Notes on Logical Notation

 

You can think of logical notation as being a way of paraphrasing natural languages so that all and only the parts of natural language that are important to logic are included. In general logic is concerned with knowing what is deducible as true given a set of other true statements. What's important to 'logic' is of course, possibly controversial. I'll just show you how to read statements in classical logical notation.

 

This comes in two parts.

 

Propositional Logic

 

The first part is called Sentential or Propositional logic because it deals with the sorts of deductive relations that exist between sentence/propositions. We know that there are various ways of putting together sentences so that the truth or falsity (the truth value) of the resulting sentence is entirely dependent upon the truth value of the parts of which it is built. The connectives that are used to build up sentences in this way are called truth functional operators. Thus we have symbols like the following.

 

A, B, ... stand for propositions. For example A could stand for 'The dog is happy'. They have to be declarative sentences in their own right; not questions, not phrases like 'the happy dog', not words, etc. They describe ways that the world is.

Sometimes you'll see P, Q, R, ... used, or p, q, r, ..., or there'll be subscripts like p1 , p2, ... They're all the same thing.

 

We also need symbols for the truth functional operators.

 

'~' stands for negation which in natural language is signalled by words like 'not'.

Other people use '', or they put a bar over the proposition being negated.

~A, A, Ā, should be read as 'not A' ('it is not the case that A'). It is true just when A is false.

 

'&' stands for conjunction which in natural language may be signalled by 'and' or 'but' or 'as well as' or ...

Other people use '', and others use '.', and others just jam the two sentence symbols together.

A & B, A B, A.B, AB should all be read as 'A and B'. It is true just when A is true and B is true.

 

'or' stands for disjunction which corresponds to words like 'or' in natural language.

Other people use ''.

A B should be read as 'A or B'. It is true just if A is true or B is true (or both)

 

'' stands for material implication which corresponds to words like 'if ... then ---' in natural language.

Other people use '' or ''.

A B, A B, A B should be read as 'if A then B'. It is true just when A is false or B is true (or both). It's identical to ~A B. Note that this is far from capturing all the implication of if ... then --- in natural language, which is why we specifically call it material implication.

 

'' stands for material equivalence which corresponds to phrases like 'if and only if' or 'just in the case that'.

Other people use '', or ''.

A B, A B, A B should be read as 'A if and only if B' (jargon alert: A iff B). It's true just when either A and B are both true or A and B are both false. It's identical to (A B) & (B A).

 

So if we've got a sentence like

A = 'Bob is happy'

and another sentence like

B = 'Bob is tall'

we can make the new sentences

~A = 'Bob is not happy'. Note: not 'Bob is unhappy', Bob might be neither. Read it as 'It is not the case that Bob is happy' to be safe.

A & B = 'Bob is happy and Bob is tall' or more naturally 'Bob is happy and tall'.

A B = 'Bob is happy or tall'

A B = 'If Bob is happy then Bob is tall', 'If Bob's happy then he's tall', 'Bob's happy only if he's tall', etc.

A B = 'Bob's happy just when he's tall', Bob's happy if and only if he is tall', ...

 

Predicate Logic

 

The second part of classical logic allows us to deal with the sort of deductions we see in arguments like

 

'Socrates is a man,

and all men are mortal,

so Socrates is mortal.'

 

The important deductive relationship includes some consideration of the properties that are said to belong to Socrates. In natural language we say that terms predicate properties of their subjects when we want to say that Socrates may rightly have the word 'mortal' applied to him because he is mortal. He has the property of being a mortal thing.

 

We also want to deal with deductive relations of statements that use quantifiers, such as 'All' or 'Some', one of which occurred in the Socrates example.

 

As it turns out, the most convenient way to treat quantified statements is to introduce variables into the formal language. In the formal language of propositional logic the formulae could be readily understood as paraphrases of English. It wasn't quite true then, however, and it is much less the case with the formal language of predicate logic: we don't have variable of the right sort in natural languages. Nevertheless, we can easily translate from natural language in to the formal language of predicate logic, make our deductions therein, and translate back out again.

 

The symbols we'll need include all the symbols of propositional logic plus these extras.

 

a, b, c, ... stand for names of things in the world. 'a' could stand for 'Socrates', for example.

F, G, H, ... stand for the predicates that could be true of things in the world. 'F' could be 'is mortal', for example.

Note that these predicates can apply to names. 'is mortal' applies to a single name for example. It is called a one-place predicate. Each predicate has a certain number of places that have to be filled before it is complete, and when it is complete it gives a proposition. To say that 'Socrates is mortal' using the above assignments we'd write 'Fa'

To say that 'Socrates is taller than Plato'. we'd write Gab (where G stands for the two-place predicate 'is taller than', a is for Socrates and b is for Plato.

And so on.

 

x, y, z, ... are variables

The predicates can be written with variables instead of names filling their places. Thus Fx, but this doesn't correspond to any part of natural language (it doesn't mean 'something is F', and we never say 'x is F' is real life. We use these sorts of things in order to talk about quantified statements.

 

There are two type of quantified statement.

 

("x) stands for the universal quantification signalled by the use of words like 'all', 'every', 'any', ...

Other people use (x),  Lx, etc.

 

A statement that 'Everything is sweet' would be written as

("x) Fx

where F is 'is sweet'; and we'd read it as 'for all x, x is sweet'. It is true if for every substitution of a name, like 'a' (where 'a' means 'Socrates'), Fa is true (ie. it is true that Socrates is sweet). It must be true for every such substitution.

 

A statement that 'All men are mortal' would be written as

("x) [Fx Gx]

where F is 'is a man' and G is 'is mortal'; and we'd read it as 'For all x, if x is a man then x is mortal'. It is true if for every substitution of a name, like 'a' (where 'a' means 'Socrates'), [Fa  Ga] is true (ie. it is true that if Socrates is a man then Socrates is mortal).

 

($x) stands for the existential quantification signalled by the use of words like 'some', 'a', 'one or more', ...

Other people use Vx, etc.

 

A statement that 'Something is sour' would be written as

($x) Fx

where F is 'is sour'; and we'd read it as 'for some x, x is sour', or 'there exists x such that x is sour'. It is true if for some substitution of a name, like 'a' (where 'a' means 'Socrates'), Fa is true (ie. it is true that Socrates is sour). It only needs to be true for one such substitution.

 

A statement that 'Some men are happy' would be written as

($x) [Fx & Gx]

where F is 'is a man' and G is 'is mortal'; and we'd read it as 'there exists x such that x is a man and x is mortal'. It is true if there is at least one substitution of a name, like 'a' (where 'a' means 'Socrates'), for which [Fa & Ga] is true (ie. it is true that if Socrates is a man and Socrates is mortal).