In recent years, philosophers have debated the question of logical pluralism: the view that there is more than one correct logic (see my interview with Greg Restall on this very issue). The idea, roughly, is that which putative logical laws hold depends on what sorts of “cases” we take logic to be about; different kinds of cases yield different (but equally legitimate) logics. A common logical monist objection is to say that a form of argument is only a logical law if it applies in all cases. If this is true, it raises the question: what argument forms do hold in all cases? At this point in the debate, a third position becomes viable, defined by the answer: none.
Gillian Russell, a philosopher of language and logic, argues both that applying in all cases is necessary for qualifying as a logical law; and that no argument form applies in all cases. As such, she believes there are no logical laws. Much of our discussion surrounds her claim that no argument form applies to all cases. Is this really true even of the law of non-contradiction, the “law” that says that ‘A and not-A’ can never be true? Of conjunction elimination (‘A and B’ entails ‘A’)? Of identity (‘A’ entails ‘A’)? Russell runs through purported counterexamples to these laws; what’s more, she illustrates a method for conjuring counterexamples to any proposed “law”.
“This sentence is false”. Is that sentence true or false? If it’s true, then what it says must hold; but what it says is that it’s false, so it must be false. But if it’s false, then what it says must not hold; but what it says is that it’s false, so it must not be false. But if it’s not false, it must be true. So if the sentence is true, it is false, and if it is false, it is true. The sentence, therefore, seems to be both true and false, which seems absurd.
Philosopher and logician Stephen Read is one of the preeminent scholars on this “liar paradox”. He is known, in large part, for rediscovering and defending a long forgotten solution to the paradox first proposed by the medieval philosopher Thomas Bradwardine. In this first half of our conversation, Read covers the paradox’s rich and influential history. It was first discovered, in its full form, in the 4th century BCE by Eubulides (who also first set down the sorites paradox). It became a central problem in the 20th century via its association with Russell’s Paradox, a major problem in the foundations of mathematics. Later in the century, two thinkers – Alfred Tarski and Saul Kripke – proposed monumentally influential theories of language and truth motivated, largely, by the paradox. But even after their contributions, the consensus is that the paradox remains unsolved. …
Some philosophers, including C.S. Peirce, have argued that the Liar is demonstrably false and not true. The argument is based on the premise that all statements implicitly assert their own truth.
At first glance, this seems plausible. If I tell someone, “I did your laundry,” it carries with it an implied “it is true that I did your laundry.” This would seem to hold for all assertions. So, Peirce argues, the Liar is really saying: “It is true that this sentence is false,” which essentially comes down to saying “this sentence is true and false.” This is no longer a paradox, but a plain contradiction, and so false. It is like saying, “I’m a cat owner that doesn’t own a cat.” That’s not a mystery, just a lie. What makes the Liar a paradox is that what it says is, on the surface of it, coherent. If it just asserts a flat-out contradiction, then it poses no problem.
That a proposition automatically asserts its own truth is an interesting notion, and it is not easy to say whether it is accurate or not. Peirce later in life argued that it was incorrect. Luckily, it is not necessary to determine whether it is accurate or not because, even if it is, it does not resolve the Liar.
It is not true that, if a proposition automatically asserts its own truth, then the Liar really says: “It is true that this sentence is false.” In that sentence, “this sentence” refers to that whole sentence.…