# Validity of Propositional Logic

I understand the whole process of validating propositional logic. However, for this following question, am not too sure of the compound propositions of each line

I am assuming

No fish are forgetful (~p)

Nobody but fish have scales (q)

--

Nobody forgetful has scales (~p => q)

is that right?

• Those sentences don't sound like something one would use propositional logic to model ... – hmakholm left over Monica Nov 8 '14 at 20:33
• @HenningMakholm it is in the exercise for PL, unfortunately so there must be some way this can be interpreted that I'm not getting – LucasCK Nov 8 '14 at 20:34
• It is not propositional logic; it is an example of Syllogism (in modern term : monadic predicate logic) : "No F are P", "All S are F"; therefore : "No P are S". – Mauro ALLEGRANZA Nov 8 '14 at 20:47

The only way I can see to shoehorn this into propositional logic would be to work with an implicit subject and let the propositional variables be:

• $S$ meaning "it has scales"
• $P$ meaning "it's forgetful"
• $M$ meaning "it's a fish"

in which case the reasoning could be written as

$$\frac{\neg(M \land P) \qquad \qquad S\to M}{\neg(S\land P)}$$

which corresponds to the classical "Cellarent" sylogism, if we rephrase it slightly as:

• No fish is forgetful
• All things with scales are fish
• Therefore: Nothing that has scales is forgetful

In modern mathematical logic, however, it would be much more natural to express it in predicate logic, where the reasoning would look like

$$\frac{ \neg\exists x(M(x)\land P(x)) \qquad \qquad \neg\exists x(\neg M(x)\land S(x))} {\neg\exists x(P(x)\land S(x))}$$

This appears to require predicate logic to validate.

Can you provide some more background on where this question comes from? For example, if this is from a textbook, what section is it associated with? What are some other problems from that section? What are some sample solutions they provide, if any?

• this was a question from my lecture notes on validity of propositional logic although it does start to look like it shouldn't belong there. No sample solutions. – LucasCK Nov 8 '14 at 21:00