What is the difference between Boolean logic and propositional logic?

Question

As far as I can see, they only employ different symbols but they operate in the same way. Am I missing something?

I wanted to write "Boolean logic" in the tag box but a message came up saying that if I wanted to write Boolean logic I should better write propositional logic. Doesn't this confirm my suspicions?

Answer 1

you can see :

• J.Donald Monk, Mathematical logic (1976), Ch.9 Boolean algebra, page 141-on :

In this chapter we give a brief introduction to the theory of Boolean algebras. As we shall see, these are algebraic structures which stand in an intimate relationship to [classical] sentential logics [emphasis added]. They will also form a source for some of the applications of logic which we shall give later.

See page 158 :

[regarding] the correspondence between Boolean algebras and sentential logics [...] We shall see that there is a full correspondence between these two kinds of mathematical objects.

Finally, see page 160 :

the following theorem, which is another kind of completeness theorem for Boolean algebras. [...] Hence we may say that the theories of Boolean algebras and of sentential logics are equivalent, in some sense.

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You can see also :

• J.L. Bell & A.B. Slomson, Models and Ultraproducts: An Introduction (1969 - Dover reprint), page 11 :

a Boolean algebra is a complemented distributive lattice with at least two elements.

Page 40 :

we impose the structure of a boolean algebra on the set $F$ of formulas of [sentential calculus] $SC$ [...] by first defining the relation $\equiv$ on $F$ by :

$$\phi \equiv \psi \ \text {iff} \ \vdash \phi \to \psi \ \text {and} \ \vdash \psi \to \phi.$$

If $\phi$ is a formula in $F$, we let $|\phi|$ the equivalence class under the relation $\equiv$ to which it belongs. Thus

$$|\phi| = \{ \psi \in F : \phi \equiv \psi \}.$$

We let

$$F/\equiv = \{ |\phi| : \phi \in F \}.$$

Now we can define the relation $\le$ on $F/\equiv$ by

$$|\phi| \le |\psi| \ \text {iff} \ \vdash \phi \to \psi.$$

Thus, we have that :

$\mathcal A = \langle F/\equiv \le \rangle$ is a complemented distriutive lattice, that is a Boolean algebra.