# Conversion from Propositional Logic to Predicate Logic

I came across a proposition which I'm having a hard time converting into predicate logic. It has been a long while since I have touched the topic.

\begin{align} &\text{Socrates is a human}\\ &\text{Socrates is mortal}\\ &\text{Therefore, some humans are mortal} \end{align}

Using predicate logic, I'd set this up as

\begin{align} &P(s)\\ &M(s)\\ &\exists x\hspace{0.1cm} (P(x) \land M(x)) \end{align}

which can be written as,

$$P(s) \land M(s) \rightarrow \exists x\hspace{0.1cm} (P(x) \land M(x))$$

However, I feel like this is incorrect. Could someone suggest how to better represent this proposition using predicate logic?

• Please replace in the conclusion $H(x)$ with $M(x)$. – Mauro ALLEGRANZA Apr 10 at 11:31
• Wuth the typo removed, the translation is correct. – Mauro ALLEGRANZA Apr 10 at 11:32
• … With the provision that $x$ does not occur free within either premise. – Graham Kemp Apr 10 at 12:17
• Nitpicking issue ... You are not "converting from propositional logic..."; you are symbolizing (expressing) some propositions in predicate logic :-) – Mauro ALLEGRANZA Apr 10 at 12:39

I wouldn't try to put these three statements into one statement. A statement is different from an argument. To indicate you are dealing with an argument, you can use the $$\therefore$$ symbol. So, I would symbolize this argument as: \begin{align} &P(s)\\ &M(s)\\ &\therefore \exists x\hspace{0.1cm} (P(x) \land M(x)) \end{align}