and that A implies B, then we know that B is true".

Exercise: try to prove modus ponens by showing that it evaluates to true for all possible ways of inserting truth values for A and B.

Spoiler Alert, click show to read:

Some more exercises:
1. are the following arguments valid?
a. p1 => p2
b. (p1 OR p2) AND (p3 OR p1) => p3
c. p3 OR p1=> (p3 OR p1) OR p2
From now on let p1 = true, p2 = false, p3 = false
d. p2 => p3
e. (p2 and (p2 => p3)) => p3

Spoiler Alert, click show to read:

Some details

This section isn't necessary to understand logic since it adds nothing on logics, but more on the way logic is often written, but it may facilitate reading about logic in other sources.

Omitting paranteheses, and precedence:
Often to make it more practical to write the rules, parantheses are omitted where they aren't needed. For instance in A AND B AND C, we get the same truth value regardless of whether we read the expression as (A AND B) AND C or as A AND (B AND C). Operator precedence is also used to remove some need for parentheses. AND can for instance have higher precedence than OR, so that when we write A AND B OR A AND C it is automatically interpreted as (A AND B) OR (A AND C).

Alternative way of writing proofs:
Another common way of writing to be aware of, is that logical arguments are often written like this:
P
==========
C

This is to be interpreted in exactly the same way as P => C. The reason for this alternative way of writing the logical arguments is because it is more readable, especially if we have longer arguments. Equivalence and implication has low precendence, so you seldom need to put parentheses around what is written to the left and right of an implication or equivalence operator (though it happens that is is needed). If we have multiple premises, we often write the logical arguments like this

P1
P2
P3
============
C

Often, people make many conclusion steps, in which case an argument can be written a bit like this:
A
therefore B
therefore C
==========
therefore my final conclusion D

This is really not a single logical argument, but multiple ones. To be more clear about what you mean, and to allow for easier validation/invalidation of the argument, these should if needed be rewritten into separate logical arguments, none of which contain a "therefore" step in the list of assumptions (premises) listed above the horizontal line. More clearly rewritten, the above becomes:
A
==
B

B
==
C

C
==
D

This is however not necessary. Two problems can occur if you fail to do it:
- if you accidentally omit the word "therefore" in some statement above the horizontal bar, it gets status as an assumption, rather than as a conclusion
- it isn't immediately clear which steps follow from which. Someone trying to check the proof for validity might need to check much more complicated combined statements. Example:
A
therefore B
therefore C
therefore D
=========
therefore E

In validating D, should we use "A and B and C" as premise, or just C? It will take much longer to evaluate (A and B and C) => D than to evaluate C => D.

Alternative symbols for operators:
Often the logical operators AND, OR and NOT are denoted by different symbols than what we've used here. The symbols include "^" (without the quotation marks" for AND, and "V" for OR, and a single quote or a tilted "L" for NOT, for instance.

More about word definitions

As mentioned above, a common source of errors in logical reasoning is to make incorrect and ambiguous word definitions. Many words in natural language (such as English, French, etc.) have multiple meanings. If I say "car", do I mean a Porsche, or a T Ford? If I just meant something with 4 wheels and an engine, it perhaps wans't important, but if I was talking about a way of getting to the beach quickly, perhaps if someone in a car rental firm misunderstood and thought I meant a T Ford, when I was thinking about a Porsche, I wouldn't get to the beach as quickly as I wanted. The multiple meanings of words can have worse consequences than that, though... The error of using the same word to denote different things at different points in a proof is called equivocation (mentioned briefly above in the section about word definitions). Here's an example of an equivocations:

Medieval Mike doesn't pay the church tax
Paying the church tax is required to follow the laws of the church
Medieval Mike doesn't follow the laws of the church
Those who don't follow the laws of the church are heretics
Heretics are murderers
Medieval Mike is a murderer
All murderers should be killed
Medieval Mike should be killed

The equivocation occurs in the step "heretics are murderers", when the word heretic, which was initially used in the meaning "those who don't follow the laws of the church". Because of the equivocation fallacy, we could make it look like Medieval Mike was a murder that should be killed.

To avoid equivocation, the most common method is to define all words unambiguously at the beginning of a discussion, or at the beginning of a logical argument. Rewritten this way, we can't prove that Medieval Mike should be killed, using the above assumptions.

Changing word definitions doesn't change the argument in most cases: Some other notes should also be made on word definitions. Once we've made all our word definitions in our argument unambigious, the argument will not change meaning if we change the words (but if the argument previously used ambiguous word definitions and we changed these to remove and equivocation, the argument will of course usually change meaning). For example:

Before:
Define cars as things that have 4 wheels and an engine
All cars have an engine
All engines have metal in them
========
Therefore all cars have metal in them

After:
Define Leprechauns as things that have 4 wheels and an engine
All Leprechauns have an engine
All Leprechauns have metal in them
========
Therefore all Leprechauns have metal in them

The meaning hasn't changed, we just use the word Leprechauns to denote what was previously called cars. However, if we present our result as if we had proven that Leprechauns have metal in them, without explaining that we're using our own, not very common definition of Leprechauns, we are of course being deceitful.

FALLACY4: More subtle forms of equivocation. A common method for hiding equivocation in fallacious arguments is to put an equivocation right at the end or beginning of the argument. Consider the following argument:
Define vehicles as things that have 4 wheels
All vehicles have an engine
All engines have metal in them
========
Therefore all vehicles have an engine in them

By making the equivocation at the beginning, it was more difficult to discover it in this case. In fact, did everyone at all spot the fallacy? Hint: try to replace the words with their definitions, and read the conclusion. The conclusion in clear text says: "all things that have 4 wheels have an engine in them". That isn't true, for example a skateboard usually has 4 wheels but no engine.

And an example of putting an equivocation at the very end:

Define vehicles as things that have 4 wheels
All vehicles have wheels
All wheel have rubber in them
========
Therefore all vehicles have rubber in them
My wooden boat is a vehicle, therefore my wooden boat has rubber in it

FALLACY 4: Putting the fallacy outside the main argument, to be able to present an irrefutable argument, while claiming to have proven more than the argument has proven.
An even more malicious method is to put the equivocation outside the actual argument, presenting a valid, unrefutable argument such as this:
Define vehicles as things that have 4 wheels
All vehicles have wheels
All wheel have rubber in them
========
Therefore all vehicles have rubber in them

...but then going around claiming to have proven that your wooden rowing boat has rubber in it. Because this claim lies outside the presented argument, how can anyone invalidate it? The way to stop someone from using this cheating method, is to explain to the other part that unless the final steps are written down too, you will ignore them because it is an attempt - deliberate or by mistake - to hide a logical fallacy to try and convince you of something that might be incorrect.

Another thing that comes with a risk of fallacy is when words are defined so that the definition contains something subjective. For instance "define ice cream as something that is tasty" means that for different people, ice cream will mean different things. Tasty is also subjective, meaning that a statement containing the word ice cream with this definition isn't a proposition, and when it isn't a proposition, it can't be used with the logical reasoning methods described in this chapter.

Replacing words with their definitions: Replacing words with their definitions is a very practical trick to see what a proposition actually says. Since a word is used to denote the phrase used to define the word, it is completely legal to always replace the word with its definition phrase. We saw this above in the example: "define vehicles as something that has 4 wheels". By inserting the definition into the conclusion "all vehicles have an engine", we immediately saw that the conclusion was unsound, which hinted to us that the argument was invalid, which made us search for, and find, the crucial equivocation fallacy - something that has 4 wheels doesn't need to have an engine - a skateboard is a good counterexample.

Attitude in a logical discussion

The above should hopefully demonstrate that writing logical expressions and being exact about every step can be time consuming. Often in discussions we can't spend that much time for the phrasing of a single statement. Therefore, most discussions tend to not be entirely logical or formal all the way through. A common attitude in philosophical-logical discussion is to follow the following rules:
1. always define any words you are asked for the definition of
2. don't call it a fallacy if your opponent isn't formal enough. Ask for a more formal version of a part you either don't understand or don't believe is true
3. if someone provides a valid proof in such detail and so formalized that is shows how every step is obtained by applying only the most basic operations of logical simplification, don't pretend the argument is invalid unless you actually believe so
4. don't stubbornly claim arguments that are true are false. If you consider the conclusion false, you may always claim the assumptions of your opponent were wrong. Remember that if someone says an argument A => B, and B is a horrible and/or unlikely thing, but A is known to be false, then you can't deny that A => B if that was true, but you are thereby not prevented from denying B
5. don't stubbornly claim arguments that are false are true. If you consider the conclusion true, try to prove it in a valid way if your first argument turned out invalid and a failure.
6. be polite, and concentrate on arguments and not on persons. Present your own arguments when asked for them, instead of answering rudely to a polite request for your argument. If you have tried to prove something that nobody has proven before, be prepared to actually have to formalize your proof quite a lot to convince anybody who has read much about logic and/or the subject in which the proof is made. Holding away your arguments or trying to hide parts or all of them by personal attacks shows that you probably have invalid arguments.

CHAPTER 2 - predicate logic

Recall from the previous chapter the statements that weren't specified enough to be propositions, for example "x likes ice cream". Not until we specify x, does the statement become a proposition. An unspecified statement of this type is called a predicate. Logic has several ways of transforming these to propositions, and several rules for how to deal with them. One method is to insert something in the place of x, one method is to use quantifiers such as "for all x, ..." or "there exists x for which ...".

Also recall from before that propositions are denoted by P(list of unspecified variables). This will be used in the examples and rules below.

Rules for quantifiers

The ":" can be read out as "such that" when it comes after "exists", and "it is the case that" when it appears after "for all".
* (not (for all x : P(x)) ) <=> exists x ( not P(x) )
The left hand side of the equivalence operator in this rule is read as "P(x) isn't true for all x", and the right hand side is read as "there exists one x for which P(x) is false". These two are equivalent statements.
* (not (exists x : P(x)) ) <=> for all x ( not P(x) )
The left hand side here means "there doesn't exist any x for which P(x) is true" and the right hand side means "for all x, P(x) is false".

FALLACY5: Mixing up logical existence with metaphysical existence. Logical "existence" isn't the same as metaphysical existence. Logical existence means that out of all things that could possibly be x in the predicate P(x), at least one must make P(x) true. So we may very well have a predicate:
P2(x) = x is green
The metaphysically non-existing Leprechauns are green, therefore we can say that "there exists x for which P2(x)". That doesn't assert the metaphysical existence of Leprechauns.

Note! Logical existence doesn't rule out that there could be more than one x, for which the predicate P(x) holds.

Proving logical existence: To prove logical existence, we can either show that there is one x for which P(x) holds. For instance, if P(x) = x is green, we demonstrate the proposition "exists x : P(x)" by giving an example of something that is green. We can also prove logical existence by showing that the opposite of logical existence would be a contradiction. The opposite of "exists x : P(x)" is "for all x : (not P(x))". Note again that logical existence isn't the same as metaphysical existence. I give here another example of this:
Let P(x) = x is an elf
Legolas is an elf => exists x : P(x)

So how do we prove metaphysical existence? The answer to this lies slightly outside the field of logic. Metaphysical existence must be defined. Some common definitions of metaphysical existence is "the object interacts with other objects", or "the object has a potential to interact, either it has done so in the past, it does so now, or it will in the future", etc. Some definitions of metaphysical existence has quite interesting consequences, such as that something that isn't seen, doesn't exist, whereas Newtonian and later relativistic physics state that gravity and other forces mean all constellations of energy-matter interact, no matter the distance, which means some of these seemingly odd definitions of metaphysical existence don't suffer from such problems. This is however off-topic and outside the scope of this quide.

Proving for all: Proving "for all x : P(x)" is done by either showing that there can't be an x for which P(x) is false, or by looking at every single possible x, and asserting that P(x) must be true for it. The latter is normally not done, since it's pretty much impossible (but not always).

Cathegories

One problem with predicates of the form P(x), is that we allow ANYTHING to be x - cars, humans, pets, suns, elves etc. This may not be what we intended. This problem can be fixed by specifying for which x the predicate is allowed to be applied to, and it may then not be applied to any other types of x. We may for instance want to prove a certain property of all cars - a property which doesn't hold for toys, pigs, elephants and elves, for instance. Now when we only apply it to the intended things, we get the desired result, but it can come with more complexity in some cases, as we shall see later.

CHAPTER 3 - modal logic

Modal logic deals with possibly, necessarily and impossible, usually as a way to handle reasoning even when we are uncertain about the truth value of some of our claims and assumptions.

Something is necessarily true if it would be a contradiction if it were to be false. If something is necessarily true, then it must be the case that it is true.

Something is impossibly true (alternatively, "necessarily false") if it would be a contradiction if it were to be true. If something is necessarily false, then it must be the case that it is false.

Something is possibly true if it isn't impossible for it to be true, but also isn't impossible that it would be false.

This may sound confusing, but it can be thought of in the following way:
- Necessarily true things are true in reality right now, and will always be in the future, and is true in any alternate universe we can think of, provided that alternative universe doesn't go against the rules of logic.
- Possibly true things are perhaps true in reality, perhaps false, at this time. There is no guarantee that they will always have to have the truth value it has right now. In alternate universes that follows the rules of logic, they could also be either true or false.
- Impossibly true things are false in reality, and must always be false in any alternative universe we can think of, provided that alternative universe doesn't go against the rules of logic.

Modal logic is slightly problematic if the world is deterministic and thus follows theoretically predictable rules, because then the notion of possibility is useless, because no other universe that follows the laws of logic, except an alternative universe which is like our own except moved slightly forward or backward in time, are even worth considering since they will never occur. If the world is non-deterministic, modal logic can be of some use, though.

CHAPTER 4 - other fallacies

A list of fallacies in the logical reasoning. Ad hominem and similar fallacies probably won't be listed here, since they aren't errors in the logical reasoning, but attempts to sabotage a discussion to hide the logical reasoning.

FALLACY1: The fallacy of redefining words in the middle of an argument, is called equivocation. See chapter 1.
FALLACY2: Failing to define words or giving them a vague meaning, to be able to assign varying meanings to them at will at arbitrary points of an argument. See chapter 1
FALLACY 3 - assuming that a valid logical argument alone proves the conclusion. A proof ISN'T an assertion that C is true. See chapter 1
FALLACY4: More subtle forms of equivocation. A common method for hiding equivocation in fallacious arguments is to put an equivocation right at the end or beginning of the argument. See chapter 1

FALLACY X: Begs the question is a fallacy where we try to prove something, by having it as one of our assumptions that that very same thing is true. Often it is disguised by adding more assumptions (called X below) to hide that our conclusion is included in the list of assumptions. The fallacy can be described as trying to reason in the following steps:
1. Prove that: A and X => A
2. Claim that therefore, A must be true, i.e. (A and X => A) => A
The first step will always succeed, because if A is false, we get an implication false => false, which evaluates to true, and if A is true, we get either false => true or true => true, both of which also evaluate to true - and an argument is valid if it evaluates to true. The second step is the fallacy, because there is nothing preventing A from being false.

FALLACY X: Trivial argument. This is related to "begs the question", see above. A => A is always true, and if our argument can be simplified to this, then it was pointless. It isn't a fallacy, since A => A always evaluates to true and therefore makes the argument A => A valid always, for example: "if the sun is yellow, the sun is yellow". However, it is pointless since it adds no information we didn't already know, and quite often tends to later be extended to become a "begs the question" fallacy, wherein we try to claim that "if A is true, then A is true. Therefore, A is true", i.e. (A=>A) => A

FALLACY X: Circular proof fallacy. Proving equivalence is done by proving A => B and B => A. Once this has been achieved, we have proven that A <=> B, i.e. that either both are true or both are false. A circular proof fallacy is to claim either of the following:
(A <=> B) => A
(A <=> B) => B
(A <=> B) => A and B
In clear text, we prove: if A then B, and if B then A. Then we claim that because of that, either of A and B or both must be true. The truth is that either both are true, or both are false. Those who make the circular proof forget the case where both are false. The first step - proving equivalence - is however not a fallacy, and not useless, since it proves that one can't be false when the other is true.

FALLACY X: Claiming that showing an argument to be invalid means the conclusion must be false. Disproving the argument means that the conclusion may be true, false or a contingent statement, but we don't know which yet.

FALLACY X: Believing that an ad hominem fallacy or other fallacy not related to the actual reasoning process makes an argument invalid. Ad hominem doesn't change the status of the logical arguments, but it's an attempt to divert attention from the logical argument.

FALLACY X: Believing that a logical reasoning fallacy automatically makes an argument invalid. for example, if we argue that:
"A, therefore B. B therefore C. A therefore not A. B and C therefore D."
Our result "A therefore D" isn't invalidated because of the fallacy "A therefore not A". However, as long as a proof contains logical reasoning fallacy, it is up to the one trying to prove something to fix it, since nobody who reads a fallacious proof can know for sure what the person who wrote it really meant, by putting the fallacy there - perhaps there was a typo somewhere that created the fallacy, or there was a fallacy somewhere else in the proof text.

FALLACY X: Burden of proof. The person who wants to make a claim has the burden of proof. Someone claiming to have proven something, must demonstrate the proof for it. If someone presents a proof containing fallacies and these are pointed out, the idea behind the proof may not be incorrect (see below), but if the fallacies can't be removed from the argument because the argument relies on these fallacies, then the argument isn't to be considered a valid proof. The opponents have no burden of trying to disprove the argument, because the person presenting the fallacious proof hasn't proven it yet.

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