# an argument that is clearly valid but invalid in a sentence logic

I was reading these paper(dont really remember the title) it stated that there are simple arguments that are clearly valid but would be counted as invalid in the sentence logic system it was using. i don't understand how this can be.. can anyone give me an example that satisfies the statement made. this is an example i saw from the paper.. "every class is easy is valid in predicate logic but not in sentence logic because you can deduce from it that philosophy is easy." i still don't understand i.e can't form an example of my own.

• We can not give you an example of something invalid in the logical system from a paper unless you tell us what that paper is. One can easily come up with a system where some "clearly valid" rules are simply not included. Just take the usual classical logic and take out the law of excluded middle. That gives intuitionistic logic where some classically valid arguments are invalid. – Conifold Dec 4 '20 at 18:19
• @Conifold yes you're right. the paper was based on klenk's logic(which is the only logic system I've ever known.) i've edited the question to add an example i saw from the paper. – Assfaw kidane Dec 4 '20 at 18:26
• "Every class is easy is valid in predicate logic but not in sentence logic because you can deduce from it that philosophy is easy." Is that really a direct quote? This makes absolutely no sense to me. "Every class is easy" is not an argument, it's a statement, and I don't see why a predicate logic take on it (as "$\forall x(C(x)\rightarrow E(x))$") would be any more or less valid than a propositional take on it (as "$C\rightarrow E$"). Without more context I have no idea what the author is getting at. – Noah Schweber Dec 4 '20 at 18:51
• @NoahSchweber Yes it's a direct quote and I see what you're saying which was why I got confused. – Assfaw kidane Dec 4 '20 at 19:01

## 1 Answer

The example you give (at least in the way you formulated it) doesn't make sense to me, because "Every class is easy" is obviously not valid in PL and neither does it allow for the deduction that "Philosophy is easy" without further axioms.

What I imagine could be meant is that sentential logic is not powerful enough to formalize the linguistic details that are needed to derive certain validities.

Take the standard textbook example

All humans are mortal.
Sokrates is a human.
∴ Sokrates is mortal.

Intuitively, this should be a valid argument, and in predicate logic, it can be formalized as

∀x(Human(x) -> Mortal(x))
Human(sokrates)
∴ Mortal(sokrates)

and proven proven to be valid using the rules of universal instantiation + modus ponens.

But in sentential logic, where there are no predicates and quantifiers, the most fine-grained formalization we can get (since there are no sentential connectives involved in any of the sentences) is

p
q
∴ r

which is obviously invalid because p ↦ True, q ↦ True, r ↦ False is a countermodel.

• this is actually a very clear explanation and example. Thank you very much. – Assfaw kidane Dec 4 '20 at 19:20
• To be fair, one can quibble with the claim about propositionalization. We can think of this as a propositional relationship between first-order formulas "is a man," "is a mortal," and "is Socrates." In the first-order context this would yield the predicate-logically-valid {∀𝑥(𝑀𝑎𝑛(𝑥)→𝑀𝑜𝑟𝑡𝑎𝑙(𝑥)),∀𝑥(𝑆𝑜𝑐𝑟𝑎𝑡𝑒𝑠(𝑥)→𝑀𝑎𝑛(𝑥))}⊢∀𝑥(𝑆𝑜𝑐𝑟𝑎𝑡𝑒𝑠(𝑥)→𝑀𝑜𝑟𝑡𝑎𝑙(𝑥)) which we could then propositionalize as {𝑀𝑎𝑛→𝑀𝑜𝑟𝑡𝑎𝑙,𝑆𝑜𝑐𝑟𝑎𝑡𝑒𝑠→𝑀𝑎𝑛}⊢𝑆𝑜𝑐𝑟𝑎𝑡𝑒𝑠→𝑀𝑜𝑟𝑡𝑎𝑙 which is propositional-logically valid. Of course this is a hack, but worth mention. – Noah Schweber Dec 4 '20 at 19:49
• Sentinel logic provides a better symbolization for Socrates mortality than presented. H: Socrates is Human M: Socrates is mortal. (P1). H -> M (P2) H (C) M This is elementary and is in many intro to logic books. – Rob Dec 5 '20 at 6:52
• @Rob Can you please point out which specific textbooks make that claim, so I know which books to never recommended? The example is wrong on a very elementary level. Your P1 translates "If Sokrates is human, then Sokrates is mortal", but this premise doesn't exist in the argument I gave, whereas the premise "All humans are mortal" doesn't occur at all in your translation, so it's not a correct formalization of the argument and hence useless in proving the validity. – lemontree Dec 5 '20 at 10:52
• The first premise of the argument is "All humans are mortal", not "If Sokrates is human, then Sokrates is mortal". These are obviously not the same sentences and you can not simply silently exchange the first sentence for the second one. The second sentence has to be deduced using rules of inference from the first, which sentential logic can not systematically do -- that's the whole point of the example. – lemontree Dec 5 '20 at 10:59