Wolfgang Schwarz :: Are logical truths true?

Are logical truths true?

Posted on Saturday, 17 Sep 2005.

Argument 1:

Hesperus is identical to Phosophorus.
By modal logic, $$m[1]$ .
Therefore, Hesperus is necessarily identical to Phosphorus.

Argument 2:

Sometimes, one is obliged to do things that are not allowed.
By deontic logic, $$m[1]$ .
Therefore, sometimes one is obliged to things that are both allowed and not allowed.

Argument 3:

Necessarily, if the moon essentially consists of green cheese, then it actually consists of green cheese.
By provability logic, $$m[1]$ .
Therefore, the moon essentially consists of green cheese.

Argument 4:

It is now 20 seconds past 19:00 hours.
It is now 30 seconds past 19:00 hours.
If it is now 30 seconds past 19:00 hours, it is not now 20 seconds past 19:00 hours.
By propositional logic, $$m[1]$ .
Therefore, the moon is made of green cheese.

What's wrong with these arguments? They are invalid: their premises are true, their conclusion false. In each case, the fallacy is to assume that a principle valid in some formal system is also valid when translated into English.

$$m[1]$ is no doubt valid in classical propositional logic: it is true on all interpretations of the propositional constants. (Interpretations, remember, are assignments of truth values, not translations into English.) But "if (p and not p) then q" is not valid in schematic English: replacing p and q by English sentences does not always result in a true sentence, as witness argument 4.

Consider the sentence $$m[1]$ . Is that a logical truth? To be a logical truth, it should at least be true. So start with the easier question: is $$m[1]$ true? This question makes no sense. Strings of symbols are only true under a given interpretation. Meaningful questions are: Is $$m[1]$ true if it means that whenever some number x equals some number y then it follows from the Peano axioms that x equals y? Is $$m[1]$ true if it means that whenever some thing x is identical to some thing y then it is metaphysically necessarily that x is identical to y? Is $$m[1]$ true if it means that the moon is made of green cheese?

In philosophical arguments -- for the epistemic theory of vagueness, for the rigidity of names, against Counterpart Theory, against Meinongianism -- it is often assumed that sentences true or valid in some formal system are still true or valid under a certain translation into English. In my view, these arguments are almost always useless. Yes, $$m[1]$ is a theorem of every normal quantified modal logic. So is $$m[1]$ . Does it follow that all things necessarily exist? Of course not. It is not a theorem of quantified modal logic that all things necessarily exist. No English sentence is a theorem of quantified modal logic. And as the 'necessity of existence' proves, the obvious translation from (normal, non-free) quantified modal logic into English does not preserve truth. In other words, the translation is not a proper translation at all. To cite a theorem of quantified modal logic in support of some philosophical claim about modality is not much better than citing some truth in French, 'translating' it into an English sentence that means something different and assuming that thereby the English sentence is justified.

If we assume or intuit or have already shown that a certain English principle -- say, that whenever one is obliged to do something, one is allowed to do it -- is true, we can ligitimately use that principle as a premise in philosophical arguments. But the premise doesn't get justified by being true in another, formal language under some translation or other, unless it is known that the translation preserve truth.

Comments

# M. on 17 September 2005, 12:00

Doesn't that undermine some naive, undergraduate conception of logic: That by the use of logic as a tool box somebody can show that a given argument stated in a natural language is valid because some "formal" reconstruction in the language of, say, the propositional calculus of it can be shown to be valid?

# ludwig on 17 September 2005, 15:44

It's good practice to formulate arguments using one language only, but if you decide to interpret the box and "obligation" in the same way, I can't see any problem with your argument II.
Furthermore, try to replace "p" in your argument IV with a NL sentence - where's the contradiction?

# wo on 17 September 2005, 23:11

I'm afraid I don't get your comments, ludwig. The problem with argument II is that the conclusion is inconsistent; so if premise 1 is true, the alleged principle of deontic logic is false. In argument IV, "p" is "it is now 20 seconds past 19:00 hours".

M., yes, I think the conception of logic taught in most intro textbooks is a bit naive. Still, in elementary logic the picture you describe seems OK: if we translate true English sentences that are neither indexical nor ambiguous or vague into propositional or predicate logic then the reverse translation of whatever can be deduced from these formal sentences is also true in English, I believe.

# Robbie on 18 September 2005, 13:03

The situation in (4) seems to me rather different from the other cases. In the context of a language that's (broadly) indexical, there's going to be type/token issues *in addition to any problems about translation*.

If you treat validity Kaplan-style (i.e. an argument-type is valid iff truth-preserving in any single context under any interpretation), then the only immediate way of securing truth-preservation for a token of a a valid argument-type is if all premisses and conclusion are assessed relative to a single context. Intuitively, that hardly ever happens: token arguments take time, for a start.

This seems to me a nice puzzle about the significance of validity (a property of argument-types) to the good-standing of token inferences; but it's nothing to do with relation between formal and natural language.

# M. on 18 September 2005, 22:58

@wo. OK, I thought you called into question a basic tenet of "philosophical logic". As to the "non-standard cases": Isn't this simply a task for the logician/semanticist to develop theories that formally preserves truth where there is real truth preserving in normal reasoning (or so it seems :)) There are in fact so many semantic aspects of NL other then indexicality, vagueness and ambiguity that we do not have any theory about (think about quantification & what , for instance, is ususally dumpend into "that is pragmatics" et c.) that logical semantics has really a LOT to do here.
BTW: That "It is now noon" is no longe true a few hours later puzzeld Hegel in PhG, it shouldn't puzzle us today - or should it?

# wo on 19 September 2005, 20:41

Hi Robbie! I agree that argument 4 is different from the others. Maybe I should have used a sorites argument instead. The view that bothers me is however present in all four cases: it's the assumption that logical theorems remain logical truths if one replaces the logical operators by corresponding English operators and inserts syntactically appropriate English expressions for the non-logical terms.

M., I largely agree, though I don't regard formal languages as uncovering the logical form of natural language, so there is no question of matching *real* truth preservation. Still, if we want to use, say, the language of quantified modal logic to think and argue about ethics or modality, some choices are obviously worse than others, and it is a good task to invent a formal system/language that at least doesn't translate obvious falsehoods into logical truths. Sadly, most modal logicians seem to care more about the ease of completeness proofs, which is why the most common quantified modal logics have no use at all for philosophy.

# lumpy pea coat on 11 October 2005, 05:41

Argument 1: There's nothing wrong here--i.e. the conclusion is not false if the premises are true. If x and y really do denote one and the same object, how could the consequent be (metaphysically) contingent?

"To be a logical truth, it should at least be true"--what sort of notion of 'truth' are you invoking? Logically, it is true. If we translate it into English in an agreeable way (as the necessity of the identity of individuals) then it *seems* at least intuitively true.

Your second theorem, that everything actual necessarily exists is only a theorem of some modal logics using an appropriate semantics--a questionable semantics as far as quantified modal logic is concerned.

Argument 3: This argument as you've given doesn't follow. Premise (1), the conditional, is true; so is premise (2) (but the necessity operator does not capture de re modality (i.e. essential attribution) because the formula is propositional); but the conclusion doesn't follow since we do not have the antecedent of (1)--hence we cannot conclude by modus ponens that (3).

Argument 4: Premise (3) is simply false unless you add the qualification '*exactly* 30 seconds past...'. Being 30 seconds past 19:00 implies being 20 seconds past 19:00. If we add the qualification then we do not get (4) (or (5)).

You're right that classical logic is explosive but that's fine since it is consistent--i.e. never do we have (p & ~p) being true. Hence we cannot conclude that anything whatever is true. If we accept that some contradictions are true then presumably we will not accept the correctness of classical logic. We might accept a paraconsistent (e.g. 4-valued) logic.

Also, how does the necessity of identities imply that the moon is made of cheese?!

# logicnazi on 12 October 2005, 08:35

First of all I want to draw attention to the fact that these sort of errors don't appear to happen in the real world. No one invokes invokes arguments like 1 to prove two things are necessarily equal. Yet many people believe statements of logic are true? What gives? Are we to believe they are unconciouslly recognizing the fallacy of their position and avoiding these sorts of arguments yet not realizing this conciously?

While I do think this post is a clever puzzle I don't think it makes any argument against the idea that logic must be truth preserving. Rather it trades on a linguistic confusion about how we use the word logic. We call deontic logic a logic merely because it is intended or claimed to be truth preserving not because we think it actually is.

In other words we use the word logic in two different ways. In one way to describe a purely formal deductive system that some people might believe is truth preserving. In a different way we use it to mean systems that are really truth preserving. All your argument really shows is that despite being termed logic in the first sense the logics you mention (at least with their accompanied translation into english since in the second sense a logic is a formal system coupled with an interpratation) simply aren't a logic in the second sense.

Or were your really trying to argue against the person saying deontic logic is termed a logic hence it must be truth preserving?

# mh on 14 October 2005, 09:43

[BTW: lumpy pea coat inserted the *exactly* in the wrong spot; it is needed in the consequent, not the antecedent of 3 in argument 4, since being 30 seconds past (exactly or not) does entail being 20 second past, but does *not* entail being exactly 20 seconds past - quite the contrary]

Some comments on your argument 4:

For the premisses to be all true, one of the following has to be the case: (i) either you give both premises 1 and 2 the *exactly* reading; then it seems the reference of "now" has to shift, or (ii) the reference of "now" is constant, then you cannot have the *exactly* reading in 1 and 2. But you need the *exactly* reading in the consequent of 3, otherwise this premise wont be true.

So what does your argument 4 show? Sure; in instantiating a valid schema of propositional logic not just *any* instantiation will do. This has been pointed out by Strawson nearly 50 years ago: merely *orthographical* identity is not a sufficient constraint. Otherwise we could even show that the schema "p. therefore: p." is not valid in English. (Indexicality and ambiguity are not the only sources here; take proper names that can be used to refer to different people.) The constraint rather has to be that all orthographically identical items have the same semantical value. The only way for the premisses in 4 to be jointly true is by violating this constraint.

Another aside: the title of your essay is somewhat misleading. It seems to be a conceptual truth (if anything is) that whatever is logically true is true. But *schemas* are neither true nor logically true, though there instances might be.

ps: "logicnazi" - now theres a name I dont like.

# wo on 14 October 2005, 10:47

yes, there should be some "exactly"s in the last argument.

I obviously wasn't clear enough about what I wanted to say. I have actually encountered argument 1 and argument 2, and several arguments of the same kind in philosophical discussions. Arguments 3 and 4 are of course silly and were meant to illustrate what's wrong with arguments 1 and 2: We can't take it for granted that mechanically translating a theorem of some formal system into English will result in a true (let alone logically true) English principle.

# trackback from logicandlanguage.net: Philosophers' Carnival XX on 10 October 2005, 07:10

Roll up! Roll up! It's the 20th Philosophers' Carnival... Welcome, one and all, to the Philosophers' Carnival - a fortnightly compilation of all that is good and true, or at least moderately amusing, in the philosophical blogosphere. This week...

Wolfgang Schwarz

Are logical truths true?

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