# Socrates Is Mortal

Previously I have talked about the need for a framework in which to discuss axiomatic systems, such framework is often called **logic**, and several such methods were developed and iterated over the years, culminating at what we now know as modern formal logic. This is a vast subject to which it is hard to do justice in just a blog post, nevertheless it is a subject worth understanding, it is a subject that once practiced helps a person think more systematically, and therefore I will try my best.

## Terms and Properties

A **term** is an object in the **domain of discourse**, like: *rainy days*, *a chicken*, or *Socrates*. Terms have **properties** which are characteristics attributed to them, such as: *blue*, or *has two legs*, or *mortal*. Formally, We can say that a chicken has two legs by denoting $\text{bipedal}(\text{chicken})$, or that Socrates is mortal by denoting $\text{mortal}(\text{Socrates})$, the result is a **proposition** a statement that has a truth value, i.e. it is either true or false, and never both.

## Propositional Calculus

Given two propositions $p$ and $q$, we can use **binary operations** to create new propositions, such as: $p \land q$ (read: p and q), $p \lor q$ (read: p or q), and $p \rightarrow q$ (read: p implies q), and we can use the **unary operator**: $\lnot p$ (read: not p). In the following **truth table** we use $T$ to denote truth, and $F$ to denote false.

$p$ | $q$ | $\lnot p$ | $p \lor q$ | $p \land q$ | $p \rightarrow q$ |
---|---|---|---|---|---|

$F$ | $F$ | $T$ | $F$ | $F$ | $T$ |

$F$ | $T$ | $T$ | $T$ | $F$ | $T$ |

$T$ | $F$ | $F$ | $T$ | $F$ | $F$ |

$T$ | $T$ | $F$ | $T$ | $T$ | $T$ |

## First-Order Logic

Instead of terms, we may use **variables**, replacing specific terms with a latin letter like $x$, instead of saying that Socrates is human, we can say that $x$ is human, in the same way: $\text{human}(x)$, and also that if $x$ is human, then it is mortal by denoting: $\text{human}(x) \rightarrow \text{mortal}(x)$. This allows us to add the **univeral quantifier** $\forall$ to express that all men are mortal: $\forall x (\text{human}(x) \rightarrow \text{mortal}(x))$ (read: for all x holds that: $x$ is human implies $x$ is mortal). in a similar way we can use the **existensial quantifier** to denote that there is an ice cream in the fridge: $\exists x(\text{isIceCream}(x) \land \text{inTheFridge}(x) )$ (read: exists $x$ such that: $x$ is an ice cream and $x$ is in the fridge).

## Rules of Inference

Given rules of inference we can deduce new propositions. Rules of inference are divided into premise propositions and a conclusion proposition. By accepting a rule of inference one accepts that if the premise propositions are true, then the conclusion must also be true. The most well known rule of inference is **modus ponens**, given two propositions: *p implies q* and *p*, we can deduce *q*. We denote:

$ \underline{p \quad \quad \quad} $

$q$

And **modus tollens** rule of inference goes like this:

$ \underline{\lnot q \quad \quad \quad} $

$\lnot p$

## Further Reading

- Aristotle’s Logic on Stanford Encyclopedia of Philosophy
- List of rules of inference on Wikipedia