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\title {Logic I \\ Lecture 01}

\maketitle

Logic I

Lecture 01

[email protected]

\def \ititle {Logic I}

\def \isubtitle {Lecture 01}

\begin{center}

{\Large

\textbf{\ititle}: \isubtitle

}

\iemail %

\end{center}

Readings refer to sections of the course textbook, \emph{Language, Proof and Logic}.

This is a course about the limits of the world. But let me start with a simple example.

\section{The Pigs of Logic}

\emph{Reading:} §1.1, §1.2, §2.1

\section{The Pigs of Logic}

Last summer you and I were working on a farm herding pigs. One evening we had almost got all of the pigs back into their sty for the night when one of them escaped and ran off down the lane.

We closed the gate to secure the other pigs and ran off after the escaped pig. After a while we came to a fork in the lane.

I said, Either it went up the left fork or it went up the right fork. You then said, It didn’t go up the left fork. So off we ran up the right fork.

\textbf{Argument 1:} \begin{quote} Either it went up the left fork or it went up the right fork.

It didn’t go up the left fork.

therefore:

It went up the right fork. \end{quote}

\textbf{Argument 2:} \begin{quote} Either it went up the left fork or it went up the right fork.

The left fork is unsuitable for pigs.

therefore:

It went up the right fork. \end{quote}

\begin{minipage}{\columnwidth}

\textbf{awFOL version of Argument 1:}

\end{minipage}

Here is our reasoning in the form of an argument.

It has two premises.

And a conclusion.

For our purposes, an argument is any number of premises followed by a conclusion. And a premise is any statement we specify as a premise; likewise, a conclusion is any statement we specify to be a conclusion.

So this is a very broad notion of ‘argument’. It wouldn’t satisfy a visitor to the argument clinic, but it will do well enough for our purposes.

This argument has an interesting property: it is logically valid. That is: there is no possible situation in which the premises are true and the conclusion false.

Of course, we might be wrong in saying that either the pig went up the left fork or it went up the right fork. Perhaps it is hiding in the hedge having a laugh at our expense. But this isn’t relevant to logical validity.

What matters for logical validity is just that the conclusion can’t be false *if* the premises are true.

Contrast this argument (Argument 2 on the handout). This argument is not logically valid.

After all, a sufficiently contrary pig could take the left fork despite its unsuitability for pigs.

So although Argument 2 is a perfectly good argument in some ways, it is not logically valid. There is a possible situation involving a contrary pig in which the premises are true but the conclusion is false.

This is a course about logical validity. We are going to investigate which arguments are valid and which are not.

It would be quite slow to do this one argument at a time. Imagine having considered this argument about the pigs, we had to consider essentially the same argument for the ducks, the geese, the cows, the sheep, the ostriches, the crocodiles, and all the other farm animals we were looking after. (Don’t ask what the crocodiles were doing on the farm.)

But luckily this won’t be necessary because we can think about many arguments as sharing a single form.

If you think about it, the pigs are a bit extraneous. When it comes to logically validity, any argument like this will be logically valid.

Really, it’s validity just depends on the ‘or’ and ‘not’ features.

As I said, this is a course about logical validity.

Logic is the study of logical validity. We want to know which arguments have this property, and what means there are of establishing which arguments are valid or not.

To make this easier, we will adopt some formal notation.

We will draw arguments like this, with the vertical and horizontal lines, and we will use symbols instead of words.

Instead of ‘this’ and ‘that’, we will also use letters like P and Q to stand for propositions. I’ll explain all this later in detail; I just wanted to show you what it will look like.

So what is this course about? It’s very simple. We are going to learn to use some formal tools in order to to investigate which arguments are logically valid and which are not.

Why Logic?

\section{Why Logic?}

why logic?

Logic is a tool for precise expression. In your future life you probably won't use logical symbols, but, if things go well, the way you formulate ideas will be guided by your familiarity with logical structures.

Studying logic also enables you to identify and explain mistakes in reasoning.

precise expression

You might think that you don't need driving lessons and that you already know how to think. That's plausible, but you are probably going to be surrounded by people who regularly make logical mistakes.

It turns out that even simple logical mistakes are quite common. Here's a question from an experiment. Which cards do you need to turn over to confirm the truth of this claim? I want you all to answer the question.

‘If a card has a vowel on one side, then it has an even number on the other side.’

(Waison & Johnson-Laird 1972)

Well, what's the answer?

The solution is: you need to turn over E and 7

‘Philosophy is thinking in slow motion.’ (John Campbell)

This course is about logical validity; there is also a sister notion, logical truth. To say that a sentence is logically true is to say that there's no possible situation in which it is false. (This is slightly different from the definition given in the textbook, but it's equivalent in that the two definitions pick out the same sentences.)

In exploring the notions of logical validity and logic truth, we are exploring the limits of the world. One reason for studying logic is just that you are curious about what is possible.

‘Logic pervades the world: the limits of the world are also its limits.’

(Wittgenstein, Tractatus 5.61)

On his album game theory, Black Thought (Tariq Trotter) mentions people of whom it is said that they

‘don’t obey no laws, not even gravity’

(Tariq Trotter, Game Theory track 10)

(The context: ‘I couldn't tell you why I think they constantly after me Maybe it's cause the news put it to me so graphically How niggaz don't obey no laws, not even gravity boy’)

Why study logic? It is easy to escape supply and demand--that law applies only to rational agents, and we are none of us entirely rational. It is slightly harder to escape gravity, which was thought to apply to all physical things although Tariq Trotter claims to have done it on Game Theory (Livin’ in a New World). The laws of logic are fundamental in the sense that absolutely everything is subject to them, even musicians.

After I heard Game Theory I tried to get the philosophy department to put Tariq forward for an honourary doctorate so that he could call himself *DR* Black Thought but in the end they gave it to Professor Peacocke instead.

2013-05-30

RE: Love and Logics

Dear Professor,

Hope you're well. My name is [name removed], a former student.

I just wanted to say thank you for your wonderful lectures on logics. I didn't know logics was so useful that it helped me to find a really nice girlfriend and enables me to win arguments all the time!

Thanks again. Haha.

[name]

Admin

Lectures: normal and fast-track
Seminar groups
Textbook
Web: https://logic-1.butterfill.com
Exercises

to: [email protected]

sent: 03:55 29/08/2008

subject: [none]

Dear Stephen,

I am retaking starting logic on tuesday, and I wonder if you could explain to me the difference between contingent, tatologous and inconsistent in terms of truth table results, as I cannot find this information anywhere else?

Many Thanks,

---

zoxiy

\section{zoxiy}

http://logic-1.butterfill.com

There are logic exercises associated with each lecture. After each lecture (or before, if you prefer), you should complete the associated exercises.

You can find links to the exercises for each lecture at: \url{http://logic-1.butterfill.com}

To complete the exercises you need to register at \url{https://logic-ex.butterfill.com} (If you don’t want to do this, you can complete the alternative textbook exercises on paper. These are also specified for each lecture at \url{http://logic-1.butterfill.com}).

Seminars will discuss exercises associated with the previous week’s lectures. As your seminar tutor will track your progress and mark your exercises, you should be sure to \textbf{complete the exercises by 2pm on the day before your seminar}.

Here is the web page describing the logic lectures at http://logic-1.butterfill.com

This is for the regular (twice-weekly) lectures

You might need to switch to the fast set of lectures, for which there are different exercises. (Choose the right page for the lectures you are attending.)

For each set of lectures, you can choose between ‘regular exercises’ and ‘fast exercises’. The ‘normal’ exercises are aimed at students who did not take a mathematical subject at A-Level or equivalent. The ‘fast’ exercises are for students who find logic relatively easy and want to focus on more difficult questions. You can switch between fast and normal exercises at any time.

When you hit the link for ‘regular exercises’, you’re told to sign in. But you can’t because you don’t yet have a password.

You need to register the first time you use logic-ex.

This is the registration form. Note that this isn’t a secure web page (no ssl).

The email you use should be your university email. Otherwise your tutor won’t know who you are and won’t grade your work.

After you sign up, you might be taken to the wrong page. So go back to http://logic-1.butterfill.com and hit the exercises link again.

Now we’ve made it into zoxiy. Here’s my motivational progress donut. It’s very motivating.

Selecting ‘follow’ will make it easier to find the exercises later, so I’ll do that.

Scroll down to see the exercises.

Here are the exercises. Each of the links is an exercise.

Click on the first exercise link to get started.

Oh look, it’s asking me to define logically valid argument. This is the first exercise, so it must be important.

Complete the exercise.

Submit the exercise by pressing the button.

After you submit, it should tell you you answered the question. This can take a while to come up. Sometimes it will tell you whether you were right or wrong, but this time it doesn’t.

You can go direct to the next exercise.

I’m not going to do this one now, I’ll save it for later.

Instead I want to go back to the list of exercises.

My progress has been updated

If I scroll down, I can also see that it recorded that I submitted the exercise here. (So I can easily see where to start from next time.)

But go back up, I want to show you something else.

Click on ‘zoxiy’ (top left) to get to the home page.

Here’s what I see when I go to \url{https://logic-ex.butterfill.com}.

There’s a link that will take me straight to the exercises I’m currently working on (the normal exercises for the normal lectures).

Also, I need to tell zoxiy who my tutor is so that she can mark my work and check my progress.

Let’s see how the page changes once I’ve done that...

I added my tutor a while ago, and in the meantime she’s graded my answer.

I can tell she’s graded my answer because there’s a message. (This message only appears when there are new grades, grades I haven’t seen yet.)

Follow the link to see the feedback.

The exercise I just submitted has already been marked. (Usually your work will be marked an hour or two before your seminar.)

Let’s see whether I got it right ...

Here’s my answer again.

And look, it’s been marked correct

Quick Intro to awFOL

\section{Quick Intro to awFOL}

\emph{Reading:} §1.1, §1.2, §1.3

\section{Quick Intro to awFOL}

John is square

Square( a )

John is to the left of Ayesha

LeftOf( a , b )

John is square or Ayesha is trinagulra

Square( a ) ∨ Trinagulra( b )

name (refers to an object)

predicate (refers to a property)

connective (joins sentences)

sentence (can be true or false)

atomic sentence (no connectives)

non-atomic sentence (contains connectives)

Our approach to studying logic will involve a formal language called awFOL. `FOL' stands for first order language, and I call this particular first-order language `awFOL` because, like nearly all first-order languages used in textbooks, it’s awful. (Where are the binary quantifiers? Why are brackets used with two completely different meanings? ...)

The language of the textbook is called ‘FOL’. ‘awFOL’ is basically the same as FOL except that you can replace symbols with words which makes typing it easier. Also 'FOL' is a really stupid name because there are lots of first-order languages. It's a bit like I ask you what language you speak and instead of saying 'Farsi' or 'English' or 'Cantonese' you say 'Language, I speak Language'. But this is trivial, it doesn't really matter what you call things. Let's move on.

As I was saying, for the purposes of logic we are going to use a formal language. In order to get a sense for this language, let's compare it to English. Take a look at this sentence, John is square.

This is a sentence.

For now a sentence is just something capable of being true or false. (In a longer course we would define what it is to be a sentence more carefully.)

In English there are names ...

... these are terms that function to refer to objects.

There are also predicates, like 'Square'.

Predicates are things that refer to properties. In this case the property is that of being square.

Take a look at this sentence.

Some properties relate several things; for example, being 'to the left of' involves two things rather than one. The expressions for these relational properties are also called predictaes.

By the way, this is also a sentence containing multiple names, 'John' and 'Ayesha'.

Now have a look at this sentence, 'John is square or Ayehsa is triangular' ... or, as I perfer to say, 'trinagulra'. (Did you spot the mistake? Well done.)

Consider the word 'or' in this sentence. It isn't a name or a predicate. It doesn't refer to an object, nor to a property.

Instead its function is to join two sentences, making a new one. We'll call things like this 'connectives'. A connective is anything that you can combine with zero or more sentences and to make a new sentence.

Here's another piece of terminology: a sentence with one or more connectives is 'non-atomic'

And, as you'd expect, a sentence with no connectives is 'atomic';

Now let's see how these sentences look in our formal language, awFOL.

Here's how the equivalent of 'John is square' looks in awFOL.

The whole thing is a sentence of awFOL.

The letter 'a' is a name; just like the English name 'John', the function of 'a' is to refer to an object (in this case, John)

And 'Square( )' is the predicate.

What about 'John is to the left of Ayesha', how can we say something like this in awFOL?

Here's the equivalent of 'John is to the left of Ayesha in awFOL'

Again, the single letters a and b are names.

And 'LeftOf( )' is the predicate. Note that, as in English, the order of the names matters. It affects who we are saying is to the left of who.

Lastly, what is the equivalent of the third sentence in awFOL?

Much as you would expect.

This is a non-atomic sentence (because it contains a connective).

Note that where the English 'or' appears, we use a special symbol. This symbol doesn't do exactly what the English 'or' does, as we'll see later.

Alles klar? Molto bene.

You might be thinking that this English sentence looks, well, ...

... a lot like this awFOL sentence. What's the point of learning a formal language? How will it help us to understand logic?

(It's a bit tricky to answer this question as I haven't yet said what logic is.)

A formal langauge enables us to avoid ambiguity, e.g.:

We need a formal language because ambiguity is awkward to deal with theoretically

\begin{quote}

This is a hospital where doctors are trained.

\end{quote}

A formal langauge also enables us to some avoid appearance--reality problems:

Appearance and reality. We need a formal language because we want a guarantee that a sentence which seems to express a proposition really does express a proposition.

\begin{quote}

Many more people have been to Paris than I have.

\end{quote}

Finally, consider these sentences.

Ayesha doesn’t know diddly squat about logic

Ayesha does know diddly squat about logic

The only difference is an extra negation in the first sentence. Normally you might think that adding a negation changes the meaning, and does so systematically. But this is not true of natural languages like English. We can construct our formal language so that it is true, thereby making our lives simpler insofar as we are interested in reflecting on inferential relations among sentences.

1.1--1.5

*1.6

1.8--1.10

zoxiy: Creating Possible Situations

\section{zoxiy: Creating Possible Situations}

https://logic-ex.butterfill.com/ex/create/qq/Red(a)

Here is the exercise. We have to make the sentence `Red(a)` true.

How do we get a red person? We can’t change the colour of people, so we just add an element until we find a red one.

The first one we add is yellow, which is no good.

You can move people around by dragging their heads.

So I’ve made space.

Now I’m going to add another element; hope it’s red.

Good, it is. But I’m not done yet.

It’s telling me that the sentence isn’t true or false, that it can’t be evaluated in this possible situation. What can I do?

I need to give the red element a name by typing here. (I’m going to call it ‘a’).

So now it has a name.

And I’m done with the exercise. But let me just show you a couple of things that might be useful with some of the exercises.

I can delete people that I don’t need like this one.

That was fun.

Let’s erase another person.

Lastly, I have to say I don’t really like this person’s face.

But I can change it by clicking on the eyes, nose and mouth. This is how I make someone happy or sad.

Now the red person is happy.

Logically Valid Arguments

\section{Logically Valid Arguments}

\emph{Reading:} §2.1

\section{Logically Valid Arguments}

An argument is \emph{logically valid} just if there’s no possible situation in which the premises are true and the conclusion false

A \emph{connective} joins one or more sentences to make a new sentence. E.g. ‘because’, ‘¬’. The sentences joined by a connective are called \emph{constituent sentences}.

E.g. in ‘P $\lor{}$ Q’,

\begin{quote}

$\lor{}$ is the connective

P, Q are the constituent sentences

\end{quote}

Consider these three sentences.

The first sentence says that John is square or Ayesha is square. The second sentence says John is not square. (I know I just told you John is square; I'm not very consistent, am I?) And the third sentence says Ayesha is square.

Note the symbol; we saw this a moment ago, it's a bit like the English 'or'.

There is also a new symbol in the second sentence, this a bit like the English 'not'.

As you recall, these symbols are called 'connectives'.

Note that the negation connective in the second sentence is making a new sentence from just one sentence. (Connectives can join any number of sentences.)

You may also recall that these first two sentences are non-atomic (they contain connectives), ...

... whereas the third sentence is atomic.

OK, so much for the sentences. So far we've been fixing terminology and getting a feeling for a formal language, which is a tool for studying logic. But I haven't said anything about what logic is.

What is logic? What is this course about?

The answer is here: it's about the notion of logical validity. An argument is logically valid just if there's no possible situation where the premises are true and the conclusion false.

Logic is the study of logical validity. We want to know which arguments have this property, and what means there are of establishing which arguments are valid or not.

Let's go though this slowly. First, what is an argument?

For our purposes, an argument is just a sequence of sentences where zero or more are identified as premises and exactly one is identified as the conclusion.

But what do we mean by premises and conclusions?

A premise is just a sentence that we say is a premise. (That's all there is to being a premise.)

Likewise, a conclusion is just a sentence that we say is a conclusion. How simple is that?

Now we're going to write a lot of arguments so it would be helpful to have a compact way of identifying premises and conclusions ...

... That is the purpose of these lines. The horizontal line specifies that the sentences to its right an argument.

And the vertical line separates the premises from the conclusion.

So this is logic: the study of logical validity. Have we understood the definition yet? Not quite ...

... What do we mean by 'possible situation'?

A possibile situation is just a way that the world is or could be. So consider the situation which is as similar to the actual situation as possible except that you are in Havana smoking a fat cigar rather than attending my lecture. This is a possible situation.

Now possible situations are huge things; in specifying a possible situation, you are specifying something as big as the actual situation, with all the trees, leaves, insects and everything. It is helpful to have a proxy for possible situations, something much simpler than a real possible situation.

For our purposes, a good proxy is often an arrangement of shapes in two dimensional space. For evaluating the argument about John and Ayesha, we can pretend that possible situations are just shapes in space. Thinking of possible situations in this way is simpler, and doesn't ignore anything relevant to this particular argument.

The final concepts in our definition of logical validity are truth and falsity. These concepts are too simple to say anything much illuminating about.

Note that whether a sentence is true or false depends on which possible situation we are talking about. In this possible situation, the first premise is true, the second premise is false and the conclusion is false.

But in this possible situation, ...

... the conclusion is true.

Incidentally, you will sometimes be asked whether a logically valid argument can have one or more false premises and a true conclusion. If you're asked that question, do think about this argument and this possible situation.

So logic is the study of logical validity. As I said before, our overall aims in this course are to discover which arguments have this property, and what means there are of establishing which arguments are valid or not. In doing this our main tool is the formal language awFOL.

2.3, 2.4