Why isn’t ‘because’ a logical connective in propositional logic?

Question

In simple terms, could someone explain why there is not a logical connective for ‘because’ in propositional logic like there is for ‘and’ and ‘or’?

Is this because the equivalent of ‘because’ is the argument of the form ‘if p, then q’, or am I missing something?

Please illustrate your answer with example(s) if possible.

Answer 1

It is because 'because' is not truth-functional.

That is, knowing the truth-values of $P$ and $Q$ does not tell you the truth-value of '$P$ because of $Q$'

For example, the two statements 'Grass is green' and 'Snow is white' are both true, but 'Grass is green because snow is white' is an invalid argument, and hence, as a statement as to the validity of that argument, a false statement.

On the other hand,'Grass is green because grass is green' is a true statement as to the validity of this as an argument, but yet again it involves two true statements.

This shows that with $P$ and $Q$ both being true, the statement '$P$ because of $Q$' can either be true or false, and hence it is not truth-functional.

Answer 2

Why isn’t ‘because’ a logical connective in propositional logic?

Is this because the equivalent of ‘because’ is the argument of the form ‘if $p$, then $q$’ ?

Exactly.

Either the connective "because" is truth-functional, in which case it is the same as "if..., then...", or it is not truth-functional, in which case we need a different way of modelling it.

See e.g. Counterfactual Theories of Causation.

See also Arthur Burks, The Logic of Causal Proposition, Mind (1951).

Answer 3

I agree with the other answers, however I want to add that the closest thing might be the turnstyle symbol $\vdash$, although this is usually read as "yields", and thus points the other way. If I write

$$A \vdash B$$ this is read as "A yields B", or "knowing A, I can prove B". If you wanted to encode because, you could probably read it backwards as "B because of A".

Note however that this is not used as part of a logical formula, but as a shorthand between formulas when writing down a proof. So $A \vdash B$ is no longer a formula, but rather a statement on how to prove $B$. (In most of the rest of mathematics, you would write $\Rightarrow$ in your proof instead, however in logic this is of course easily confused with the implication inside formulas)

Answer 4

You can define things however you want. (Be careful; you may accidentally be inconsistent.)

Either:

• "because" is logically equivalent to a binary operator
• or it's not

If it is, it's probably the same as "only if" (Or take your pick of the other 15 operators). Adding a "because" overload would create an additional word to remember: unneeded complexity. We like simplicity.

If it isn't, you can define a binary function because(a, b) however you want.