The Laws of Logic

Logical Tautologies

Arthur Pap

Let's look at logical arguments which logicians have called tautologies.


Take


            If q, and (if p, then q), then p.


That is saying that if q were the case, and if p entailed or implied q, then p would have to be the case too. That is, if q then p (in short-hand). In order for q to be the case, p would need to be the case whether or not q is actually the case.


However, we can split the compound


          If q, and (if p, then q), then p.


into parts.


It may well be the case that this implication is correct (or, in its shorter version, that p implies q) even if p (the antecedent) is false. That is, a false proposition can still imply things. It still has implications. After all, the compound formula begins with the locution “If q” and also contains “if p”. In other words, it's an implicational conditional. It tells us what is implied by what even if both p and q are false.

That could also mean that if the antecedent, p, is false, then the entire implication (or conditional) must also be false. That is, if p is false, even though it implies q, then the whole conditional implication must be false.... but wait. In addition, the entire conditional above may be false, and p is false, yet the implied q may still be true!

Clearly it's useful here to distinguish truth from correctness. That is,

      If q, and (if p, then q), then p.

is a correct implicational conditional; though it's not a true one. It's false because p is false, and p implies q. Indeed the conditional is correct (the inference pattern is correct); thought false (one or more propositions are false) even if the consequent, q, is true (when taken on its own).

However, logicians also say that the conditional itself is true (as Arthur Pap does) and not simply correct. Here the notions of truth and correctness are fused. That means that that the overall conditional is deemed true, despite one of its propositions being false. It's the inference pattern that's true. Not necessarily one or all of its propositions.

Here's another tautology:

      If not-q and (if p, then q), then not-p

It's not the case that q is the case. Therefore q is false. However, p implies q. Yet q, as we've said, isn't the case. Thus if p implies something that's not the case (q), then p implies q. Therefore p mustn't be the case either. That is, if p were true, it would imply q. Yet it has already been stated that q is false or not-q. Thus not-q itself implies not-p (though only in conjunction with the parenthesized conditional (i.e., “if p, then q”).

Inferences Involving Classes

Instead of using p's and q's (propositions) in logic, logicians sometimes think in terms of classes instead, as with the Aristotelian syllogism. Take this example:

       i) For all classes AM, BS, and A:
      ii) if all AM are BS
     iii) and some A are AM
     iv) then some A are BS.

Or:

     i) If all African-Americans (AM) have black skin (BS)
    ii) and some Americans (A) are African-Americans (AM)
    iii) then some Americans (A) have black skin (BS).

Illicit Conversion

However, don't make the mistake of illicit conversion. That is, from

     All AM are BS

it doesn't follow that

    All BS are AM.

It doesn't follow that because

      All African-Americans (AM) have black skin (BS)

that

     All those who have black skin (BS) are African-Americans (AM).

Clearly, there are many people who have black skin who aren't also American.

Illicit conversations can happen when logical matters are discussed as well.

For example, it's often said that

     i) All tautologies are formal truths, hence logical truths.

However, that doesn't mean that

     ii) All logical truths are tautologies.

The inversion of “All tautologies are logical/formal truths” is “All logical/formal truths are tautologies” is illegitimate for the simple reason that not all logical/formal truths are tautologies. There is more to logic than tautologies (as against the position advanced in Wittgenstein’s Tractatus).

It other words, it must be said that some logical truths, or inferences (as with mathematics), supply us with additional information (though not facts). Indeed with additional truths.

Here's another example of illicit conversion. Take the statement
 

All logical truths are true by sole virtue of the meanings of their constituent terms (in particular, their logical constants).

It doesn't follow from that
 

All statements that are true just by virtue of the meanings of the terms (hence, that don't require empirical verification and that can't be falsified) are logical truths.

You can't invert “All logical truths are true by sole virtue of the meanings of their constituent terms” and get “All statements that are true by virtue of their meanings alone are also only logical truths”.

For example, the well-known example of


         All bachelors are unmarried.

This is indeed true because the interpretations of its constituent meanings make it true. Nonetheless, the words “bachelors” and “unmarried” are hardly logical in nature. And if they aren't logical in nature, then “All bachelors are unmarried” can't simply be true because of logic (or it can't solely be a logical truth).

Nonetheless, as Quine made very clear, “All bachelors are unmarried” can be made logical; or it can be translated into a logically true statement, simply by writing

       All unmarried men are unmarried.

You don't need to understand what “married”, and therefore “unmarried”, means. All you need to understand are the strictly logical terms: “all” and “not” (or “un”). (Though “are” is not strictly speaking a logical word... then again, in a certain sense, neither is “all”.)

And just as it was said that not all logical, or formal, truths are tautologies, it can also be said that not all necessary, or a priori, truths are strictly analytic. (Of course analytic truths have been seen as tautological too; though this this was largely discounted earlier on.)

Basically, the argument is that some truths are necessary, or can be known a priori, because of the way the world is; or, on some accounts, because of the way all (human) minds work.

Take this example from Arthur Pap:

       No event precedes itself.

This isn't true because of its “constituent meanings” and it's not a tautology (as such). However, it is about world; though it can also be said that one need not consult the world in order to know that it's true. Though surely if that's the case, then how can it be about the world at all? Perhaps the best that can be said that once one gets to know the world (as it were), and one gets to understand and use the words contained in the statement, then one knows that it's necessarily true and that it can be known a priori. That is, it requires previous experience of the world and of language. Yet once that experience and knowledge is in, you don't need to check to see that it's true – indeed that it's necessarily true.

Despite saying all that, the “meanings of the constituent words” can still be said to make it true. After all, once we understand the word “precede”, then it will become clear that an event cannot preceded itself or that x cannot precede x. However, meanings are still important (as they are to most logical truths). Nonetheless, as said earlier, the word “precede” isn't logical in nature.

Defining the Logical Constants

Wittgenstein once said that “some things can only be seen, not said”. This is true of the logical constants. When you define the logical constants, you do so by using other logical constants.

For example,

       All things have property P.

That can be defined as:

      Not-(some things don't have P).

Strictly speaking, “all things” is a use of the universal quantifier (which isn't a logical constant). Anyway, this universal quantifier too is defined by using the logical constant known as negation (or “not-” in this case).

Now take:

       p and q

This is a propositional conjunction which uses the logical constant “and”. It can be defined in this way:

      Not-(not-p or not-q).

To say that both p and q are true is also to say that it's not the case that both p and q are false. Or that “not-p or not-q” is false because both p and q are true. So here a compound proposition which uses the logical constant “and” is defined in terms of the logical constant of negation ( or “not-” in this case).

Finally we have:

      p or q

This is a compound proposition which uses the logical constant of disjunction (or “or” in this case). That in turn is defined in this way:

    If not-p, then q

Here again the constant of negation is used to define another constant. That is, if either p or q is true (though not both), and p is false (or “not-p” in this case), then q must be true because the compound proposition (“p or q”) says that either p or q is true. So if p is false, then q must be true.

Saul Kripke's 'Identity and Necessity'

Saul Kripke begins his paper by stating his general metaphysical position. That is, although it is a contingent a posteriori fact and a discovery of astronomy that Hesperus and Phosphorus are one and the same thing, their identity is still necessary. Our epistemological and scientific findings are irrelevant to Hesperus and Phosphorus's numerical identity. Perhaps the problem is that ‘Hesperus’ and ‘Phosphorus’ are not proper names at all; but substitutes for definite descriptions. That is, 'Hesperus' really means "the star seen at night…" So if 'Hesperus' were an abbreviated description and not a bona fide name, then what would be an example of a bona fide name? According to Bertrand Russell, only demonstratives like "this" and "that" are genuine names. They're genuine names because they're not dependent on descriptions. They are essentially contentless. More precisely, according to Russell, "this" and "that" are uninterpreted, indescribable and unconceptualised sense data; they are the objects of our own "immediate acquaintance".

But even ‘this’ and ‘that’ must rely on some kind of descriptive content; at least for the speaker. That is, even though he doesn’t have a name or even an explicit description, he must have still individuated the this or the that. Otherwise how will he know what he's in fact referring to? “Which this?” or “Which that?” This is certainly the case for the hearers. Though how does the speaker himself distinguish various this’s from various that’s? After all, in an act of ostensive definition one could be pointing at the brown on the table, or the cup on the table, or whatever. So ostension alone can't individuate a this from a that. And if it’s all a question of sense-data, how does the speaker know that the hearer will have the same kinds of sense-data? Even sense-data for the speaker can't in and of itself individuate a this or a that. Sense-data presupposes individuation; otherwise it wouldn’t be the data of something. Although, according to traditional sense-data theorists, we move from sense-data to the objects in the external world. Though without prior individuation, how would the sense-data theorist distinguish between relevant and irrelevant bits of sense data? Presumably when the theorist has sense-data of, say, a table, he'll also have sense data of, say, the things on the table, the colour of the table, and the objects in his general field of vision.

One can see why Kripke was concerned to argue that proper names have no descriptive content because the definite descriptions of Hesperus and Phosphorus didn't coincide. So it followed, to Kripke, that proper names mustn't rely on their descriptive content. Indeed they have no descriptive content at all; otherwise how could they be in fact identical. Similarly, how could we know that they are one and the same thing? Therefore proper names can't and mustn't rely or depend on any descriptive content.

Further into the paper Kripke again tackles the problem of the necessity of certain identity statements. He begins with the example of a pain and whether or not it's identical with a particular brain state. (This is tackled in greater detail later on in Kripke’s paper.) He then tackles the identity of heat with molecular motion and water with H
2O. Again, he admits that the identification of heat with molecular motion and water with H
2O are both contingent a posteriori discoveries. However, this has no effect on their necessary identities. Indeed there's another contingent fact about these necessary identities. Kripke says that of course we can "imagine heat without molecular motion" and a mental state "without any corresponding brain state". None of this affects the necessary identities. (Note: Kripke believes that the identity of heat with molecular motion is necessary; though he doesn't think the same about the identity of a particular mental state with a particular brain state.)

Heat is molecular motion; whereas, say, pain is a result or a product of molecular motion. There could of course be the feeling of heat without molecular motion. Though heat would still be molecular motion. The feeling of heat in our sensory receptors isn't actually heat. Therefore we could have an equivalent feeling from, say, light waves or sound waves (as Kripke also argues). 

The same is true of H
2O and water. There may be other examples of stuff that has the macro-qualities of water; though it would not thereby be water. In Kripke’s case, the macro-properties of water aren't the standard by which we determine or define water. That standard falls within the ambit of water’s micro-properties – that is, H
2O molecules. It is these micro-properties that make water a natural kind, not water’s macro—properties; which may, after all, be shared by other substances. Water is also H
2O whether or not we discover this to be the case. 

Why doesn’t all this apply to mental states and brain states? Because mental states are defined exclusively in terms of their phenomenal qualities; unlike water. That is, if we come across phenomenal qualities that don’t coincide with particular brain states, then such mental qualities aren't necessary identical to such brain states. There is, however, a contingent identity between mental states and brain states. There is no distinction between macro- and micro-properties when it comes to mental states.

After this Kripke discusses definite descriptions again. Take "the inventor of bifocals" (i.e., Benjamin Franklin). Kripke's argument is that someone else - other than Benjamin Franklin - might have been the inventor of bifocals. The reason for this, as we shall see, is that "the inventor of bifocals" is a definite description; whereas 'Benjamin Franklin' is a proper name. Kripke christens "the inventor of bifocals" a "non-rigid designator”; whereas 'Benjamin Franklin' is a ‘rigid designator’.

Rigid designators necessarily designate the things or persons they designate. They can't designate any other things or persons. They designate these things or persons in all possible worlds. The description "the inventor of bifocals", on the other hand, could designate someone other than Benjamin Franklin at another possible world. Indeed we 
needn't go to another possible world to meet an alternative inventor of bifocals. In this world someone else might have invented bifocals. So "the inventor of bifocals" is a non-rigid (or flaccid) designator. It's non-rigid because the description refers to different persons at different worlds and might have referred to a different person even in our own world.

So, again, Kripke says that we can't rely on “the inventor of bifocals” to refer or identify Benjamin Franklin. What we can rely on is the proper name ‘Benjamin Franklin’. For a start, someone may not know that Benjamin Franklin was the inventor of bifocals; even if they know who he is in some kind of other way. Indeed at another possible world someone else might have been the inventor of bifocals. So the proper name needs to apply to Benjamin Franklin at all possible worlds. This in turn implies that he must have some kind of essence that's unchanging at different possible worlds. The proper name refers to the essence of Franklin; whereas definite descriptions capture only contingent or accidental properties of him.

Because of their non-contentful status, proper names must refer to Benjamin Franklin at all possible worlds. If we relied on definite descriptions, we may pick out someone who's not in actual fact Benjamin Franklin. So, yet again, Kripke wants to guarantee a necessary relation between reference and referent.

What is an example of a rigid designator? Kripke offers the example of "the square root of 25" which designates the number 5. Why is this designator rigid? Because at every possible world "the square root of 25" would designate the same thing - viz, the number 5. There can be no other object of designation when the designator is rigid.

Although "the square root of 25" sounds a little like a description, in fact it's not. It's a rigid designator. It must always refer to the same entity – viz., the number 5. That designator couldn't refer to anything else because of the precision of the quasi-description that is in fact a name. The same is true of, say, “the number below 6”. That too must refer to the number 5. However, “Johns favourite number”, if it is 5, isn't a name because it could refer to other numbers in our world and at other possible worlds. We can of course be wrong about “the square root of 25”; though that would simply be a fact about us. 

Why is “the square root of 25” the same as, say, “Tony Blair”? Again, the former appears to be in some sense descriptive; though the name “Tony Blair” doesn’t seem to be descriptive, at least not at a prima facie level.

When Kripke talks about rigid designators, he doesn't mean to say that the referents of these things need exist at all possible worlds. They're not necessary beings. What is necessary, however, is that the rigid designator would refer to the same entity at all possible worlds; even if in fact it only actually exists in one possible world – say, ours. The referents of rigid designators needn't be like, say, universals. 

Kripke gives his own example of necessary existents: mathematical entities. If we get back to rigid designations of non-necessary beings, such as Benjamin Franklin, then the name “Benjamin Franklin’ must designate Benjamin Franklin “in any possible world where the object in question does exist, in any situation where the object would exist”. Of course, if Benjamin Franklin didn't exist, the name would have no designation. It would have no referent.

Kripke isn't saying, however, that Tony Blair must be called ‘Tony Blair’ at all possible worlds. He may have a different name at other possible worlds. However, our name ‘Tony Blair’ refers to their Tony Blair even if their Tony Blair isn't actually called “Tony Blair”. Our name doesn't refer to their names, it refers to the object, Tony Blair, at all these possible worlds - even in the ones in which Tony Blair is named, say, “Harry Buttock”. 

Kripke is in essence emphasising the importance of objects rather than names. More than that, he's emphasising the essences of objects which make it possible for Tony Blair to exist at different possible worlds - even in those at which he has a totally different name. He'll still be the same object. Moreover, he'll still have the same essence.

Now it will be interesting to see how Kripke departs from David Lewis in his view of possible worlds. According to Kripke, Lewis actually believes that, say, Nixon exists at many possible worlds. The man Nixon actually has some kind of existence at other possible worlds. Kripke, on the other hand, claims that there are only “counterparts” of Nixon at other possible worlds. These counterparts “resemble Nixon more or less, but none can really be said to be Nixon”. Indeed, at a prima facie level, it's hard to imagine what Lewis means by Nixon’s, as it were, multiple instantiation (or exemplification). On Lewis’s account, the man Nixon would be some kind of universal that's multiply instantiated at many possible worlds. Then we would have - wouldn’t we? - a particular that is also a universal at the same time. If that were the case, our Nixon wouldn't be the prototype: he'd instead be some kind of non-spatiotemporal Nixon universal outside of time. Kripke thinks that we can solve our modal problems by simply positing Nixon counterparts, rather than numerous trans-identical instantiations of the American ex-president.

According to Lewis, Nixon is multiply instantiated; though these Nixon duplicates don't also actually duplicate Nixon’s earthly life. The man is duplicated; though his actions etc. aren't. And these alternative actions and events actually occur at other possible worlds. According to Lewis, Nixon himself, not a counterpart, actually may not have suffered the Watergate scandal. This of course prompts the question: Why did Lewis insist on making Nixon multiply instantiated? Does this hinge on his realism about possible worlds?

As I said earlier, the names of numbers are seen as rigid designators. The name ‘9’ refers to the object 9 at all possible worlds. Now it's Kripke’s intention to make proper names rigid designators too. He wants the name ‘Nixon’ to refer to the same thing or person at all possible worlds. How does he argue his case?

Firstly he argues that it's easy to imagine Nixon doing things which he didn’t actually do. Perhaps we could imagine him having been awarded the Nobel Peace Prize for Literature. Perhaps his counterpart, not his instantiation, received this prize at another possible world. There's no problem with such possibilities. They aren't logically impossible. Can we imagine the man himself being different? Kripke argues that “we cannot say ‘Nixon might have been a different man from the man he in fact was’”. If Nixon had been a different man, we might ask: In what sense is he still Nixon? Could Nixon have changed all his properties and still remain Nixon? Clearly not. Could Nixon have had some of his properties changed and still remain Nixon? Possibly; though we would perhaps need to make a distinction between essential and contingent properties. We couldn’t change any of Nixon’s essential properties without changing Nixon into someone or something else. Or, more precisely, we would make Nixon cease to exist. Here we're faced with a Liebnitzian position in which all of a thing’s properties are essential to that thing. Or, tautologically, all the essential properties, however many, are essential to a thing or person.

Perhaps we can safely say that Nixon wouldn't have been Nixon had he a different brain. The brain of Nixon, therefore, was essential to Nixon. How does all this show us that ‘Nixon’ rigidly designates Nixon? If Kripke did indeed have certain or many essential properties, then a counterpart at another possible world that didn't have these essential properties couldn't be designated by the name ‘Nixon’. It would be designating something or someone else – a non-Nixon. If ‘Nixon’ designates anything, that thing must be Nixon and not a non-Nixon. On the other hand, the description "the thirty-seventh President" doesn't entail any essential properties; or, indeed, any properties at all except the property being the 37^th President of the United States. Anything or anyone might have fulfilled the role of being the thirty-seventh President of the United States. Can anything or anyone be Nixon the man? How can x be y if x and y are discernable objects? x would only be y if they were indiscernible objects – that is, if they shared all their properties (including relational ones).

Kripke offers us an analysis of the technical terms he'll be using in the remainder of the paper. 

Firstly he asks: "What do we mean by calling a statement necessary?" His answer is: Firstly, the statement is true. Secondly, "it could not have been otherwise". Contingent truth, on the other hand, is a matter of a statement being true; though it could have been the case that it isn't true. Kripke says that these are metaphysical issues. He then discusses a priori truth and says that such a thing "can be known independently of all experience". Because of the concern with our knowledge of these statements, they're assigned to the realm of epistemology. Questions of a priori truth are epistemological because they're concerned "with the way we can know certain things to be in fact true". As Kripke was well aware, traditionally it was thought that all necessarily true statements could be known a priori. Of course Kripke questions this assumption. In fact he offers his own alternative. Some things or statements may be necessarily true; though only knowable a posteriori (that is, our knowledge depends on experience). Kripke offers his own example: the Goldbach conjecture. This conjecture claims that every even number is the sum of two primes. Because this is a mathematical statement, it must be necessarily true (if it is true). However, the Goldbach conjecture isn't known a priori. Here Kripke qualifies the notion of the a priori. It's not simply a question of what is known independently of experience; but also what "can be known independently of experience". 

Another addition to the a priori argument, in relation to Goldbach's conjecture, is that part of its - possible - truth would be our ability to prove it true if it were true. Kripke denies this too. It's been known since Gödel, Kripke argues, that within certain mathematical systems there's at least one theorem that's not provable within that system. So there can be no absolute and total guarantor of truth within any mathematical system. This means, again, that not all mathematical truths are provable. Therefore they certainly aren't known to be true a priori. (Gödel’s stance on mathematical systems may be applicable to systems of various descriptions outside of pure mathematics.)

Essentialism

Kripke now goes into a different subject (though it ties in with everything else): the notion of essential properties. What are essential properties? According to Kripke, they're those properties that "are such that [an] object has to have them if it exists at all". If a particular object didn't have these properties, it wouldn't even exist as that object. There's another way of expressing this. If a particular object didn't have these essential properties, "it would not be this object". Kripke gives a possible example of an essential property of a lectern. He says that it must be made of wood, rather than ice. Wood is an essential property of the lectern.

What does it mean to claim that this lectern has wood, rather than ice, as an essential property? The essentialist claims, according to Kripke, "that this lectern could not have been made of ice". This claim is further glossed by saying that "in any counterfactual situation…we would have to say also that it was not made from water". If this lectern were made of water, it wouldn't be this lectern.

Now Kripke reverts back to arguments about the a priori. That is, to the argument that necessity doesn't depend on being known a priori. It terms on the lectern "we cannot know a priori whether this table was made of ice or not". However, and this is Kripke's primary point, "given that it is not made of ice, it is necessarily not made of ice". Kripke expresses this in symbolic logic:

P ⊃ P
P--------------P

This means: If the lectern isn't made of ice (P), then it's necessarily () the case that the lectern isn't made of ice (P). The lectern isn't made of ice (P). Therefore the lectern is necessarily not made of ice (P).

The important point to be extracted from the above is that the conclusion  P is known a posteriori "since one of the premises on which it is based [i.e., P] is a posteriori".

Kripke then gets back onto the notion of rigid designators. He offers us a more detailed characterisation of them. First of all he talks about two rigid designators, ‘a’ and ‘b’. Both rigid designators designate the same thing, viz, x. He says that “in every possible world, a and b will both refer to this same object x, and to no other”. So if both ‘a’ and ‘b’ designate the same object, then “there will be no situation in which a might not have been b”. He goes on to say that that “would have to be a situation in which the object which we are also now calling ‘x’ would not have been identical with itself”. This would be a necessary identity between two names that designate the same object, namely x. 

To get back to the example that opened the paper, Kripke says that “one could not possibly have a situation in which…Hesperus would not have been Phosphorus”. So if both names rigidly designate the same object, say, Venus, then both names are necessarily identical.

Kripke pre-empts the possible critiques of his position. He says that some “people tend to regard identity statements as metalinguistic statement”. To put this very simply. Metalinguistic statements are statements about sentences and names rather than things and events. So instead of

Hesperus is Phosphorus.

We have

“'Hesperus’ and ‘Phosphorus’ are names of the same heavenly body.”

Kripke isn't talking about the necessary identity of names, . He's talking about the necessary identity of things. Of course the identity of names may have been false. We may have called Phosphorus ‘Juniper’ and Hesperus ‘Klink’. Though if they both rigidly designated the planet Venus, then their designations would be necessarily identical. However, the names themselves aren’t necessarily identical. How could they be? Names, signs and inscriptions are of course arbitrary and contingent. 

Kripke gives an excellent example of what’s at issue here. Take the statement “2+2=4”. If we're talking names exclusively, this statement wouldn't be necessarily true, or perhaps even true at all. If we're talking about the accepted designations of these inscriptions, then the statement is necessarily true. Kripke elaborates. He says, “’2’ and ‘4’ might have been used to refer to two different numbers” (to the ones they do now refer). If the inscription ‘2’ referred to the mathematical object 3, then the statement “2+2=4” would be necessarily false. In this instance, “2+2=4” should be “2+2=6” because, again, the inscription ‘2’ refers to the object 3.

The opponent of Kripke is still not entirely convinced. He says: “’Look, Hesperus might not have been Phosphorus’”. Kripke’s adversary then goes onto say that if “things had turned out otherwise, they would have been two different planets…so how can you say that such a statement is necessary?””.

Kripke dissects this opposition. He says that there are two things that the adversary could mean. The adversary can mean that he can't know a priori that Hesperus is Phosphorus. Of course we can’t; and Kripke has already conceded that. Alternatively, the adversary could mean that there could be circumstances - or perhaps possible worlds - in which Hesperus wouldn't have been Phosphorus. This is all a question of mixing up names and the objects of names. How could we say that two numerically identical things - namely Hesperus and Phosphorus - might not have been the same? Again, the adversary’s problem is easily explained. He says that at another possible world the people of that world may well have named Venus ‘Phosphorus’; though, contrarily, used ‘Hesperus’ as a name for Mars. In that case, “Phosphorus is Hesperus" wouldn't have been a necessary identity statement. Again, we're not talking about names: we're talking about things. Whatever names we give Venus, if they rigidly designate the same object, Venus, they'll form necessary identity statements. Hesperus and Phosphorus are numerically identical, no matter what names we use. We're talking about their designations, not the names of those designations.

Names and Descriptions

For Kripke's enterprise it's important that names are clearly distinguished from descriptions. In certain parts of the tradition, they were thought to be closely related. As Kripke says, it was thought that "we fix the reference of the term 'Cicero' by use of some descriptive phrase, such as 'the author of these works'". After the reference has been fixed, according to Kripke, it was the case that 'Cicero' rigidly designated the man who wrote these works. Though if the descriptive phrase were important or necessary for the fixing of the name, then if someone else wrote the works mentioned earlier, then he would be Cicero. According to Kripke, we "do not use [the name] to designate whoever would have written the works in place of Cicero". In fact it seems, at a prima facie level, ridiculous to think that someone else could have wrote the works of Cicero. However, what Kripke wants to argue is that the name alone fixes the reference, not a description.

Traditionally, the name and a description used to fix the reference were taken to be synonyms. Kripke argues that we can't depend on the description. If we did, at another possible world someone else could have written the works written by Cicero. Therefore the name 'Cicero' would apply to that person if name and description were synonymous.

Kripke then offers his wider critique of identifying names and descriptions. He says that “suppose that we do fix the reference of a name by a description”. What would be the consequence of this for a theory of reference? He argues that name and description would still not be synonymous. The name would still rigidly refer to the object in question “even in talking about counterfactual situations where the thing named would not satisfy the description in question”. Indeed Kripke goes further than this. He says that the “reference of names is rarely or almost never fixed by means of description”. Is this because the relation between description and referent is contingent; whereas the relation between name and referent is necessary?

Kripke gets to works on examples. 

Take ‘heat’ and ‘the motion of molecules’. Both terms could be seen to refer to the same thing. That heat is the motion of molecules is a scientific fact. It is an a posteriori judgment. The motion of molecules isn't “contained in the concept” - as Kant would have put it - of [heat]. As Kripke put its, “scientific investigation might have turned out otherwise”. However, the discovery was indeed contingent or a posteriori; though the connection between heat and the motion of molecules is necessary. (Note: not between the names ‘heat’ and the desctiption ‘the motion of molecules’). Regardless of our knowledge, our words, etc., there's a necessary connection or identity between heat and the motion of molecules.

Kripke, thankfully, offers us many possible arguments against his general thesis. For example, what if an increase in the motion of molecules didn’t cause sensations of heat in our sensory receptors; but, instead, the slowing down of molecules did? In that case, so the adversary argues, heat wouldn't be identical to the motion of molecules.

Another argument against Kripke thesis would be this. What if there were no people on this planet? If there were no people on this planet, then there would be no sensations of heat. Would we, in that case, say that heat didn't exist in this counterfactual world? Kripke would argue that heat still exists regardless of human sensory receptors. Why does he think this? Because if there were fires on this uninhabited planet, these fires would still heat up the air. Kripke’s general conclusion is that heat isn't necessarily identified by the feelings of certain sensations (those of heat). Indeed more strongly, heat has nothing to do with sensations, strictly speaking. (This could be seen as the opposite position to Berkeley’s idealism.)

Kripke offers another counter-argument against his general thesis. In yet another counterfactual situation, the creatures on our planet don’t get the sensation of heat when they're exposed to things that cause us to feel heat. In this counterfactual situation, Kripke imagines creatures that get visual sensations when they're exposed to sound waves.  Kripke here is also changing the example. Instead of heat being connected with sensations of heat, we now have sound waves being connected to visual sensations. Should we now say that sound waves would be light (visual sensations)? No. Light would still be necessarily identical to streams of photons; just as heat is necessarily identical to the motion of molecules. In both cases, the existence of counterfactual creatures and human sensations would be irrelevant to the necessary identifications.

Kripke backs up his position by saying that the terms ‘heat’ and ‘the motion of molecules’ are both rigid designators of the same thing. Both ‘heat’ and ‘the motion of molecules’ refer to the same thing or process at all possible worlds. In accordance with previous explanations, sensations can't be identical to particular things or processes at all possible worlds - as we've seen. To use Kripke exact argument, because “heat is in fact the motion of molecules, and the designators” ‘heat’ and ‘the motion of molecules’ are both rigid, then “it is going to be necessary that heat is the motion of molecules”.

Where are Kripke’s adversaries going wrong? In a sense, the answer to this is quite simple. We've identified something that is contingent to be part of the definition of heat. That contingent fact is that on this planet human beings happen to be sensitive to the motion of molecules. When we experience the motion of molecules we feel heat. So we identify heat, and therefore the motion of molecules, with our experiences of heat (say, of things being hot). Heat, and therefore the motion of molecules, “causes such and such sensations”. We identify heat with these sensations; whereas the only real necessary identity is between heat and the motion of molecules.

To clarify his point, he goes back to the Cicero example. Here too we identify a contingent property of Cicero with Cicero: that of writing such and such works. Kripke concludes that ‘Cicero’ and ‘heat’ must be used as rigid designators. ‘Cicero’ always stands for the thing Cicero; and not for any of his descriptive properties. ‘Heat’ always designates the motion of molecules; and not other contingent properties (e.g., heating up our hands to cause the sensation of heat).

Kripke’s adversary wrongly thought that heat could be identified with something that isn't the – increased – motion of molecules. His identification of heat with the sensation of heat is a contingent, not a necessary, identification.

To get this point across Kripke identifies a non-necessary - that is, a contingent - identification: the identity of a pain with a particular brain state. The Identity Theorists, contrary to Kripke, claim that there's a necessary connection between a particular pain and a particular brain state; or, more generally, between pains and brain states. What the identity Theorist commits himself to, according to Kripke, is that if we have pain state X, then we must be in brain state Y. Similarly, if we are in brain state Y, we must experience pain state X. Kripke I think would accept that there may well be a connection – not an identity – between pain state X and brain state Y; though that connection can't be necessary. This simply means that I can experience pain state X and not be in brain state Y. It's logically possible to feel pain state X and not have the corresponding brain state Y. The conclusion of this is that the Identity Theory must be wrong. Pain state X isn't identical to brain state Y. The Identity Theorist may say, according to Kripke at least, that the identity between pain state Y and brain state X is “contingent”. Kripke argues that the Identity Theorist can't claim that the relation is contingent. He says that the Identity Theorist must believe that “we are under some illusion in thinking that we can imagine that there could have been pains without brain states”.

Reference

Saul Kripke, ‘Identity and Necessity’, from Identity and Individuation (1971)

Do Logic and Existence Go Together?

It's said by some (or most) logicians that “logic must handle every possible state of affairs” and hence that it “can't imply the existence of anything” (Dale Jacquette). That almost sounds like a non sequitur. Yes, logic must handle “every possible state of affairs”. Nonetheless, how does it follow from this statement that logic can't imply the existence of anything? Why can't logic be able to handle every state of affairs and imply the existence of something (or one thing)?

Is it because if logic is applicable to everything, then implying the existence of something would pollute its ability to handle all states of affairs (note the jungle of quantifiers here)? Or is is it that the case that something (or these things) would somehow make logic contingent (or empirical) in nature? Nonetheless, implying (or allowing) the existence of something that's contingent (or empirical) isn't the same as arguing that logic itself is contingent (or empirical). Logic can still be applied to the the contingent (or empirical) even if isn't itself contingent (or empirical).

Does it mean, instead, that if logic implies the existence of anything (or even something) that it would somehow depend on that something? And, if logic did imply the existence of anything, then its logical purity would somehow be sullied?

In that sense, quantificational logic (or first-order logic) is far from being pure. Quantifiers in logic have existential import (or have ontological commitment). That is, a quantificational proposition is about the existence (or non-existence) of something (or of many things). Even free logic accepts abstract objects of various kinds. It can also be said that logical statements about self-identity have existential import. That is, the proposition (x) (x = x) has existential implications. And, more obviously, so too does, (∃y) (y = y).

It seems to follow from the acceptance of quantificational logic that an empty universe is excluded – nay, it's logically impossible. However, do these facts about quantificational logic necessarily apply to the more generic “logic” we began discussing? Perhaps quantificational logic is actually a deviant logic!

Lewis Carroll's Premises Paradox

The British writer, mathematician and logician Charles Lutwidge Dodgson (which was Lewis Carroll’s real name) worked in the fields of geometry, matrix algebra, mathematical logic and linear algebra. Dodgson was also an influential logician. (He introduced the Method of Trees; which was the earliest use of a truth tree.) For some time Dodgson was Mathematical Lecturer at Christ Church, Oxford University.

And, of course, Dodgson (under the name Lewis Carroll) also wrote Alice in Wonderland, Alice Through the Looking Glass and many other books and poems.

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Lewis Carroll (I’ll use Dodgson’s better-known pen name from now on) formulated his premises paradox in 1895. (The argument was advanced in Carroll’s paper ‘What the Tortoise Said to Achilles’; which was published by the journal Mind.) This paradox refers to the possibility of infinite premises being required in order to reach a single conclusion.

Firstly, two or more premises are usually linked to a conclusion in a logical argument.

So how are they linked?

Now that can be a question of the philosophical nature of the link between premises and conclusion: whether it’s an example of entailment (or logical consequence), implication (or material implication) or whatnot. However, that wasn’t the point that Lewis Carroll was making. He was making a purely logical (not a philosophy-of-logic) point.

To put this simply: in order to justify (or explain) how any given premises are related to a conclusion (or how the premises entail or imply the conclusion), then a further premise will need to be brought into the argument in order to do so. Moreover, another premise will be required in order to tell us (or show us) how (or why) it’s the case that if the premises are true, then the conclusion must follow and also be true.

Now that added explanation (or justification) will itself be a premise within the argument.

Of course the same problem will repeat itself.

Just as we brought in a premise to link two, three or more premises to a conclusion: now we have to say how this new premise is itself linked to those previous premises. Or, alternatively, we’ll need to know how all the premises (when taken together) are linked to the conclusion.

A solution has been offered to this logical paradox.

That solution is simply to say that no added premises are needed in the first place. That is, the link between the premises and the conclusion doesn’t need to be explained (or justified) by a further premise.

So why is that?

The (or one) answer is to say that the premises and conclusion are simply linked by a rule of inference. That rule itself explains (or shows) the relation between the premises and the conclusion. Nonetheless, that sounds (at least at a prima facie level) like a cop-out. To simply say that premises are linked to a conclusion by a rule of inference doesn’t appear to be saying anything… much. Surely that rule -again! — will need to be explained by a further premise.

It can also be argued that saying that premises are linked to a conclusion by an inference rule is itself a further premise.

Yet that’s unless that rule of inference somehow works without any justification or explanation.

In other words, it’s not a justification: it’s just a rule. It’s something that “can be shown but not said” (to use Wittgenstein’s much-quoted phrase); just as the nature of the logical constants can’t be said, only shown.

Does the Mind-as-Computer-Programme Idea Support Mind-Body Dualism?

“Yes, I am a confirmed "dualist" at this point in my journey, i.e., I understand my mind to be separate from my body/brain, analogous to the way an application program and its data are separate from the computing infrastructure they depend on.” - Chasw

Does believing your mind to be separate from your body/brain automatically mean you're a dualist? That would depend on what the word 'separate' means. The mind simply being (fundamentally) different from the body wouldn't commit anyone to dualism.

Dualism is a commitment to the idea that mind and brain are fundamentally different ontological categories or 'substances'. That creates the problem of mind-body interaction. If mind and brain (body) were fundamentally different, then how could brain-to-mind and mind-to-brain interactions be explained?

Dualists are also committed to minds being separable from brains, not just being separate and different. I don't think the 

mind = programme/data 

and the 

brain = hardware

comparison works here because, after all, even programmes and data are physical. Programmes and data depend on physical syntactic devices which encode data/info and which themselves rely on physical processes to implement that data (as well as to send that data, electronically, around the hardware/infrastructure). The programme-hardware/mind-brain analogy works only to show us that the same programme can be instantiated in different kinds of hardware: from brains to computers to coke machines. In itself, it's not an argument for dualism; which is specifically about the mind being of a different substance/category than the brain. When it comes to computer programmes and their different hardware juxtapositions, there are no such deep ontological problems as there are for mind-brain/body dualism.