On Philosophy

July 14, 2007

Properties Of Sentences

Filed under: Logic — Peter @ 12:02 am

Truth and falsity can both be understood as properties that sentences can have. And they are not the only properties that seem to apply to sentences, the property of being unknown truth value and of being linguistically underdetermined are also things it seems appropriate to say of sentences. But the problem with these properties is that if we can say definitely that the sentence is true or false then certainly it is not of unknown truth value, or linguistically underdetermined. On the other hand certainly being of unknown truth value doesn’t imply that the sentence is neither true nor false. Thus describing the relationships between these properties is not quite possible within the confines of standard logic.

To overcome this problem we can introduce a new binary relation, ⇒. Intuitively ⇒ means something like “can be derived from” or “can be proved from” or “can be known on the basis of”. In terms of proving claims it works pretty much the same way as standard implication (→), because we can assume that (a⇒b)→(a→b). What differentiates it from standard implication then is that it says nothing about the truth of the claims it holds for, a⇒b should be read as “if a can be known/proved then b can be known/proved from a”.

Let me give an example of how this makes a difference. The claim ∀x(Sx→(Tx≡~Fx)) is an attempt to say that if a sentence is true than it is not false, and that if a sentence isn’t false then it is true (Sx means x is a sentence, Tx means x is a true sentence, and Fx means x is a false sentance). But might be problematic, because then we might be able to proceed from a failure to be able to know that a sentence is false to the conclusion that it must be true (whether this is actually possible depends on what additional rules of inference we permit). And it certainly prohibits the possibility of truth gaps or sentences that are both true and false. Now I don’t want to take a stance on whether the previous sentence is one we should accept as an axiom, but it isn’t the only possibility. We could, for example, adopt ∀x(Sx→(Tx⇔~Fx)) instead. From this we can conclude that if we know a sentence to be true or to be false then we can know it not to have the opposite truth value as well. However, it says nothing about sentences of which we don’t know the truth value of. Thus it permits truth value gaps and sentences that are both true and false, so long as we are not in a position to determine the truth of such sentences (which is exactly what we would expect of such sentences).

And so the introduction of ⇒ allows us to claim that Tx &rArr ~Ux and Fx &rArr ~Ux (where Ux means x is of unknown truth value) without simultaneously claiming that no sentence us of unknown truth value, assuming we accept ∀x(Sx→(Tx⇔~Fx)) as an axiom.

Now if we are going to try to stick as closely as possible to first order logic obviously then the predicates will apply to objects. And logically objects are simple. But obviously we will want to appeal to the structure of the sentence at times. What we need then is to establish a one to one correspondence between sentences, as strings of symbols, and the mathematical objects to which our sentence predicates apply to. The following is sufficient: ∃!x(Tx⇔(Eval(“φ”,{})=#t)). It claims that for every sentence φ there is a single object, x, such that the truth of x (Tx) allows us to conclude that the sentence is true and that being able to conclude that the sentence is true allows us to conclude that Tx holds. Of course I haven’t explained what the Eval function is, but for the moment it can be understood simply as a way of determining the truth status of the sentence. Since it would be a pain to write that sentence out every time we wanted to refer to a particular sentence object we will let sφ denote the unique object that fulfills the above condition.

Now with ⇒ and that bit of notational convenience out of the way I can describe the connections between the various predicates that can hold of sentences. Here Tx means that the sentence x is true, Fx means that the sentence x is false, Ux means that the sentence x is of unknown truth value, and Lx means that the truth of x is linguistically underdetermined. Let me first describe how the predicates apply to logically complex sentences, by cases (this doesn’t cover all cases, just the obvious ones):
(Usφ∧Tsψ)∨ (Tsφ∧Usψ)∨ (Usφ∧Usψ)⇒Usφ∧ψ
(Lsφ∧Tsψ)∨ (Tsφ∧Lsψ)∨ (Lsφ∧Lsψ)⇒Lsφ∧ψ
(Usφ∧Fsψ)∨ (Fsφ∧Usψ)∨ (Usφ∧Usψ)⇒Usφ∨ψ
(Lsφ∧Fsψ)∨ (Fsφ∧Lsψ)∨ (Lsφ∧Lsψ)⇒Lsφ∨ψ
And between the predicates

And finally let me say a few words about Eval. Now normally logicians follow Tarski and make the right side of the truth biconditional, Tx≡…, a decomposition of the sentence into the truth of its fundamental components. But this method cannot handle self reference or loopy reference, nor does it easily accommodate predicates such as “unknown”, which in a sense compete with truth when evaluating the sentence, despite the fact that they do not exclude the possibility that the sentence really has a particular truth value. Fully detailing Eval is a complicated proposition, but it is easy to give a high level description of how it works. Eval then is a function which maps a sentence object and a set to one of four special values, #t, #f, #u, or #l, which correspond intuitively to the four predicates discussed here holding of that sentence. Eval(s,q) then performs that mapping as follows: If s∈q then Eval maps the pair to #l. If s corresponds to a complex sentence then Eval is used to determine what its simpler components map to, with the exception that the second parameter of this use of Eval is q∪{s}, and then those results plus the above inference rules are used to determine what Eval should yield for s. For example if s corresponds to ~φ then Eval(sφ,q∪{s}) is evaluated, and if it yields #t then the function returns #f, and so on. And, finally, if s is primitive then we assume that there is some, possibly non-logical, procedure for determining the truth of such an s, and that the results of such considerations are incorporated into Eval so that it gives the appropriate results (as with the Tarksian method).

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