Grades of Truth-functionality

Truth-functionality is a notion at the heart of classic propositional logic. A standard explication of the notion runs roughly as follows:

An n-ary connective C is truth-functional iff it corresponds to a truth-function.

More precisely, C is truth-functional iff there is a mapping from the truth-values of any n sentences S₁ … S_n to the truth-value of the concatenation of C with S₁ … S_n.

According to that explication, the following operators are equally truth-functional:

T It is true that …

TF It is either true or false that …

EX It is expressible in English that …

The two latter operators are only special in corresponding to a constant truth-function, namely the function that maps both True and False unto True.

But one may sense an important difference between TF and EX: the former connective actually operates on the truth-values of the embedded sentence such that the truth-value of the complex sentence is a result of that operation. But the latter connective does not operate on the truth-value of the embedded sentence at all. That it corresponds to a constant truth-function is not the result of its being sensitive to the truth-value of the embedded sentence; in some sense, it does not depend on that truth-value.

A definition of truth-functionality that captures the described difference between TF and EX can be given in terms of the explanatory connective ‘because of’:

An n-ary connective C is truth-functional iff the concatenation of C with any sentences S₁ … S_n has the truth-value it has because of the truth-values of S₁ … S_n.

This definition will classify the standard connectices of propositional logic as truth-functional (a true negation, for instance, is true because the negated sentence is false, and a false negation is false because the negated sentence is true). And it also classifies TF as truth-functional (’It is either true or false that snow is white’ is true because ‘Snow is white’ is true; had the latter been false, then the former had been true because of that.)

But EX does not count as truth-functional in the defined sense (’It is expressible in English that snow is white’ is true, but not because ‘Snow is white’ is true).

While I do not think that these considerations show that the standard definition of truth-functionality is in any way flawed (it is a technical notion after all), the alternative definition captures differences between operators that might as well be associated with the term ‘truth-functional’.

(See the papers-section for a more detailed exposition of my proposal.)

Posted by Benjamin.

Logical Symbols in HTML

In case you want to include some logical symbols in your posts and / or comments, you can use the converter for funny characters. Simply click on the appropriate symbol and you will get the corresponding html code. (Found via Theorem(e).)

Posted by Miguel.

Really-Really-On-Topic Theories of Truth

A T-equivalence for a sentence x is an instance of ‘S is true iff p’ in which ‘S’ is replaced with a quotational designator of x and ‘p’ with a sentence synonymous with x. A truth theory for L is a theory that has, for every sentence of L, a T-equivalence as a theorem. A theory of meaning for L is a finite theory knowledge of which could suffice for knowing the meaning of every sentence of L.

Theories of truth are not theories of meaning (this is widely acknowledged, though Davidson seems to go back and forth). Proof: Let L be a toy language comprising only the sentence ‘Snow is white’. Let T be a theory comprising the axiom ‘“Snow is white” is true iff snow is white’. T is a truth theory for L. But knowing that ‘Snow is white’ is true iff snow is white cannot suffice for knowing that ‘Snow is white’ means that snow is white, since a sentence can be true iff snow is white without meaning that snow is white. ‘Blood is red’ is such a sentence. In general, the information that a sentence S is true iff p is not sufficient to infer that S means that p, since every sentence that has the same truth value as ‘p’ will be true iff p, regardless of whether it means that p. Call this the Very Simple Coextensionality Problem (VSCP). A theory suffers from VSCP if the knowledge that it provides about a sentence S could be had about any sentence that is coextensional with S. No theory suffering from VSCP can be a theory of meaning.

Proper truth theories will not have axioms for whole sentences, but axioms for sub-sentential expressions. Such theories do not suffer from VSCP. Since they provide information about what, say, the predicate of an atomic sentence S is true of, they provide knowledge about S that one cannot have about every sentence that is coextensional with S. But these theories

Knights and Knaves

Most of you will know the knights and knaves of Raymond Smullyan’s logic puzzles, the notorious truth-tellers and liars (actually, ‘liar’ is not wholly adequate, but anyway). In a true classic among these puzzles, you have to determine which one of two paths to choose (one leads where you want to be, one leads to your doom). Recently, a new solution to the puzzle has been published. You can find it here.

Posted by Benjamin.

Listen to Philosophy

As most of you probably know, this years John Locke Lecture by Bob Stalnaker has been made available as mp3 on the Oxford philosophy page here. I found this to be a great thing, and I really hope it will set a new trend. It not only gives you the chance to listen to one of todays pre-eminent philosophers while, say, cleaning the kitchen; you can actually pause him or make him repeat those bits that you didn’t get straight away. Wouldn’t it be great to be able to rewind and replay some of the talks you have heard at conferences over the years? Well, the good ones, anyway…

So here is some more philosophical stuff that can be listened to on the World Wide Web. Of course, there is Philosophy Talk, a weekly show run by John Perry and Ken Taylor with well over 100 episodes by now, all available from their archives. And there’s Philosophy-Bites, where Nigel Warburton and David Edmonds present ‘pod casts of top philosophers interviewed on bite-sized topics’. Their latest episode, ‘aired’ on October 8, features Anthony Kenny on his New History of Philosophy, and past episodes include Tim Williamson on vagueness, Simon Blackburn on moral relativism, and Tim Crane on mind and body, to name but three. But there’s more. On Learnoutloud, you can listen to the whole of Plato’s Republic (or download it, well over half a gigabyte). They also showcase a 7 minute piece of Wittgenstein’s Tractatus, and a whole lecture series on Heidegger’s Being and Time (personal favourite: The One, part II).

So to all those setting up conferences with great philosophers: record and post!

Posted by Miguel.

Minimalism

Paul Horwich’s minimal theory of truth consists of all propositions of the form:

<p> is true iff p.

As was pointed out by Anil Gupta (Philosophical Perspectives 1993), Horwich’s minimal theory does not logically imply certain general facts about truth. Horwich (Truth, 2nd ed.) acknowledged Gupta’s point, but he thought that those general facts nevertheless are non-logically implied by the theory. What is involved, he thinks, is the following non-logical but truth-preserving rule:

If S is a set of premises all which attribute a certain property P to a proposition, such that every proposition is attributed that property by one of the premises, then S non-logically entails that every proposition has P.

Panu Raatikainen (Analysis 2005) assumed that what Horwich had in mind is the omega-rule, but he correctly points out that the omega-rule would not be of help to Horwich, because it is only applicable to a denumerable infinity of premises. But there are far more propositions than natural numbers.

Panu is right about the usefulness of the omega-rule. But doesn’t his point provide clear evidence that the rule Horwich alludes to just isn’t the omega-rule? Instead, it is the rule stated above; that rule requires for its application more premises than there are natural numbers (and as many premises as there are propositions). So, it is not the omega-rule, and the rest of Panu’s criticism somehow misses its target.

To see whether the generalization-problem can be solved by Horwich, one should forget about the omega-rule and instead focus directly on the rule above. The crucial questions are: Is the rule truth-preserving? And can it explain how finite minds arrive at the required generalizations starting from Horwich’s theory?

Posted by Benjamin.

The King is dead. Long live the King!

As many of you will already know (but some may not), The Journal of Philosophical Logic is no longer supported by the Association of Symbolic Logic. After problems with Springer, the complete editorial board decided to leave the journal behind and to found a successor, The Review of Symbolic Logic. Springer will try to keep JPL alive (if only for financial reasons), but the fate of the journal is unclear.

So go and spread the news! (For more information, go here.)

Posted by Benjamin.

Credits

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