Systems Of Reasoning (S1E03) : Semantics, consistency and soundness.

I am going to make this blogpost a bit different. I am going to make it a bit “fun”, and less proof/theorem.

Any propositional variables can be assigned a truth ($T$) or falsehood ($F$) value through a mapping $f : \Phi \rightarrow \{T,F\}$. Where $\Phi$ is a set of all propositional variables.  We can show that it’s more general than this, i.e. $\Phi$ can be a set that contains all eff
—– HOW?—-
The values can be retrieved through $f_x$, where we have

1. $f_{x}p = f(p)$ for some propositional variable $p$
2. $f_{x} \sim B = \sim f_{x} \sim B$ for some wff $B$.
3. $f_{x} [B \vee C] = f_{x} B \vee f_{x} C$ for some wffs $B, C$.

Facts

1. For some wff $A$, we write $\vDash A$ and say that $A$ is a tautology if $f_{x} A = T$ for any assignment ($x$) I make to the internal statements. That is to say that $A$ evaluates to true probably because of its structure and not its content.
2. If a wff $A$ always evaluates to F, then we say $A$ is a contradiction.
3. $A$ implies $B$ iff $\vDash A \Rightarrow B$
4. $A$ is equivalent to $B$ iff $A \equiv B$ is a tautology.

Systems Of Reasoning (S1E02) : The Axiomatic Structure.

This is an episode in a series on mathematical logic approached with some rigour. Here, we will (still) be closely following the book by Peter B. Andrews: An Introduction To Mathematical Logic and Type Theory. We will look at the axiomatic structure of a logistic system we’ve been working on. The previous blogpost can be found here: (S1E01): The Rules.

An axiom is a statement that is taken to be true. The system we’ve been building from (S1E01) (called $\mathcal{P}$) consists of well-formed formulas having one of the following

(S1E02) (Sc0) axiomatic schemas:

1. $A \lor A \Rightarrow A$
2. $A \Rightarrow B \lor A$
3. $A \Rightarrow B \Rightarrow (C \lor A) \Rightarrow (B \lor C)$

We then also have just one primitive rule of inference:

1. Modus Ponens
For some wff $Q$, if it is accepted and is of the form $A \Rightarrow B$, then we can infer $B$, given $A$ holds. $A,\; and \;B$ are wffs.
Example: Let $Q =$ If it is an evergreen tree, then it doesn’t loose leaves in winter and its leaves are always green.

System Of Reasoning (S1E01): The Rules.

The Pilot.

This is an episode in a series on mathematical logic approached with some rigour. Here, we will be closely following the book by Peter B. Andrews: An Introduction To Mathematical Logic and Type Theory. In this episode, we will:

Part 1