Beginners Guide to Logic

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Beginners Guide to Logic

Postby Obw » Fri Sep 22, 2006 6:44 pm

[There are various parts to this guide, with new parts being added slowly. Each new part is in a new post. Scroll down if you have already read this first post.]

With the help of Faust, here is an ILP primer on basic Philosophical Logic. There are two sections to this post – 1) A basic introduction to logic and 2) A refresher section for people who learnt the basics but want to be reminded of things. The first section is patronising and the second is not, because I wrote the first and I’m not a very nice person. Mistakes in there are therefore all mine.

If you are a new student of logic, you should start with [1] and then, once understood, move on to [2]. We hope to introduce more guides like this in future. This is a tentative start.

There is a lot that has been written about logic and you would do well to buy yourself an introduction to philosophical logic. This guide can only act as a motivator or as a very basic introduction but will leave you with many questions. If you just want to pretend to appear clever, you would do well to learn this guide by heart and that will put your ability to win arguments above 90% of internet users today.

Here we go…get yourself a cup of coffee or a strong whiskey.

Section [1]


When we talk about logic, in terms of Philosophy here at ILP, we’re actually talking about the study of argument. Argument is logic, logic is argument. Philosophers care a great deal about arguments, and hence, one cannot hope to be a philosopher without knowing about logic even if you then decide to ignore logic (notable example: Nietzche).

However, there’s a problem. Everyone has got an opinion, and just how do we tell good arguments from bad ones?

It should be mentioned that to a philosopher, an argument is interesting only in so far as what is being said and not how it is being said. So, we could shout and swear at one another, but the philosopher only cares about what we are shouting and swearing and more often only about the structure of what we are saying, such as the order in which we say things. In this sense, an argument is not actually a shouting match. To be successful at philosophy, you need to teach yourself to argue without becoming emotional (or, at least, not too emotional!).

For our purposes here, we shall describe an argument as a sentence or a set of sentences which lead up to some other sentence. This last sentence may or may not be justified by the sentences which came before it. Confused yet? Good! The way to spot the last sentence is to look for a word like ‘therefore’ or ‘so’ or ‘hence’. The sentences which lead up to this kind of word are known as premises. The sentence which comes after the ‘and so..’ or ‘therefore’ word is called the conclusion. For for a Philosopher, an argument is:

Premises, followed by a ‘therefore’ sort of word, and a conclusion.

Not too tricky, right? Keep an eye out for the words like ‘therefore’ or ‘thus’ because these warn you that the conclusion is about to arrive. Unfortunately, reading a paper or talking to your friend will not always yield arguments as simple or straightforward as this. Sometimes you have to cut through lots of rubbish to find the argument itself. This is why learning about arguments is so helpful – because you will actually become better at expressing yourself to others while you understand others much better too (probably even better than they understand themselves…).

The next step for a philosopher is to ask how we can tell if the conclusion, once it is identified, is a logical consequence of the premises or not. Do the premise-sentences justify us accepting the conclusion-sentence? How do we know? How can we tell?

Firstly we need to find out if the argument is valid or not. Uh oh – what is validity?


Validity: A judgment made about the form of an argument. Validity has nothing to do with truth. An argument is said to be valid only “by virtue of it being a substitution-instance of a valid argument form.”

What the hell does that mean? Don’t worry – it’s part of Formal Logic, which is described further down the page – but for now just keep reading. Take another sip of coffee.

When we talk about validity, we are actually not talking about words, sentences or phrases. We are talking about the structure or the form of the argument. For example:

"Obw is a man, and since all men are mortal Obw must be mortal too."

We can display this natural language argument as follows:

Premise 1: “All men are mortal.”
Premise 2: “Obw is a man.”

Conclusion: “Obw is mortal.”

This argument has a form which you will one day be able to recognise (see Formal Logic below once you understand the definitions). First, let’s cover some definitions:

In order to analyse the above argument we need to be aware of some basic definitions. This is a bit like learning algebra in a maths class - but much more fun.

Mathematicians use symbols like X and Y to hold the place of numbers. I'm sure you remember having to learn that 2x + 2x = 4x or something like that. If not, don't worry. Maths people use these symbols to represent numbers. In Philosophy we don't need to use maths symbols. We have our own special types of variables (things which can represent other things).

In Philosophy, our variables reserve space not for numbers, like in maths, but for sentences. To be precise - we don't call them sentences, we call them propositions.

But just what is a proposition?

A proposition is best imagined as something absolutely identical with the meaning of a sentence rather than a sentence itself. In this way, a proposition can be the 'general gist' of two sentences, even when these sentences are actually different. For example:

Sentence 1: "Obw is more handsome than Faust."
Sentence 2: "Faust is uglier than Obw."

These two sentences can be described as expressing the very same proposition: I.e. that Obw is one some way better looking than Faust, or any way you want to word it. The proposition is the meaning both sentences allude to. Even though the sentences are different.

Now you know what a proposition is. Does it feel good? Good! Sometimes, logic types call propositions sentences. That's because they enjoy confusing you.

Let's go back to Philosophical variables. We mentioned that maths people use X and Y and so on. We also mentioned that in Philosophy, we want to have variables not for numbers, but for propositions. Logicians use letters to reserve space for propositions, and although there are many different ones, you only need to learn about three right now:

p, q and r.

That wasn't so painful! As Philosophers, we can use p, q and r to represent our propositions, or the propositions of others' arguments. These have a special name: sentential variables (sentential because they deal with sentences (propositions)).

Ok, back to argument form. So we mentioned that validity is all about the form of an argument and nothing to do with the truth of an argument. You should think of the whole argument-analysing process as having two distinct steps:

1) You figure out if the argument is valid.
2) If it is valid, you figure out if the premises are true. If they are, the conclusion is gonna be true too.

It’s like a logic machine. Validity is the oil. Without oil, the machine doesn’t work. If your argument has validity, and you put truth into it, it will spit truth out. If you put rubbish into it, it will spit rubbish out. If it’s not valid, there’s just no oil, and it isn’t going to spit anything out – it might just fall apart. It’s not even worth touching without big gloves and a plastic hat.

So let’s consider what we have just learned:

- This means that even if the premises of an argument are false, the argument can still be valid. Hmm, interesting.
- This means that whenever the premises of a valid argument are true, the conclusion will be true too.
- This means that if an argument is valid, it is impossible for the premises to be true and the conclusion false.
- This means that an invalid argument is where it’s possible that the premises could be true AND the conclusion false. Oh… Ok, I am beginning to get this now.

We have another word to describe an argument that is valid AND has all true premises. Given that a valid argument with true premises MUST have a true conclusion (thanks to its validity!), we can call this argument sound. A sound argument, therefore, is an argument that is valid and an argument that has true premises. If the argument is valid, but some premises are false, it is a valid argument only. If the argument is not valid, we don’t even need to worry about the premises!

Deductive argument: A deductive argument is a valid argument. You will often see philosophers talk about valid arguments as being deductively valid arguments. This comes from the word ‘deduce’. The conclusion is deduced from the premises.

Ok – so you have a basic appreciation of what it is for an argument to be valid, but how can we formally show one argument to be valid and another not valid?

Formal Logic and Validity

Formal logic is all concerned with argument forms, or forms of argument. We mentioned earlier that a valid argument is an argument that has a valid argument form. But, I didn’t tell you what a valid argument form is yet:

If we consider the example I gave right at the start – that of myself, Obw, being a man and being mortal, we can see this argument to be valid – if the premises are true then the conclusion has to be true too. It would be impossible for those premises to be true, and the conclusion false. So it has to be valid. This is the force of logic.

Of course, the premises might not be true. On the internet, nobody knows you are a dog. I might be a Dog, and Obw might be the name my owner gave me. You can’t be sure. But you can be sure the argument is valid – and validity doesn’t care if the premises are true or not, only the form of the argument.

Here's the argument again:

P1: All men are mortal
P2: Obw is a man

C: Obw is mortal.

The other premise could be false too – maybe some men are immortal (like God, perhaps, if he exists). Yet again, this changes nothing about the validity of the argument. Only whether or not we would call the argument sound or simply dismiss it in its current form.

Here’s the Formal Logic definition of validity:

An argument is considered valid if and only if, it is an example of a valid logical form.

Ooer! So it seems that we are saying there are things called ‘valid logical forms’ and that as long as an argument shares one of these forms, it can be considered valid. Hey- that’s really cool. This means all I need to do is learn the valid logical forms and learn how to recognise them and I will kick ILP ass. Quite correct.

Formal Logic, then, is concerned only with valid logical forms of argument. This can be described as ‘investigating formal validity’.

How to Identify Logical Argument Forms

Now you know that the logician, and the philosopher, has the job of recognising valid forms of argument. Let’s consider a new argument:

P1: If Obw is a man then he is mortal
P2: Obw is a man
[therefore]
C: Obw is mortal.

Using our sentential variables (remember these? The proposition-holders p, q and r) we can represent the form of this argument. Do you notice how the first premise P1, has a sort of structure:

IF… obw blah blah… THEN… blah blah.

If something, then something else.

Let’s put in place our sentential variables to reserve space for some propositions, whatever they might be (we don’t care, we’re just investigating validity right now):

If p then q.

Now let’s do it for the whole argument, and investigate the form:

Code: Select all
P1: If p then q.
P2: p.
[therefore]
C: q.


Interesting! Now we have a bare-bones glimpse of the inner workings of our original argument. We’ve stripped away the details and got right down to the mechanics – the form. When we formalise an argument like this, we must be careful to use the same variable to represent the same sentence all the way through the argument. Such as ‘p’ in this case represents ‘obw is a man’ and ‘q’ represents ‘obw is a mortal’.

So how do we tell valid forms of argument from invalid forms? Well, a form of argument is valid if, and only if, every example of that argument form is itself valid. You better read that again. Let me explain..

Valid argument forms are structures of argument that will never fail to lead us to create valid arguments as examples. This is called the substitutional criterion of validity. Here is an example. Stay with me. You will get it soon, I promise.

In Maths, 1x + 1x = 2x. For every time the variable ‘x’ is used in this equation it will always be true that two of them added together will leave us with 2 in total. This will be true even if x represents footballs, cars or planets. So with regards to arguments, we are saying that a valid argument form will be valid whatever it is talking about – Obw being mortal, Faust being ugly, Ben smelling of old socks or anything else. If it is a valid argument form, every example of that form being used anywhere in any argument will also always be valid. That’s how you know it’s a valid argument form. Formally, we call such examples of the form in action ‘substitution-instances’ of that form. It doesn’t matter what you call it, as long as you understand the principle beneath it.

Conversely, then, in order to show that an argument form is invalid we only need to show how another example of that argument form is clearly not valid. This is called ‘refutation by counterexample’ as is the most straightforward yet powerful type of argumentative put-down.

Here are some arguments for you:

P1: If p then q
P2: q
[therefore]
C: p.

Notice the subtle difference here? Let’s put the flesh back on the bones:

P1: If all ILP members are English then Obw is an Englishman.
P2: Obw is an Englishman.
[therefore]
C: All ILP Members are English.

Does something look wrong here? Try asking yourself:

- Is the argument an example of the logical form shown above?
- Are the premises true? (I actually am English, if you didn’t know)
- Is the conclusion false?

Here are the answers:

- Yes
- Yes
- Yes

Let’s look at another example then:

P1: If p then not q
P2: Not p
[therefore]
C: q.

Hmmm. Interesting. Something smells though. Here is the natural language argument that has this form, but broken down:

P1: If all ILP members are Canadian then it is not the case that Obw is an American
P2: Obw is not Canadian
[therefore]
C: Obw is an American.

Once again, ask yourself some questions about this argument:

- Is the argument an example (a ‘subtitution-instance’) or the argument form in question?
- Are the premises true?
- Is the conclusion false?

And the answers:

- Yes
- Yes
- Yes

So are the arguments valid or invalid? Check back with the definitions. Does that mean the argument forms are valid or not? Check back with the definitions.

If you understand these questions and answers then you understand something more about argument forms and validity.


Summary:

- An argument is: premises, a ‘therefore’ sort of word, and a conclusion.
- An argument can only be valid if, and only if, it would be impossible for the premises to be true, and the conclusion false.
- An argument is invalid if, and only if, it is possible for the premises to be true and the conclusion false.
- A sound argument is a valid argument with true premises.
- In Formal Logic, an argument is valid in virtue of it being a substitution-instance of a valid argument form.
- An argument form is valid if and only if every such substitution-instance of that particular form is also valid, regardless of what it is an argument about.
- An argument form is invalid is there is some substitution-instance of that particular form which is itself invalid.
- A counterexample to an argument form is a substitution-instance of that particular form which is itself an invalid argument.


Next time.. (maybe) Inductive validity, formal symbolic logic, QL.

If you have questions that a re-reading of this does not answer, please feel free to PM me.
=====================
Section [2]

Logic is the study of the relations between propositions. Specifically, logic ascertains the connection between the truth of certain propositions and another proposition, the latter of which is, in logic, called a conclusion. The conclusion of any argument is itself a proposition, and can be used as a premise in another argument. The only difference between propositions that are not conclusions and those that are conclusions is their position in a given argument.

A proposition, or statement (these words can be taken, for present purposes, as synonymous) is distinct from a sentence, in that two different sentences may contain the same proposition, and two propositions may be made by the same sentence, depending on the context. And example of the former is this pair of sentences: “This room is completely dark” and “There is no light in this room.” An example of the latter is “I am a boy”, which differs according to who is making the utterance, or - across contexts - whether it asserts “I am not a girl” or “I am not a man”. It is a convention in logic to use the terms “proposition”, “statement” and “sentence” as synonyms, but it must be remembered that “sentence” is used in a technical sense here. Only declarative sentences qualify as “logical” ones, and even then they must contain a statement. This is mere nomenclature, and is not the subject proper of logic.

Propositions are the building blocks of arguments. Logic is not primarily concerned with the truth of the propositions in an argument, but with how the truth of conclusions is related to the truth of premises. The main concern of logic is not truth, but validity. The truth of premises must be previously accepted for the purposes of a logical argument, and only the dependence of the truth of the conclusion on the truth of the premises is the purview of logic. Valid arguments are those in which the premises cannot be true and the conclusion false. Said another way, if the premises of a valid argument are true, then the conclusion must be true.

A valid argument, then, guarantees that if the premises are true (and they may, in the event, not be) then the conclusion is also true. But validity does not guarantee the truth of the premises itself. So, a valid argument may yield a false conclusion, if one or more premises are false. It may also yield a false conclusion under these same circumstances. Only the relationship between the conclusion and the premises is at stake here, and not the truth of any of them.

How this relationship is discovered is through the process of inference…….
Last edited by Obw on Wed Sep 27, 2006 3:11 pm, edited 1 time in total.
<i>"One may learn how to think about the arguments of Plato and Aristotle from someone whom one might not like to have as a friend."</i> - Nussbaum
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Beginners Guide to Propositional Logic

Postby Obw » Wed Sep 27, 2006 2:50 pm

BEGINNERS GUIDE TO PROPOSITIONAL LOGIC PART 1

Welcome back. Last time we talked about the basics of logic and we discussed what it might mean for an argument to be valid, and the nature of that validity among other things. We also covered some basic definitions such as 'sound'.

This time I want to turn our attention to something a little more specific - a type of logic called Propositional Logic. Propositional Logic is the most friendly form of being logical, and it's vital for all critical Philosophers to learn.

We covered alot of this before, but now I will be less general. If you understood most of what I was banging on about above, you will not have any problems learning how to use PL (Propositional Logic) to your crushing advantage in debates and in forming and expressing your own opinions.

Without further ado..

Propositional Logic

PL is a formal language, so it's a way of showing normal (natural) language sentences in symbols. These symbols allow us to represent sentences easily and to then connect them to one another so as to represent the grammar of an argument and then study it and ultimately...

...test for validity! Five bonus points if you knew I was going to say that.

PL uses special little machines called logical connectives for doing the hard work. There are only five logical connectives in the formal language PL. These are:

v - "Or" - "Either something or something else"

& - "And" - "Both something and something else"

<-> - "If and only if" (this is normally shown as a double headed arrow, but my keyboard doesn't know PL) - "something if and only if something else"

-> - "If something then something else"

~ - "Not" - "It is not the case that something"

It is worth noting that four of these logical connectives require us to have two sentential variables or sentential constants whereas "~ (Not)" requires only one.

Code: Select all
What is a sentential constant?

If you recall, I earlier wrote about sentential variables and referred to the symbols that can reserve space for our sentences or propositions. these are symbols like:

p, q and r.

However, we have another form of these, in capital letters, which we call sentential constants. These, rather than being variables which can hold any sort of sentence, are constant and only hold one specific sentence:

P, Q and R.

Ok, so bear that in mind.


There are two types of connectives in PL - binary and unary. A binary connective is one which requires two sentences and a unary requires only one. Try to guess which connective is the only unary connective in PL.

In English we use and to join two propositions together, such as :

Obw is a man and Obw is mortal.

If we want to express this in PL, we might write:

P & Q

To be perfect, we should also add some brackets around this 'formula', but for now we don't need to worry with that.

By joining these two sentential constants together, we have formed a logical conjunction. A conjunction is just a fancy way of saying we have connected one thing to another thing.

In formal logic language, we can say that we have used (P & Q) to formulise the natural language conjunction 'Obw is a man and Obw is mortal'.

Conversely, when we employ the use of v (Or) we create disjunction. P v Q is a disjunction, as we are saying 'Either P or Q."

When we use <-> (If and only If) then we are creating a biconditional. P <-> Q is a biconditional.

To use ~ (Not) is to create a negation. ~P is a negation of P.



Every time you express a part of PL in formal terms you are creating an atomic or compound formula. A formula is just a mixture of atomic things. We call them atomic because one of the most irreducible things in life is the atom. Similarly, atomic PL forumlae cannot be reduced further. (No- there are no quarks in PL! - bad physics joke).

Therefore, P is an atomic formula of PL. Because it's correct, we also call it well formed. A well formed atomic formula of PL. You don't really need to remember that.

A combound formula, as noted above, is a mixture of atomic formulas. So, two atomic formulas together in the conjunct:

(P & Q) - a well formed compound formula of PL.

You can then combine compound formulas together to make more complex formulas, such as (P & Q) v (Q & R). Confused? Don't worry.

It's easy to get mixed up during the stages of constructing these formulas. We have to be careful, as philosophers, to avoid mixing any uncertainty into our formal logic cake. The ingredients have to be exact at all times. To help us in this pursuit, we have some simple rules to follow.

There are four rules in all, two for our single unary connective (~) and two more for our binary connectives (&, v, <->, ->).

To help illustrate these rules, let's look at some new sentences. Ah! Natural language - feels good eh:

a) Either I will log out of ILP and watch a movie or I will continue reading this crap.

b) I'll log out of ILP and either I will watch a movie or I will continue to read this crap.

(You don't have to be "logged in to ILP" to "read this crap" after all.)

Ok, this is the exciting bit. Let's try to formulise these natural language sentence into the Formal Logic language PL.


HOW TO FORMULISE (FORMALISE!) IN PROPOSITIONAL LOGIC

The rules for binary connectives:

1) A binary connective must connect two formulas
2) If either of these formulas is itself a compound formula then you must put it in brackets first.

You should be familiar with these rules from maths - (1+2)x3 has to be separated from 1+(2x3), for obvious reasons (if they're not obvious, send me a pm and I will explain).


Bearing the above two binary connective rules in mind, try this out for yourself on the sentences above. Take (a) first. I have also given it a go further down the page, and you can check your results with mine.

Step 1: Establish where the propositions are and what they are. How many 'atomic' or 'simple' sentences are there? This helps us find out how many sentential constants we need to use (i.e. Do we need P Q and R or just P and Q - or more?).

Step 2: Draw up a key, like you see on maps. So state clearly P = Blah and Q = blah blah.

Step 3: Find out what connectives we need to use. (do we need an '&'? How about a '<->'?). Write a list of what we need. Are there conjunctions or disjunctions? Any biconditionals?

Step 4: Write the formula out in full. To represent a conjunct, remember to use brackets. It's important to use brackets because we need to pay attention to scope. Uh-oh - what the hell is scope? Don't panic - I'll tell you shortly.

Do the same procedure for sentence (b). Once you have both written in formal PL, compare them to my own below:

OK, here's what I get when I try to formalise sentence (a):

Step 1: I see three separate sentences. I see :

1) I'll log off ILP
2) I'll watch a movie
2) I'll stop reading this crap

This tells me I am going to need three sentential constants - P, Q and R. (not p,q r as these are variables and used for forms and not specific arguments or sentences).

Step 2: Here's my key:

P = I'll log off ILP
Q = I'll watch a movie
R = I'll stop reading this crap

Step 3: I see only two logical connectives being used in both sentences: & and v. The first is a conjunct and the second is a disjunct.

Step 4: Here's my full formula:

a) (P & Q) v R (either "P and Q" or "R")

b) P & (Q v R) ("P" and "either Q or R")


How did yours compare? Did you miss out the brackets? If so, don't worry, I will tell you why they're important now. If you didn't get the same results in some other way, don't worry. Go back to the start of this section and re-read. Most people don't get this first time.

Scope

The difference between formula (a) and formula (b) is the effect the connectives have on the formula. In both (a) and (b) we have the same two connectives but in different places - either in or outside the brackets. When a connective is inside brackets, it only connects the formulas inside the brackets. When a connective is outside of the brackets, it connects the brackets contents with everything else.

This crucial difference is known as scope. Here's the formal definition:

The scope of a particular connective is the connective itself combined with what it actually connects.

In this way, the scope of the connective '&' in (a) is only "(P & Q)". The scope of 'v' in (a) is, however, is the whole sentence "(P & Q) v R". To practice, you can check what the scope of connectives in (b) is. Any problems, give me a pm.

Within the framework of considerations surrounding this idea of scope, we can define other important parts to a PL formula. One such part is known as the main logical connective. The main logical connective can be defined like so:

In any given formula of any particular complexity the main logical connective is that connective which has scope over the entire formula.

If you think of scope as a hierarchy of 'levels', the connective which can 'see' all the parts of the formula is the big daddy of connectives in that formula. In comparison, the smallest scope is held by the lowest 'level' connective, such as the sub-sub-sub brackets of a formula, where a connective might only affect two atomic formula and nothing else below it.

Why the hell do I care? Ok, well basically this idea is important because once you identify the main logical connective, you are able to find out what form of formula you are about to deal with. For example, if the main logical connective happens to be '&' then you know this formula, however complicated it might look, is a conjunction. What would the formula be if you identified '<->' as the main logical connective? Have a look at the definitions for the logical connectives again if you are not certain.

Great, now you know about the two rules for binary connectives and a bunch of other useful stuff.

Next time: The two rules for unary connectives. Man I am knackered.
<i>"One may learn how to think about the arguments of Plato and Aristotle from someone whom one might not like to have as a friend."</i> - Nussbaum
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