What is Logic?

Logic is the systematic study of the form of valid inference, and the most general laws of truth. A valid inference is one where there is a specific relation of logical support between the assumptions of the inference and its conclusion.

There is no universal agreement as to the exact scope and subject matter of logic, but it has traditionally included the classification of arguments, the systematic exposition of the ‘logical form’ common to all valid arguments, the study of proof and inference, including paradoxes and fallacies, and the study of syntax and semantics.  Historically, logic has been studied in philosophy (since ancient times) and mathematics (since the mid-19th century), and recently logic has been studied in computer science, linguistics, psychology and other fields.

Most people tend to think of themselves as logical. Telling someone you are not being logical is normally a form of criticism. To be illogical is to be confused, muddled, and irrational.

We all reason. We try to figure out what is so, reasoning on the basis of what we already know. Logic is the study of what counts as a good reason for what and why.

Here are two bits of reasoning – logicians call them inferences.

1. Rome is the capital of Italy, and this plane lands in Rome; so the plane lands in Italy.
2. Moscow is the capital of USA; so you cannot go Moscow without going to USA.

In each case, the claims before the ‘so’ – logicians call them premises – are giving reasons; the claims after the ‘so’ – logicians call them conclusions.

The first piece of reasoning is fine; but the second is pretty hopeless and simply false. The premise had been true – if say, the USA had bought the whole of Russia (not just Alaska) and moved the white house to Moscow, the conclusion would have been true. It would have followed from the premises; and that is what logic is concerned with.  It is not concerned with whether the premises of an inference is true or false. That is somebody else’s business (in this case the geographer’s). It is interested simply in whether the conclusions follow from the premises. Logicians call an inference where conclusion really does follow from the premises valid. So the central aim of logic is to understand validity.

Kinds of Validity:

1. Deductive validity
2. Inductive validity

 

Let us consider the following three inferences.

1. If the burglar had broken through the kitchen window, there would be footprints outside; but there are no footprints; so the burglar did not break in through the kitchen window.
2. Jones has nicotine stained fingers; so Jones is a smoker.
3. Jones buys two packets of cigarette a day; so someone left footprints outside the kitchen window.

The first inference is a very straightforward one. If the premises are true so must the conclusion be. Or to put is another way, the premises could not be true without the conclusion also being true. Logicians call an inference of this kind deductively valid.

Inference number two is a bit different. The premise clearly gives a good reason for the conclusion, but is not completely conclusive. After all Jones could have simply stained his hands to make people think that he was a smoker. So the inference is not deductively valid. Inferences like this are said to be inductively valid.

Inference number three seems to be pretty hopeless by any standard.

 

Inductive validity is very important notion. We reason inductively all the time; for example in trying to solve problems such as why the car has broken down, why a person is ill, or who committed a crime.

Despite this, historically much more effort has gone into understanding deductive validity – may be because logicians have tended to be philosophers or mathematicians (in whose studies deductively valid inferences are certainly important) and not doctors, detectives or mechanics.

So what is a valid inference?

We saw where the premises can’t be true without the conclusion also being true.

But what does that mean? In particular, what does the can’t mean?

In general can’t can mean many different things. Consider for example: Mary can play the piano, but John can’t; here we are talking about human abilities.

Compare: ’You can’t go in here: you need a permit’; here we are talking about what some code of rules permits.

 

It is natural to understand the ‘can’t’ relevant to present case in this way; to say that the premises can’t be true without the conclusion being true is to say  that in all situations in which all the premises are true, so is the conclusion.

But what exactly is the situation? What sort of things go into their makeup and how do these things relate to each other?

And what is it to be true?  Now there is a philosophical problem.

‘Situation’ and ‘Truth’ are complex concepts in philosophy which philosophers incessantly struggle to grapple with.

 

 

 

Aristotelean Logic

Aristotle’s collection of logical treatises is known as Organon. Of these treatises, the Prior Analytics contains the most systematic discussions of formal logic. In addition to Organon, the Metaphysics contains relevant material.

Subject and Predicates

Aristotelean logic begins with the familiar grammatical distinction between subject and predicate. A subject is typically an individual entity, for instance a man, or a house or a city. It may also be a class of entities, for instance all men. A predicate is a property or attribute or mode of existence that a given subject may or may not possess.

For example an individual man (the subject) may or may not be skillful (the predicate), and all men (the subject) may or may not be brothers (the predicate).

The fundamental principles of predication are:

1. Everything is what it is and acts accordingly. In symbols:

A is A.  For example, an acorn will grow out of an oak tree and nothing else.

2. It is impossible for a thing both to be and not to be. A given predicate either belongs or does not belong to a given subject at a given time. Symbolically: Either A or non-A.

For example, a society must be either free or not free.

These principles have exercised a powerful influence on subsequent thinkers. The twentieth-century intellectual Ayn Rand titled the three main divisions of her best-selling philosophical novel Atlas Shrugged after principles above, in tribute to Aristotle.

Syllogisms

According to Aristotelian logic, the basic unit of reasoning is the Syllogism.

It is of the form

Some A is B.

All B is C.

Therefore, some A is C.

 

Every syllogism consists of two premises and one conclusion.

Each of the premises and the conclusion is one of the four types.

Universal affirmative:          All A is B.

Universal negative:               No A is B.

Particular affirmative:          Some A is B

Particular negative:              Some A is not B.

The letters A, B, C are known as terms. Every syllogism contains three terms. The two premises always share a term that does not appear in the conclusion. This is known as the middle term.

 

A more comprehensive format of syllogism:

All [some] As are [are not] Bs.

All [some] Bs are [are not] Cs.

So, all [some] As are [are not] Cs.

 

In order to classify the various types of syllogisms, one must take account of certain symmetries. In particular “no A is B” and “no B is A” are equivalent,

as are “some A is B” and “some B is A”.

Furthermore, the order of the two premises in a syllogism does not matter.

Allowing of these symmetries, we can enumerate a total of 126 possible syllogistic forms. Of these 126, only 11 represent correct inferences.

For example, the form

   all A is B, all B is C, therefore all A is C

represents a correct inference, while

    all A is B, all C is B, therefore some A is C does not.

The classification of syllogisms leads to a rather complex theory. Medieval thinkers perfected it and developed ingenious mnemonics to aid in distinguishing correct from the incorrect ones.

Theory of Definition

In the older logic a definition is the delimitation of a species by stating the genus which includes it and the specific difference or distinguishing characteristic of the species. A typical definition of man as rational animal. The genus is the animal genus and the distinguishing characteristic is rationality. (What has been stated in capsule form is the Aristotelian theory of definition).

Aristotelian analysis, do seriously promulgate the four traditional rules of definition:

1. A definition must give the essence of that which is to be defined.
2. A definition must not be circular.
3. A definition must not be negative when it can be in the positive.
4. A definition must not be expressed in figurative or obscure language.

Certainly these rules have serious use as practical precepts. They rule out as definitions statements like:

Beauty is eternity gazing at itself in a mirror.   KHALIL GIBRAN, The Prophet,                            which violates Rule 4, or:

Force is not a kinematical notion, which violates rule 3.     {notio/motion ? }

THE SENTENTIAL CONNECTIVES

 

We need to develop a vocabulary which is precise and at the same time adequate for analysis of the problems and concepts of systematic knowledge.

We must use vague language to create a precise language.

We want to lay down careful rules of usage of certain key words: ‘not’, ‘or’, ‘and’,

‘If … then …’, ‘if and only if’, which are called sentential connectives.

Negation and conjunction.

We deny the truth of a sentence by asserting its negation. We attach word ‘not’ to the main verb of the sentence.

Sugar causes tooth decay. Negation: Sugar does not cause tooth decay.

However, the assertion of negation of a compound sentence is more complicated.

‘Sugar causes tooth decay and whisky causes ulcers’

Negation: ‘It is not the case that both sugar causes tooth decay and whiskey causes ulcers’.

In spite pf apparent divergence between these two examples, it is convenient to adopt in logic a single sign for forming the negation of a sentence. We shall use the prefix

‘-‘, which is placed before the whole sentence. The negation of the first example is written:     – (Sugar causes tooth decay).

The negation of the second example is – (Sugar causes tooth decay and whisky causes ulcers)

The negation of a true sentence is false, and negation of false sentence is true.

NEGATION

.

P	-P	Q	-Q
T	F	T	F
F	T	F	T

 

The word ‘and’ is used to conjoin (combine) two sentences to make a single sentence which is called the conjunction of two sentences.

‘Mary loves John and John loves Mary’ is the conjunction of sentence ‘Mary loves John’ and sentence ‘John loves Mary’. The ampersand sign ‘&’/ (‘^’) is used for conjunction.

The conjunction of any two sentences P and Q is written as P & Q.

The conjunction of two sentences is true if and only if both sentences are true.

There is no requirement that two sentences be related in content or subject matter.

Any combinations, however absurd are permitted.

 

CONJUNCTION

P	Q	P & Q
T	T	T
T	F	F
F	T	F
F	F	F

 

Disjunction:

The word ‘or’ is used to obtain the disjunction of two sentences. The sign ‘V’ is used for disjunction. The disjunction of any two sentences P and Q is written P V Q.

The disjunction of two sentences is true if and only if at least one of the sentences is true.

DISJUNCTION

P	Q	P V Q
T	T	T
T	F	T
F	T	T
F	F	F

 

Implication: Conditional Sentences.

The expression ‘if …, then ….’ Is used to obtain from two sentences a conditional sentence. A conditional sentence is also called an implication.

 

IMPLICATION

P	Q	P èQ
T	T	T
T	F	F
F	T	T
F	F	T

 

 

 

Conditional:

P è Q

P implies Q

If P, then Q

The conditional statement is saying that if P is true, then Q will immediately follow, and thus be true. So the first row naturally follows the definition.

Similarly, the second row follows this because we say “P implies Q’ and then P is true and Q is false, then the statement “P implies Q” must be false, as Q did not immediately follow P.

The last two rows are tough ones to think about, so let us look them individually.

Row # 3           P is false, Q is true.

Think of the following statement.

If it is sunny, I will wear my glasses.

If P is false and Q is true, then it is saying that it is not sunny, but I wore glasses anyway. This certainly does not invalidate my original statement as I might just like my glasses. So if P is false, but Q is true, it is reasonable to think “P implies Q” is still true.

Row #4           P is false, Q is false.

Using the example about sunglasses, this would be equivalent to it, not being sunny and me not wearing my glasses.

Again this would not invalidate my statement that if it is sunny, I wear my glasses.

Therefore, if P is false and Q is true, “P implies Q” is still true.

Continuing with sunglasses, the only time you would question the validity of my statement is if you saw me on a Sunny day without my glasses (P true, Q False).

Hence the conditional statement is true in all but one case, when the front (first statement) is true but the back (second statement) is false.

• Conditional is a compound statement of the form “If P then Q”
• Think of the conditional as a promise
• If I do not keep my promise, in other words Q is false then the conditional is false, if the promise is true.
• If I keep my promise, then Q is true and the promise is true, then the conditional is true.
• When the premise is false (i.e. P is false) then there was no promise, hence by default conditional is true.

 

 

Equivalence:  Biconditional Sentences.

• The expression ‘if and only if’ is used to obtain from two sentences a biconditional sentence. A biconditional sentence is also called an equivalence and the two sentences connected by ‘if and only if’ are called the left and right member of the equivalence.
• The biconditional

P if and only if Q       (1)

Has the same meaning as the sentence

P if Q and P only if Q      (2)

And (2) is equivalent to

If P then Q, and if Q then P.    (3)

Rules of usage for conjunction and implication tell us that (3) is true just when P and Q are both true or both false. Thus the rule “A biconditional sentence is true if and only if its two members are either both true  or both false.

As a matter of notation it is written P çè  Q for biconditional formed from sentences P and Q.  It can also be said Q is necessary and sufficient condition for P.

When a conditional statement and its converse are combined, a biconditional statement is created.

“P if and only if Q”, notation   P çè Q

P çè Q   means P è  Q and Q è P

           EQUIVALENCE

P	Q	P çèQ
T	T	T
T	F	F
F	T	F
F	F	T

 

Summary of connectives and Truth Tables

Disjunction	P V Q	P or Q
Biconditional	P çè Q	P if and only if Q
Conditional	P è Q	If P then Q
Conjunction	P ^ Q     (P  & Q)	P and Q
Negation	~ P   or   – P	Not P

 

Truth tables (F = false, T = True)

 

P	Q	P V Q	P & Q	P è Q	P çè Q
T	T	T	T	T	T
T	F	T	F	F	F
F	T	T	F	T	F
F	F	F	F	T	T

 

P and   ~P have opposite truth values.

Tautologies

A tautology is true for all possible assignments of truth values to its components.

 

A tautology is also called a universally valid formula and logical truth.

A statement formula which is false for all possible assignments of truth values to its components is called a contradiction.

 

Three Well-Worn Arguments for the Existence of God

From the book “An Incomplete Education”

…

NOTE: During our Sunday discussion we will see how logicians analyze these arguments about the existence of God.

 

These old chestnuts mark the point at which philosophy — which supposedly bases its arguments on reason — and theology – which gets to call in revelation and faith – overlap. The results as you will see, sounds an awful lot like wishful thinking.

THE COSMOLOGICAL ARGUMENT:

This one dates all the way back to Aristotle’s theory of motion and encompasses Thomas Aquinas’ version, known as the argument from contingency and necessity.

We know from experience that everything in the world moves and changes, said Aristotle (or simply exists said Aquinas), and everything that moves, or exists, has a mover, i.e., a cause, something that precedes it, and makes it happen. Now, we can trace lot of things in the world back to their immediate causes, but there is always another cause behind them and another behind them. Obviously said Aristotle,

we cannot keep tracing effects back to causes indefinitely; there has to be one cause that isn’t, itself caused by something else, or one entity that existed before all the others could come into existence. This first cause, the Unmoved Mover, is God.

The cosmological argument, widely accepted for centuries, started running into snags when Hume decided that the whole principle of cause and effect was a mirage.

Later Kant made matters worse by pointing out that there may be cause and effect in this world, we do not get to assume that the same holds true out there in the Great Unknown.

Today, critics counter the cosmological argument by pointing out that there is no reason to assume we cannot have an infinite series of causes, since we can construct all sort of infinite series in Mathematics. Also that the argument never satisfactorily dealt with the question of any four-year-old knows enough to ask, namely, Who made God?

 

THE ONTOLOGICAL ARGUMENT:

 

This is an example of old philosopher’s dream of explaining the nature of universe through sheer deduction; also of how slippery a priori reasoning can get.

The argument, which probably originated with St. Anslem back in the Middle Ages

and  which hit its peak with Descartes, Spinoza and Leibniz,

the Continental  Rationalists of the seventeenth century runs as follows:

We can conceive of Perfection (if we couldn’t, we would not be so quick to recognize imperfection) and we can conceive of a Perfect Being. God is what we call that Being, which embodies all imaginable attributes of perfection, the Being than which no greater Being can be conceived. Well if you are going to imagine a Perfect Being, it stands to reason that He exists, since a Perfect Being that did not exist would not be as perfect as a Perfect Being that did, and isn’t, therefore, the most Perfect Being you can imagine.

(Is He?)  Hence by definition, God Exists. If you are still reading at this point, you may have already noticed that the ontological argument can be criticized for begging the question; that is, it assumes at the outset, the very thing it purports to prove.

Still, when you think about it, the argument is not nearly as simpleminded as it appears. Just where did you get your idea of a Perfect Being if you are so sure no such thing exists?

THE TELEOLOGICAL ARGUMENT, OR THE ARGUMENT FROM DESIGN:

Simply by looking around, you can see that the world is a strange and wondrous place, something like an enormous machine with millions of perfectly made perfectly interlocking parts. Now, nobody but an underground filmmaker would claim that such a structure could be the result of mere chance.

For metaphysicians from Plato and Aristotle to eighteenth-century Enlightenment thinkers, enamored of mechanical symmetry of the universe, and nineteenth-century ones, enamored of biological complexity of same, the idea that there had to be a Mind behind all this magnificent order seemed pretty obvious. The teleological argument survived for so long partly because the world is pretty amazing place, and partly because the argument’s validity never depended on the idea that God is omniscient or omnipotent, only He is a better planner than the rest of us.

However as Hume, the great debunker, was to point out, even if we could assume the existence of a Cosmic Architect who was marginally better at putting it all together than we are, such a mediocre intelligence, which allowed for so many glitches in the plan, would hardly constitute God. And then along came the mathematicians again, pointing out that, according to the theories of chance and Probability, the cosmos just might be an accident after all.

 

 

Decision Theory:

Great Expectations

Pascal Wager’s analysis using Probability

Inductive Reasoning

Practical Reasoning

 

You can choose to believe in the existence of monotheistic (a Christian) God; you can choose not to. Let us suppose that you chose to believe. Either God exists or God does not. If God exists all well and good. If not then your belief is a minor inconvenience and not a big disaster.

Now suppose on the other hand, that you chose not to believe in the existence of God.

Again either God exists or not. If God does not exist, all well and good. But if God does exist? Big problem! You are in for a lot of suffering in the afterlife, may be for all eternity. So any wise person ought to believe in the existence of God. It is the only prudent course of action.

The argument is now usually called Pascal’s Wager, after the 17^th century philosopher and Mathematician Blaise Pascal, who first put it forward.

What is one to say about the wager?

Let us think a little about how this kind of reasoning works. When we perform actions, we often cannot be sure of the results, which may not be entirely under our control.

But we can usually estimate how likely the various possible results are; and just as importantly we can estimate the value to ourselves of various results.

Conventionally, we can measure the value of an outcome by assigning it a number on the following scale, open ended in both directions.

…….. -4,  -3,  -2, -1, 0, +1, +2, +3, +4 ………..

Positive numbers are good, and further to the right, the better.

Negative numbers are bad, and further to the left, the worse,

0 is a point of indifference: we do not care either way.

Suppose we want to go on a bike ride. It may, however rain.  A bike ride when it is not raining is great fun, so we would value that at say +10.

But a bike ride when it is raining can be pretty miserable, so we would value that say at     -5. What value should be put on the only thing that is that is under our control:

going on the ride? We could just add the two figures, -5 and 10, but that would be missing an important part of the picture. It may be that it is most likely to rain, so although the possibility of rain is bad, we do not want to give it too much weight. Suppose the probability of rain is 0.1, correspondingly the probability of no rain is 0.9. Then we can weight the values with appropriate probabilities to arrive at an overall value.

0.1 x (-5) + 0.9 x 10   = 8.5

This is equal to 8.5 and is called the expectation of the action in question, going for a ride. {‘Expectation’ here is a technical term; it has virtually nothing to do with meaning of word as used normally in English}.

In general let a be the statement we perform some action or other.

Suppose there are two possible outcomes o1 and o2.  (It o for orange, not zero)

Let v(o) the value we attach to O being true. Then the expectation of a, E(a) is the number defined by:

Pr(o1) x V(o1) + Pr(o2) x V(o2)     {Pr  = Probability}

But how to make a decision whether to go for a bike ride or not.

We know that overall value for going on a ride is 8.5.

What is the expectation of not going on a ride?

Again, either it will rain or it will not rain – with the same probabilities.

The two outcomes are:

• That it will rain and I stay at home.
• That it will not rain and I stay at home.

In both of these cases I derive no pleasure from a bike ride. It might be slightly worse if it does not rain. In that case I might be annoyed that I did not go. But in neither case is it as bad as getting soaked. So the values might be 0 if it rains and -1 if it does not. I can now calculate the expectation of staying at home:

• x 0 + 0.9 (-1)

This comes to -0.9 and gives me the information I need; for I should chose that action which has the highest overall value, that is, expectation. In this case going has expectation 8.5 whilst staying at home has value -0.9. So I should go for a ride.

 

Thus given a choice between a and –a, I should chose whichever has a greater expectation. (If they are the same, I can simply choose at random, say by tossing a coin). In the previous case there are two possibilities. In general there might be more (say, going for a ride, going to the movies, and staying at home). The principle is the same, though: I calculate the expectation of each possibility, and choose whatever has the greatest expectation. This sort of reasoning is a simple example from the branch of logic called decision theory.

Now let us take Pascal’s Wager: In this case there are two possible actions: believing or not: and there are two relevant possibilities: God exists or does not. This info can be presented in the form of the following table.

	God exists	God doesn’t exist
I believe (b)	0.1\ + 10^2	0.9\ – 10
I don’t believe (-b)	0.1\ – 10^6	0.9\ + 10^2

The figures on the left of backward slashes are the relevant probabilities, 0.1 that God exists, and 0.9 that God does not exist. (Whether or not I believe has no effect on whether or not God exists, so the probabilities are same in both rows). The figures to the right of slashes are the relevant values. I do not much mind whether or not God exists; the important thing is that I get it right; so the value in each of these cases is +10^2. (Perhaps one’s preference here might not be exactly the same, but it does not matter too much, as we shall see.) Believing when God does not exist, is a minor inconvenience, so gets the value -10. Not believing, when God does exist, is really bad, it gets the value -10^6.

Given these values, we can compute the relevant expectations.

E(b)       =   0.1 x 10^2  +  0.9 x (-10)     ≈             0

E(-b)      =  0.1 x (-10^6)  + 0.9 x 10^2     ≈    -10^5

You may think that the precise values I have chosen are somewhat arbitrary and artificial; and so they are. But in fact, precise value do not really matter too much. The important one is       the -10^6. This figure represents something that is really bad. (Sometime, a decision theorist might write this as – ∞).  It is so bad that it will swamp all other figures. Even if the probability of God’s existence is very low. That is the punch in Pascal’s Wager.

The wager might look fairly persuasive, but in fact it makes a rather simple decision-theorist error. It omits some relevant possibilities. There is not just one possible god, there are many;    a Christian god (God), Islam’s Allah, Hinduism’s Brahman, and lots more that various minor religions worship. And a number of these very jealous gods. If God exists, and you believe in Allah – or vice versa – this is even worse. For in both Christianity and Islam, believing in false gods (Shirk) is worse than being a simple non-believer.

Let us draw up a table with some more realistic information.

	No God exists	God exists	Allah exists
No Belief (n)	0.9\+10^2	0.01\ -10^6	0.01\-10^6
Believed in God (g)	0.9\-10	0.01\ +10^2	0.01\-10^9
Believe in Allah (a)	0.9\-10	0.01\ -10^9	0.01\+10^2

If we compute the expectations on even this limited amount of information, we get:

E(n)= 0.9 x 10^2 + 0.01 x (-10^6) + 0.01 x (-10^6) ≈ -2 x 10^4

E(g)= 0.9 x (-10) + 0.01 x (10^2) + 0.01 x (-10^9)  ≈ – 10^7

E(a)= 0.9 x (-10) + 0.01 x (-10^9) + 0.01 x (10^2)  ≈ – 10^7  .

Things are looking pretty bleak all around.                                                                                                             But it is clear that theistic beliefs are coming off worst.                                                                                                 You should not have any of them.

 

Noor Salik