Extensional logic includes standard scientific and mathematical logic. In this sets and objects are determined by their members. Intensional Logic is the logic of informal everyday human reasoning, where the order and context of sets or objects affects how they are determined. As an example let us specify two sets: “creatures with hearts” and “creatures with kidneys”, extensionally these two sets refer to exactly the same group, thus the two specifications are identical. Intensionally however, it is obvious that the two statements have different meanings, since different aspects of reality have been highlighted by the words.

Extensional logic allows us to substitute equivalent objects, so if :

K = sqr(9) = 3 = (2+1)

we can substitute any of these in an equation for any other without a problem. Contrast that with:

26th June 2001 = my birthday = three days after my discharge from hospital =ten years after the day uncle Fred passed away

These dates are extensionally identical, but when I invite people to my party, I specify the date as 26th June, to specify the date as “three days after my discharge from hospital” would be to refer to the same date but to say something confusing. Intensional logic thus disallows substitution, due to the limited knowledge and frame of reference of persons. Extensional logic is beyond psychology, intensional logic is wrapped up in, and intertwined with, the subjective viewpoints of people.

The distinction between these two logics can become apparent in statistics and probability. Consider a person, Jack who is a member of three different groups (among many), white adults, working men, residents of the UK. Each of these groups can have an average expected age of death, which is different in each case. This doesn’t mean that Jack has three different expected death ages, he obviously has just one, but by referring to him intensionally as a member of each different group, one at a time the probability appears to change. This shows the context/viewpoint dependent nature of intensional logic.

“Once upon a number”, by John Allen Paulos
In slightly more formal terms, truth is not preserved across an intensional operator. By this I mean something that converts a statement of fact into a statement of an intensional state: typically a mental state such as belief.

('Intension' and 'intention' are pronounced the same but mean different things. An intention is one kind of mental attitude to a state: that of wanting or planning to see it occur in the future; so intention may be an example of intension. However, inevitably, the familiar spelling 'intention' is often used to mean the completely different idea of intension, in which case you just have to bear in mind that philosophers are using it as a term of art much at variance from its usual meaning.)

For example, "Oslo is the capital of Norway" and "Copenhagen is the capital of Denmark" are plain extensional statements. Symbolize them by Cap(O, N) and Cap(C, D). Belief is an operator that turns this fact into another fact: if Bush believes X, then that's a fact about Bush. Let's symbolize it Bel(B, X). But the X in this proposition is not a simple name like 'Copenhagen'; rather it is itself another proposition.

So Bel(B, Cap(O, D)) means "Bush believes that Oslo is the capital of Denmark".

If in addition "the summit is in Oslo", you can't substitute "Oslo = the capital of Norway" into Bel(B, Cap(O, D)). It might be true that (i) Bush believes the summit is in Oslo, and (ii) Bush believes that Oslo is in Denmark, but not (iii) Bush believes that the capital of Norway is in Denmark.

Equality is not the only thing whose truth value fails to be preserved across intension. Boolean algebra can also fail. This explains an apparent paradox of infallibility versus humility. If you write a book, carefully, you don't know of any spelling mistakes (or typos) in it. If you look and don't find any, you are usually humble enough to suppose that there could well be some there and you have just missed them. You don't believe that the book is entirely free from error. It's just that you don't know of any errors in it.

Consider all the words in the book, W(1), ... , W(n). For each W(i) you believe it is correct. Symbolize this as
Bel(Corr(W(i))).
So we have the chain
Bel(Corr(W(1))) & ... & Bel(Corr(W(n))).
This sounds, when spoken out loud in English, that you're claiming perfection. But a claim of perfection is actually a different construct, namely
Bel( Corr(W(1)) & ... & Corr(W(n)) ).
You can't play with the bracketing while preserving truth.

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