Thinkers Forum USA
Cordially invites all participants to the monthly Meeting/Discussion
On Sunday, February 24, 2019
Time
11: 55 AM
To
2: 30 PM
Speaker
Noor Salik
Topic
What is Logic?
Moderator
Dr. Fayyaz Sheikh
Location
Saffron Indian Cuisine
97 RT 303, Congers, N.Y. 10920
845 767 4444
Brunch served after lecture
Outline of topic for discussion
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.
- Rome is the capital of Italy, and this plane lands in Rome; so the plane lands in Italy.
- 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:
- Deductive validity
- Inductive validity
Let us consider the following three inferences.
- 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.
- Jones has nicotine stained fingers; so Jones is a smoker.
- 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:
- 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.
- 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 s 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:
- A definition must give the essence of that which is to be defined.
- A definition must not be circular.
- A definition must not be negative when it can be in the positive.
- 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.
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 ç==è 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 ç==è Q
P ç==è 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.
Pascal Wager will also be discussed, if time allowed.
US Berkley