Reply To: The Liar's Paradox


One of the charms (is that the word I want?) of logic is that it leads to topics such as the one under discussion here. This topic, however, properly belongs either to metalogic or the philosophy of logic (it’s not always easy to distinguish one of these from the other). Metalogic, as the name suggests, takes formal logical systems and makes them the object of formal study. Philosophy of Logic (as in Susan Haack’s book, Philosophy of Logics – note the plural) also takes the various logics as objects of study but the questions it raises are of a philosophical rather than a formal nature. And then, arising from the perceived limitations of modern logic (such as the Paradoxes of Material Implication) we have the development of Modal Logic which then ramifies into a whole system of so-called deviant logics – epistemic logic, deontic logic, 3-valued logic, multi-valued logic and the seemingly paradoxical fuzzy logic!

To return to the topic under discussion: a number of approaches have been taken to the so-called Liar Paradox. One approach is to outlaw self-referentiality so that propositions are not allowed to refer to themselves. This (probably) works but it has a disturbingly ad hoc air about it which may be intellectually unsatisfying. Another approach, to which I am drawn, is to hold that all propositions make an implicit claim to truth so that the Liar sentence simultaneously asserts and denies the same thing and so is a contradiction and so is false. But there are other attempts at a solution and no general agreement is to be found on how this is to be handled. These attempts at a solution and some others are discussed in this Wikipedia article

When I was a student I found these kinds of questions practically irresistible, much to the detriment of my other studies and my social life. While I would encourage people to think hard about such topics, it probably isn’t a good idea to do so at the same time that one is attempting to learn the nuts and bolts of a primary formal system. Furthermore, to take part in current discussions you really need to familiarise yourself with modern Proposition and Predicate Logic, some Modal Logic, some metalogic and some philosophy of logic. If you want a short DIY introduction to elementary modern logic to get you started, send me an email ( and I’ll get my material to you.

Finally, to see how logic can allow one to focus on aspects of philosophical discussions that might otherwise remain out of sight, consider the following snappy 6-line proof for the existence of God (I think I stole this from Charles Hartshorne but I’m not 100% certain):

Let G stand for the proposition ‘God exists’; let § stand for the phrase (modal operator) ‘it is necessary that….;’ and let * stand for the phrase (modal operator) ‘it is possible that…’

Line 1: *G (It is possible that God exists)
Line 2: If G then §G (if God exists, then God necessarily exists)
Line 3: If *G then *§G (If it is possible that God exists, then it is possible that God necessarily exists) We arrive at this by applying the * operator to each side of line 2
Line 4: *§G (it is possible that God necessarily exists) We get this by putting lines 1 and 3 together, using the modal version of the rule called Modus Ponens
Line 5: §G (God necessarily exists) We get this from line 4 by using a modal logic rule that allows us to dispense with all but the last modal operator in front of any given proposition.
Line 6: G (God exists) This comes from the most basic rule in modal logic; whatever is necessarily true is true

The premises of this argument are plausible; it doesn’t seem unreasonable to think that it is possible that God exists. Similarly, if God isn’t to be thought of as a rock or a tree, it seems reasonable to think that if He exists at all, He exists necessarily. Grant these two premises and the rules of modal logic (up to a certain level) and the conclusion (God exists) pops out!

I like to tease my atheist friends with this bit of modal prestidigitation. The serious side of the argument is that it forces one to look again at whether the premises are, in fact, obviously true (perhaps it’s not even possible that God exists or perhaps God might exist but not necessarily!). On the more purely logical side, it leads to a consideration of the plausibility of the various systems of modal logic, in particular, the rule used in line 5 that allows you to reduce a proposition prefaced by a mixed string of modal operators to just the last operator. Here, as elsewhere on the borderlands of logic, our intuitions tend to run into the sand.

There is much more that could by said on this and related topics but not by me tonight!