What is difference between truth-functional inconsistency and non-truth-functional inconsistency?

Question

The statement:

Jim is a bachelor and Jim (the same Jim) is married.

If we take,

P: Jim is a bachelor.

Q: Jim is married.

This statement is of the form P ∧ Q with truth-table:

$$\begin{array}{|c|c|c|} \hline p&q&p∧ q\\ \hline T&T&T\\ T&F&F\\ F&T&F\\ F&F&F\\\hline \end{array}$$

Well, truth-table clearly shows that it is truth-functional contingent, but at the same time it is impossible in a logical or conceptual sense. So, it is obviously inconsistency (and not contingent). But it is not truth-functional inconsistency, it is non-truth-functional inconsistency.

What is difference between them?

Answer 1

See page 62 of Schaum's Outline :

The statement ‘Jim is a bachelor and Jim (the same Jim) is married’, for example, has the propositional form ‘"$B \land M$", and hence is truth-functionally contingent, as this truth table reveals.

But this statement [it means : in natural language] is inconsistent, not contingent; what it asserts is impossible in a logical or conceptual sense. Its inconsistency, however, is non-truth-functional, being a consequence of the semantics of the expressions ‘is a bachelor’ and ‘is married’, in addition to the logical operator ‘and’.

As you noted, the truth-table for "and" applied to the statement $B \land M$ does not show a contradiction (nor a tautology) and thus the truth-functional symbolization of the natural language statement is contingent.

In order to show the inconsistency of the natural language statement, it is not enough the truth-functional symbolization available with propositional logic.

What we need is a "deeper" level of analysis that consider also

the semantics of the expressions ‘is a bachelor’ and ‘is married’, in addition to the logical operator ‘and’.

This will available wit predicate logic where we can analyze the "atomic" sentences with a subject-predicate logical form :

$\text {Bachelor}(\text {Jim})$ and $\text {Married}(\text {Jim})$.

With the tools of predicate logic, we may further express the "logical or conceptual" link implicit into the semantics of the expressions ‘is a bachelor’ and ‘is married’, that means to consider the "definitional axiom" :

$\text {Bachelor}(x) \text { iff } \lnot \text {Married}(x)$.

In this way, from $\text {Bachelor}(\text {Jim}) \land \text {Married}(\text {Jim})$ we may derive the non-truth-functional [i.e. not expressible in propositional logic] contradiction :

$\lnot \text {Married}(\text {Jim}) \land \text {Married}(\text {Jim})$.

See page 63 :

if a wff is truth-functionally contingent, then it is contingent only so far as the operators represented in the wff are concerned. Some specific statements of the form will be genuinely contingent, while others will be non-truth-functionally necessary or inconsistent as a result of factors not represented in the wff.