r/logic

Yo so this logic shit seems cool to me and I wanna learn it (preferably free).What would be the best way?Know it might seem like this post is satire but I genuinely want to learn.

I'm just starting to study logic and am having trouble understanding this. For context, it's in my math textbook. After this topic, the book moves on to sets and then functions

It is well known that every logic, i.e. a structural Tarskian consequence relation, is characterizable in terms of some family of matrices, namely its Lindenbaum matrices. Three years ago a user on Maths Stack Exchange asked whether there are logics that cannot be characterized by a single matrix rather than a family thereof. That is a natural question. Maybe by some artful combination of the matrices in some characteristic family?

Since nobody seems to have answered their question, I thought I’d do that here. The answer is that there indeed exist logics not characterizable by a single matrix. I shall give an example and show that that is the case for it.

First, recall that if a logic ⊢ is characterized by a single matrix, it is then uniform, meaning that whenever

(1) Γ U Δ ⊢ α,

(2) Δ is consistent w.r.t. ⊢ (meaning it doesn’t ⊢-imply all formulae), and

(3) Var(Δ) ∩ Var(Γ U {α}) = ∅,

then

(4) Γ ⊢ α.

Exercise for the reader: show this! Hint: suppose ⊢ has a characteristic matrix, and suppose (4) is false. Can you construct a Δ for which (1)-(3) are not all true?

Now let B be the 2-valued Boolean algebra on {T, F} with, say, infimum (conjunction) and complement (negation). Let B(T) be the matrix consisting of B plus {T} as its designated set, i.e. the natural matrix for classical logic, and B(F) mutatis mutandis with designated {F}. Let ⊢ be the logic characterized by the pair of these matrices.

(Notice that since the logic characterized by B(F) alone is, so to speak, the logic of preservation of falsehood, at least to the extent that classical logic is the logic of preservation of truth, then ⊢, as their intersection, is the logic of preservation of any uniform value the premises may have. It is the logic of equivalence!)

Observe that we have

(5) p & ~p ⊢ q & ~q

in this logic. But, {p & ~p} is actually consistent w.r.t. ⊢, since any valuation onto B(F) assigns it designated. Moreover, premise and conclusion have no variables in common. Hence, if ⊢ were uniform, we should have

(6) ⊢ q & ~q,

putting ∅ = Γ. But, we do not, since any valuation onto the classical matrix refutes (6) of course. Hence, by the above result, ⊢ has no characteristic matrix.