Jeff has posted about adding a validity predicate to arithmetic here. I have been thinking about this, and have a different twist. Assume that we add a logical validity predicate Val(x, y) to arithmetic (in what follows, F is the Godel code of formula F). Val(F, G) holds iff the argument from F to G is logically valid. Now, one rule that a logical validity predicate ought to satisfy is:
VS2: (Val(F, G) & F) entails G.
The second rule that a logical validity predicate ought to satisfy is:
VS1: If F entails G, then Val(F, G)
The trick here is that we need to decide, when applying VS1, which sense of 'entails' we have in mind. Should we conclude that Val(F, G) holds if:
- G is derivable from F in first-order logic?
- G is derivable from F in first-order logic supplemented with VS1 and VS2?
- G is derivable from F plus arithmetic?
- G is derivable from F plus the T-schemas?
There are extant arguments that show that options 3 and 4 are inconstent (Beall & Murzi [unpublished] and Shapiro  show this for 3, and Whittle  shows this for 4). But, of course, is is rather implausible that either arithmetic or the T-schemas are logically valid (of course, their arguments do show that other interesting notions of validity are inconsistent). To remind you, the derivation of a contradiction for case 3 goes like this. Diagonalization provides a sentence P such that:
P is arithmetically equivalent to Val(P, #)
where "#" is some arbitrary contradiction. We then reason as follows:
- P Assumption.
- Val(P, #) 1, Diagonalization.
- # 1, 2, VS2.
- Val(P, #) 1 – 3, VS1.
- P 4, Diagonalization.
- # 4, 5, VS2.
As Jeff notes in his earlier contribution, 1 is consistent (in fact, we don't need to add a new predicate at all, since the relevant notion of validity is definable in PA). Thus, if 'entails' means derivable in first-order logic, then we can consistently add the rules above to arithmetic.
As a result, the intuitive rules for logical validity (unlike, strikingly, the intuitive rules for the truth predicate) are truth-preserving and consistent. But are they themselves logically valid? In other words, can we add versions of VS1 and VS2 to arithmetic where 'entails' means derivable from first-order logic plus VS1 and VS2? Interestingly, the answer is "no", as the following derivation demonstrates. Let Q be the conjunction of the axioms of Robinson arithmetic, and let jn(x, y) the recursive function mapping the Godel codes of two formulas onto the code of their conjunction. Diagonalization provides a sentence P such that:
P is arithmetically equivalent to Val(jn(P, Q), #)
We now reason as follows:
- P & Q Assumption.
- Q 1, logic.
- P 1, logic.
- Val(jn(P, Q), #) 2, 3, logic.
- Val(P&Q, #) 2, 4, logic.
- # 1, 5, VS2.
- Val(P&Q, #) 1 – 6, VS1.
- Q Assumption
- Val(jn(P, Q), #) 7, 8, VS2.
- P 8, 9, logic.
- P & Q 8, 10, logic.
- # 9, 11, VS2.
A few observations about the proof:
- We only apply arithmetic (diagonalization in the move from 3 to 4 and from 9 to 10, recursive arithmetic in the move from 4 to 5 and from 7 to nine, since these depend on the arithmetic fact that jn(P, Q) = P&Q) within the scope of an assumption of Q. Thus, the system in which the proof occurs does not assume that arithmetic is valid (or even true).
- The proof does not show that this version of the rules VS1 and VS2 are inconsistent. Instead, it shows that these rules allow one to prove that arithmetic is inconsistent (in this manner, the result is very different from the standard proof of the inconsistency of the T-rules).
Anyway, this is kind of cool. Just as the Liar paradox (or, if you want to be fancy, Tarski's theorem) shows that the T-sentences governing the truth predicate can't be true, this shows that the rules for the logical validity predicate can be true, but can't be logically valid.
Beall, J. & J. Murzi [manuscript], “Two Flavors of Curry Paradox”, online at:
Shapiro, Lionel, , “Deflating Logical Consequence”, Philosophical Quarterly 60(*).
Whittle, B. , Dialetheism, Logical Consequence, and Heirarchy, Analysis 64(4): 318 – 326.
[Edited for readability - rtc]