What is Logic? Have you ever thought about the question? Have you ever tried to come up with your own definition? If, yes, then you are interested in a discipline that dates back to the 4th century BC, practiced worldwide by man and woman whose professional title is simply: Logicians.
Logic as a subject itself find its deepest roots in Philosophy. In my effort to understand Logic, I came across the book: Logic: A Very Short Introduction by Graham Priest, and so I read it and this post is a summary of my own of the book. It will make understanding the basics of Logic a piece of cake for you. Some basic terminology is introduced along with syntax that requires little to no mathematical background. In fact, I believe, after grasping this book, you will have much easier time pursuing disciplines such as Computer Science, Mathematics and Philosophy.
The Concept of Validity
Validity is a cornerstone concept in the study of Logic. There are two main types of validity:
- Deductive Validity
- Inductive Validity
But to understand each, you first have to understand some basic terminology:
- Premise: The part of the inference that gives reason to something
- Conclusion: The part of the inference that get its reason and ground from the premise(s).
- Inference: Snippets of reasoning. They are made of a set of premises and a conclusion
The whole concern of Logic, as Graham Priest puts it, is whether the conclusion follows from the premise(s). And when it does follows from the premise(s), Logicians call the whole inference Valid.
So, the central aim of logic is to understand validity.
Graham Priest – LOGIC: A Very Short Introduction
When is an inference Deductively Valid?
- An inference is deductively valid, when there isn’t a single case where the premise(s) are true and the conclusion isn’t also true.
And When is an inference Inductively Valid?
- An inference is inductively valid, when the premise(s) give good reason for the conclusion, but not good enough to draw a final conclusion upon them.
Truth Functions
- Truth Values:
In any inference, sentences can either be True or False–hence, the concepts of Truth and Falsity. If a sentence is true, then we assign it the truth value T. Else, we assign it the truth value F. As you can guess, T and F stand for True and False, respectively.
Logical Operators:
- Grammar Form: OR | Syntax Form: A V B, where A, B are sentences.
When a sentence has the logical operator OR within it, we call it a Disjunction. And, we call sentences at both sides of the ‘OR’ disjuncts. - Grammar Form: AND | Syntax Form: A & B, where A,B are sentences.
When a sentence has the logical operator AND within it, we call it a Conjunction. And, we call sentences at both sides of ‘OR’ conjuncts. - Grammar Form: NOT | Syntax Form: ¬S, where S a sentence.
If S is a sentence, then ¬S is called the Negation of S.
While we are at it, let us define what a sentence is based on traditional grammar:
- A sentence, at its simplest form, is composed of: Subject + Predicate.
According to Merriam-Webster’s Dictionary:
- a subject is: that of which a quality, attribute, or relation may be affirmed or in which it may inhere
- a predicate is: something that is affirmed or denied of the subject in a proposition in logic
Note that you can also chain inferences. For instance: Let W be the sentence ‘Ben Wajdi is a writer’. Let N be the sentence ‘Ben Wajdi was born somewhere in North Africa’. Let Y the sentence ‘Ben Wajdi is a North African-born Writer’. Let F be the sentence ‘Be Wajdi is French’.
Now, if we chain all of the above inferences in the following way:
Truth Conditions:
Truth Conditions for Negation:
- S has the truth value T, if ¬S has the truth value F
- ¬S has the truth value T, if S has the truth value F
Truth Conditions for Disjunction:
- The disjunction Q v R has the truth value T, if either one of the disjuncts has the truth value T
- The disjunction Q v R has the truth value F, only if both disjuncts have the truth value F
Truth Conditions for Conjunction:
- The conjunction Q & R has the truth value T, if each of the conjuncts has the truth value T on its own
- The conjunction Q & R has the truth value F, if either one of the conjuncts has the truth value F
Truth Tables:
| S | ¬S |
| T | F |
| F | S |
| Q | R | Q v R |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
| Q | R | Q & R |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Quantifiers & Names
Modern Logicians call words like ‘nobody’, ‘someone’ or ‘everyone’ Quantifiers and when they appear in a sentence they are distinguished from words like ‘Ben Wajdi’ or ‘David Cohen’ because these are Names. And even if both Quantifiers and Names can both serve as subjects, they tend to work differently.
So How do they differ?
- a situation comes furnished with a stock of objects
- the relevant objects can be people, or any other collection of objects depending on the situation
- all the names we generate about this situation refer to one of the objects in this collection
- thus if we write W for ‘Wajdi’, w refers to one of these objects
- and if we write A for ‘author’, then the sentence wA is true in the situation just if the object referred to by w has the property expressed by A.
The Particular Quantifier
- if we use a quantifier like ‘someone’ and say “someone is an author’ then this is true in the situation that there is an object or other in the relevant domain, (an object x in the collection of objects such as) that happen to be an author
- we denote ‘some object x is such that’ as ∃x
- thus, if we want to write ‘∃x x is an author’ we can write it simply as: ∃x xA
- => Logicians call ∃x a Particular Quantifier
The Universal Quantifier
- Now, let’s talk about another type of quantifier: ‘everyone’
- For instance: if we say ‘everyone is depressed’
- For the sentence to be true, all the objects in the collection of objects must have the property ‘depressed’
- in other words, ‘Every object, x, is such that x is Depressed’
- Let’s denote the property ‘depressed’ as D, then we can write the whole inference as: ∀x xD
- Logician call ∀x a Universal Quantifier
And so, names and quantifiers work differently. The best evidence is that Logicians will write ‘Wajdi is an author’ and ‘Someone is an author’ differently–∃w wA and ∀x xA respectively. And the key takeaway from all of this is that an inference when examined merely through its grammatical form it can mislead us when judging its validity.
Quantifiers play a central role in many important arguments in mathematics and philosophy.
Graham Priest – LOGIC: A Very Short Introduction
Graham Priest then walks us through a popular argument for the existence of God: The Cosmological Argument (very interesting read it in the book, pages 21-22)
Descriptions
We’ve seen already that at its simplest form, a sentence is a subject and a predicate. We’ve also seen that a subject can be a name or can be a quantifier. But, it also can be what logicians call a Definite Description.
A definite description has the form of ‘the thing satisfying such and such a condition’. Example: ‘The man who wrote the first novel in history’.
Th author reminds as that thanks to ‘one of the founders of modern Logic’ English Philosopher and Mathematician Bertrand Russel, we can write the above sentence as:
- Rewrite ‘the man who wrote the first novel in history’ as ‘the object, x, such that x is a man and x wrote the first novl in history’
- Now let’s write ix for ‘the object, x, such that’
- Then, our previous sentence becomes ‘ix(x is a man and x wrote the first novel in history)
- And, if we write M for ‘is a man’ and N for ‘wrote the first novel in history’ then we can write the whole thing as => ix(xM & xN)
- The general syntax for a description is: ixcx, where cx is some condition containing occurrences of x.
We know that definite descriptions can take the role of subjects. And we also know that subject + predicate make a sentence. Hence, if we write the predicate ‘was born in Spain’ as U, then the sentence ‘the man who wrote the first novel in history was born in Spain’ bceomes ix(xM & xN)U.
We can write µ as a shorthand for ix(xM & xN), then the whole sentence/inference becomes: µU
Remember, in the last section we talked about the differences between Names and Quantifiers. Rest assured if you have questions going in your mind about the classification of Descriptions because they are considered Names.
==> And so, in the above example, the sentence µU is true only if the object referred to by phrase/description µ has the property expressed by U.
The author here warns us that even though Descriptions are Names, they are a special kind of Name.
Unlike ‘proper names’ like ‘Wajdi’ and ‘Ben’, Descriptions carry information, and often properties, about the object within it. For instance, the Definite Description ‘the man who wrote the first novel in history’ carries within it two properties about the object: he is a man + he wrote the first novel in history.
This is a special case of something more general, namely: the thing satisfying such and such a condition, satisfies that very condition. This is often called the Characterization Principle (a thing has those properties by which it is characterized).
Graham Priest – LOGIC: A Very Short Introduction
==> Here, the author talks about the Ontological Argument for the existence of god. Go on and read it in the book if you’re interested, I didn’t cover it here for the sake of brevity.
Something else you need to know about descriptions: if µP is a sentence, µ description and P a predicate, and if the object referred to within the description µ does not really exist, then µP is false.
But here again, Graham Priest warns us that it is not always the case, and that this is when we start encountering some weird behavior regarding Descriptions. Sometimes, it turns out, the object referred to can be kind of ‘imaginary’ or ‘not real’ and still the property P attached to it can still hold and be true. An example the author present is about ‘Zeus’. After all, ‘Zus’ didn’t really exist, nor did he live on ‘Mount Olympus’, yet still the property of ‘was worshiped by ancient Greeks’ is still true. And so, the author reminds us that there are some cases, where the object does not exist yet still there are truths about it.
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