On an axiomatic definition of $P\Rightarrow Q$

Question

One thing that often confuses beginners in logic is that

$P\Rightarrow Q$ is TRUE when $P$ is FALSE, whatever Q is.

One (weak) reason is that we are mostly interested in the case where $P$ is TRUE, and with the above property, we have the nice formula $P\Rightarrow Q=(\textrm{non }P)\textrm{ or }Q$.

I am trying to justify the property by an axiomatic approach (to avoid to define $P\Rightarrow Q=(\textrm{non }P)\textrm{ or }Q$). For convenience, I will use $T$ for TRUE and $F$ for FALSE. Defining $P\Rightarrow Q$ means choosing a result in $\{T,F\}$ for all 4 possibilities of $P$ and $Q$ in $\{T,F\}$. So, I will use the functional notation $c(P,Q)$ to denote the resulf of $P\Rightarrow Q$. Clearly, if we want to express that $Q$ is the consequence of $P$, we should have the axioms:

1. $c(T,T)=T$: if $P$ is TRUE, then $Q$ is also automatically TRUE.
2. $c(T,F)=F$: $Q$ cannot be FALSE when $P$ is TRUE.

In addition, I would add

1. Contraposition Principle: $c(P,Q)=c(\textrm{non }Q,\textrm{non }P)$
2. Equivalence Axiom: $P\Leftrightarrow Q =(P\Rightarrow Q)\textrm{ and }(Q\Rightarrow P)$

where non, and and $\Leftrightarrow$ have their usual definition. Writing $a=c(F,T)$ and $b=c(F,F)$, I obtain the following truth table:

$$ \begin{array}{cccccc} P & Q & c(P,Q) & c(Q,P) & c(P,Q)\textrm{ and }c(Q,P) & P\Leftrightarrow Q \\ T & T & T & T & T & T \\ T & F & F & a & F & F \\ F & T & a & F & F & F \\ F & F & b & b & b & T \\ \end{array} $$

so the Equivalence Axiom forces $b=T$. The Contraposition Axiom would have the same consequence.

Question: My axioms say nothing about $a$. What could be a natural axiom to add to the list to obtain $a=T$?

Answer 1

Here's an axiom that sounds silly at first, but I think it'll grow on you:

• $\Rightarrow$ is not a symmetric relation.

That is, there exist $x$ and $y$ such that $x\Rightarrow y$ but not $y\Rightarrow x$.

Why is it desirable to define $\Rightarrow$ in such a way that it is not a symmetric relation? Simply because the graphical symbol "$\Rightarrow$" is not spatially symmetric; it has a pointy bit on the right side but not on the left. It would be perverse and misleading to use an asymmetric symbol to denote a symmetric relation. (If we want to denote a symmetric relation, we should choose a symmetric symbol like $\Leftrightarrow$!)

Answer 2

Alternatively, in a order theoretical approach, you can define a binary relation $\leq$ which is reflexive, antisymmetric and transitive, thus obtaining a poset $(2, \leq)$.

Note that $ v \vDash P \rightarrow Q$ precisely when $[[P]]_v \leq [[Q]]_v$.

Answer 3

"I am trying to justify the property by an axiomatic approach."

It is worth commenting that there is a somewhat different, and more natural, approach to justifying the truth-functional conditional.

Consider a natural deduction system for propositional logic. If this has the rules (a) Modus Ponens, (b) Conditional Proof, plus standard rules for other connectives and standard structural rules that allow you to chain proofs together, then it is easy to show that the resulting system has a truth-functional conditional obeying the familiar table. So that nicely shows the cost of rejecting the truth-functional account -- we either have to reject Modus Ponens or Conditional Proof (but then would we still be dealing with a conditional?) or we have to reject some background assumption that otherwise seems appealing.

This should be spelt out in any standard logic text that gives a natural deduction treatment of basic logic. Of course, you can trade in a natural deduction system for an equivalent axiomatic system: but the price is -- surprise, surprise -- a loss of intuitive appeal, "naturalness". And more than that, emphasis on axiomatic systems arguably betrays a mistaken view of what logic is about: it is fundamentally about inference, and patterns of good inference, rather than about running up a catalogue of logical truths. The natural deduction approach aims to capture good inference patterns -- and then we find that a plausible account of good inferences with the propositional connectives makes the (indicative) conditional come out truth-functional.