Expressing one logical constant in terms of two others

[Theophilus7]

I was just wondering whether anyone here would know how to go about expressing '&', 'v' and '<->' in terms of '¬' and '->'?

There may be one or two logicians among you

 

[:æ:]

I was just wondering whether anyone here would know how to go about expressing '&', 'v' and '<->' in terms of '¬' and '->'?

There may be one or two logicians among you

Admittedly, I'm not formally trained in logic, but if I am not mistaken, the characters you're talking about do not represent constants but rather they represent logical operators.

 

[PapaLandShark]

Oooo...Logical Constants. I, also, am not formally trained...but I once stayed a night at the Holiday Inn!

That being said...I popped out and found a decent article on the subject:

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called "logical constants."[1] Thus, for example, the sentences

Every boy loves some girl.​

and

Some boy loves every girl.​

are thought to differ in logical form, even though they share a common syntactic and semantic structure, because they differ in the placement of the logical constants "every" and "some". By contrast, the sentences

Every girl loves some boy.​

and

Every boy loves some girl.​

are thought to have the same logical form, because "girl" and "boy" are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the "logical constants" of a language from its nonlogical expressions.
While it is generally agreed that signs for negation, conjunction, disjunction, conditionality, and the first-order quantifiers should count as logical constants, and that words like "red", "boy", "taller", and "Clinton" should not, there is a vast disputed middle ground. Is the sign for identity a logical constant? Are tense and modal operators logical constants? What about "true", the epsilon of set-theoretic membership, the sign for mereological parthood, the second-order quantifiers, or the quantifier "there are infinitely many"? Is there a distinctive logic of agency, or of knowledge? In these border areas our intuitions from paradigm cases fail us; we need something more principled.
However, there is little philosophical consensus about the basis for the distinction between logical and nonlogical expressions. Until this question is resolved, we lack a proper understanding of the scope and nature of logic, and of the significance of the distinction between the "formal" properties and relations logic studies and related but non-formal ones. For example, the sentence

If Socrates is human and mortal, then he is mortal.​

is generally taken to be a logical truth, while the sentence

If Socrates is orange, then he is colored.​

is not, even though intuitively both are true, necessary, knowable a priori, and analytic. What is the significance of the distinction we are making between them in calling one but not the other "logically true"? A principled demarcation of logical constants might offer an answer to this question, thereby clarifying what is at stake in philosophical controversies for which it matters what counts as logic (for example, logicism and structuralism in the philosophy of mathematics).

For anyone interested you can find the full article here.

 

[StrugglingSceptic]

I was just wondering whether anyone here would know how to go about expressing '&', 'v' and '<->' in terms of '¬' and '->'?

Check the truth table for -> and see how you can insert negations to get the truth tables for & and v.

Otherwise:

1) ((p & q) <-> ~(p -> ~q);

2) (p v q) <-> (~p -> q);

3) (p <-> q) <-> ((p -> q) & (q -> p))

Using (1)

((p -> q) & (q -> p)) <-> ~((p -> q) -> ~(q -> p)).

There may be one or two logicians among you

Are you doing axiomatic propositional calculus? It is fairly common for the language of this calculus to only have the connectives -> and v.

If you are interested, it is possible to express every connective in terms of a single neither-nor connective.