English has a special meta-programme for encoding compound messages. The following sentences are examples of its handiwork. Study them.

    Note the pattern. These sentences are compound sentences. They have not one verb phrase as fulcrum, but two quite separate ones. Each compound sentence comprises an independent sentence (as the grammarians call it), here depicted in bold type, and (let us call it) a subsidiary string in italics. The subsidiary string is in each case formed by prefixing the dependent sentence with a subordinating conjunction. Commit this technical vocabulary to memory, and make sure you have it taped before you read on. Here is some mnemonic advice.

    To see each of these subordinating conjunctions dealt with in detail, just click on the link. Thereafter use your browser buttons to navigate. But first, if you have not yet read it, you will need some General Information. And if you prefer not to understand too deeply, and merely wish to absorb some crude rules of thumb for formalization, then you will no doubt ignore the rest of these pages and go straight to The Headlines, which will pop up in a separate window.

     


    Although..

    Consider the message m1, encoded by the sentence

    [1]    Although it was very hot, Her Majesty wore a cardigan.

    m1 is built up from the two prior messages m2 and m3, encoded respectively by the sentences

    [2]    It was very hot

    [3]    Her Majesty wore a cardigan.

    Status:

    'Although' accords the dependent message m2 the status of a conceded, but discounted truth.

    Typically, the concession will be to some other protagonist to the conversation. We are discussing Her Majesty's renowned conservatism in matters sartorial, and I am of the opinion that even Her Majesty would not be wearing the damn cardigan at yesterday's Trooping of the Colour, given the unbelievable heat-wave. So when you affirm m3, I have my doubts. I affirm m2. But you stick to your guns. You concede to me that m2 is true, but you discount it, now affirming m1.

    Affirmation:

    Here, the dependent message m2 is conceded. And the independent message m3 is affirmed outright. So the speaker is committed to the truth of both prior messages. And is not committed to the truth of anything else1....

    Formalization:

    ...and so we formalize as a conjunction. Let 'P' stand for the proposition that it was very hot, and let 'Q' stand for the proposition that Her Majesty wore a cardigan. Or, as we write in our elegant shorthand:

    P:    It was very hot

    Q:    Her Majesty wore a cardigan

    And we formalize as [P Q].

    In the mistaken framework of the boy Hodges, we say that '--although--' operates as a truth-functor.

     


    Because..

    Consider the message m1, encoded by the sentence

    [1]    Because it was very hot, Her Majesty stripped off.

    m1 is built up from the two prior messages m2 and m3, encoded respectively by the sentences

    [2]    It was very hot

    [3]    Her Majesty stripped off.

    Status:

    'Because' accords the dependent message m2 the status of an explanatory truth. Here, m2 is cited as an explanation of the truth of m3.

    Affirmation:

    Here, the dependent message m2 is implicitly affirmed (as an explanatory truth). And the independent message m3 is affirmed outright. So the speaker is committed to the truth of both prior messages. And this time she is committed to something else: an explanatory connection, causal or rational, between m2 and m3...

    Formalization:

    ...and so we cannot here formalize adequately, either as a conjunction or as anything else. We can extract something weaker, a conjunction entailed by m1. But we cannot capture everything of logical import.

    P:    It was very hot

    Q:    Her Majesty stripped off

    And we can extract the conjunctive information [P Q] .

    But Nota Very Bene Indeed: This does not capture the logic of m1. For m1 entails more than that bare conjunction - it also entails that there is an explanatory connection between our proposition P and our proposition Q.

    In the mistaken framework of Hodges, we say that '-- because--' does not operate as a truth-functor.

     


    Since..

    Consider the message m1, encoded by the sentence

    [1]    Since there are no trees in Orkney, it is a desolate place.

    m1 is built up from the two prior messages m2 and m3, encoded respectively by the sentences

    [2]    There are no trees in Orkney

    [3]    It is a desolate place.

    Status:

    'Since' accords the dependent message m2 the status of a premise, from which the independent message m3 is inferred, or (more often) deduced.

    Often, as here, deduced with the aid of other, unstated premises, which provide the ground of the inference. The obvious ground here being the proposition that places without trees are desolate.2

    Affirmation:

    Here, the dependent message m2 is implicitly affirmed (as a premise), from which the independent message m3 is deduced. So the speaker is committed to the truth of both prior messages. And again she is committed to something else: that m3 follows from m2.

    Formalization:

    ...and so we cannot here formalize adequately the whole message. Deduction is not to be captured by mere talk of truth values, being a movement of the mind from one proposition to another. We can extract something weaker, namely the truth of both m2 and m3 . But we cannot capture the logic of m1 in The Propositional Calculus.

    Almost always, though, when you are formalising an actual argument in which a 'since'-message appears, all that you will need is indeed just the conclusion m3. The proponent of the argument requires m3 as a premise for her further reasoning, and just happens to have also thrown you a little sub-argument, as a reason for believing that necessary premise.

    Here goes:-

    Q:    Orkney is a desolate place

    And we can formalize as: Q .

    Nota Very Bene Indeed: This does not capture what is going on in m1, which does something really quite complicated. It announces a premise, performs a deduction, and affirms the conclusion of that deduction. And be sure to get very clear how 'since' differs from 'if'. 'If' does not affirm a premise as true, and then draw a conclusion. Instead 'if' accords the dependent message the status of an hypothesis, and announces that it is being treated as true, whether or not it really is, and deduces the conclusion thus hypothetically. The proponent of such a message helps himself to the inference whilst avoiding commitment to the truth of it's premise. The proponent of a 'since'-message is committed to the truth of that premise

    In the mistaken framework of Hodges, we say that '-- since--' does not operate as a truth-functor.

     

     


    Unless..

    Consider the message m1, encoded by the sentence

    [1]    Unless I have made a mistake , the answer is 42.

    m1 is built up from the two prior messages m2 and m3, encoded respectively by the sentences

    [2]    I have made a mistake

    [3]    The answer is 42.

    Status:

    'Unless' accords the dependent message m2 the status of a waiver. It has the effect of withdrawing commitment from the full-blown affirmation of m3. The speaker does not want to commit herself fully to the independent message m3. But instead wishes to offer only a partial commitment, a commitment understood to be withdrawn in the case where the dependent message m2 is true.

    Thus, here I want to allow that I may have made a mistake, in which case I do not offer the answer of 42 with my full personal imprimateur. The answer may well still be 42, despite my presumptive error, but if I have made a mistake, don't blame me if you rely on my answer.

    Some other examples:-

    Unless the taloids catch him in a three-in-a-bed romp, Blair will be re-elected

    Unless it rains, we'll go to the seaside

    Unless the bough breaks, baby will be safe in the cradle

    In each case, the speaker is committed to the truth of the independent message,that Blair will be re-elected, that we will go to the seaside, that baby will be safe in the cradle, except in the circumstances specified by the dependent message.

    Affirmation:

    Here, the dependent message m2 is not affirmed at all. The speaker acquires no commitment to either its truth or its falsity. And the independent message m3 is only affirmed presumptively, presumptive on the falsity of the dependent message. So the speaker is committed to the truth of m3 only in the case that that m2 is false. In other words, she is committed to either m2's being true, or m3's being true. And is not committed to the truth of anything else3....

    Formalization:

    ...and so we formalize as a disjunction.

    P:    I have made a mistake

    Q:    The answer is 42

    And we formalize as [P Q].

    In the mistaken framework of the boy Hodges, we say that '--unless--' operates as a truth-functor.


    Whether or not..

    Consider the message m1, encoded by the sentence

    [1]    Whether or not you like them, carrots are good for you.

    m1 is built up from the two prior messages m2 and m3, encoded respectively by the sentences

    [2]    You like them

    [3]    Carrots are good for you.

    Status:

    'Whether or not ' does exactly what it says on the tin. It accords the dependent message m2 the status of an irrelevance, and announces that the truth or falsity of m2 has no bearing on the truth of m3....

    Affirmation:

    ...and so, of course, the dependent message m2 is not affirmed at all. And the independent message m3 is affirmed outright. So the speaker is committed to the truth of m3, and not committed to anything else at all.

    Formalization:

    ...and so we formalize as a single proposition.

    Q:    Carrots are good for you

    And we formalize as: Q.

    In the mistaken framework of the boy Hodges, we say that '--whether or not--' operates as a truth-functor.


    Despite the fact that..

    Consider the message m1, encoded by the sentence

    [1]    Despite the fact that Mike Tyson is a convicted rapist, this administration let him in.

    m1 is built up from the two prior messages m2 and m3, encoded respectively by the sentences

    [2]    Mike Tyson is a convicted rapist

    [3]    This administration let him in.

    Status:

    'Despite the fact that' accords the dependent message m2 the status of a truth acknowledged to count against the determination of the truth reported by the independent message m3.

    The acknowledged truth that Mike Tyson is a convicted rapist would naturally count against letting him in to the country. But the decision went the other way.

    Affirmation:

    And so, of course, the dependent message m2 is implicitly affirmed. The speaker acquires a commitment to its truth. And the independent message m3 is, of course, affirmed outright. So the speaker is committed to the truth of both m2 and m3, and not committed to anything else4.

    Formalization:

    ...and so we formalize as a conjunction:

    P:    Mike Tyson is a convicted rapist.

    Q:    This administration let him in.

    And we formalize as: [P Q]

    In the mistaken framework of the boy Hodges, we say that '--despite the fact that--' operates as a truth-functor.

    One further matter: I have been ever so slightly naughty with the grammar here5.


    If..

    Consider the message m1, encoded by the sentence

    [1]    If Oswald didn't shoot Kennedy, somebody else did.

    m1 is built up from the two prior messages m2 and m3, encoded respectively by the sentences

    [2]    Oswald didn't shoot Kennedy

    [3]    Somebody else did.

    Status:

    'If' accords the dependent message m2 the status of an hypothesis, and announces that it is being treated as true, whether or not it really is. Now there are many6 reasons why one might wish to do that, to hypothesize a message. But here the reason for thus hypothesizing the dependent message is in order to deduce the truth of the independent message m3. Another way of putting it is that 'if'-messages of this kind are condensed arguments, where the speaker performs the deduction or inference whilst witholding commitment from the premise.

    Make sure that you understand the difference between m1 and the argument encoded by

    Oswald didn't shoot Kennedy. Therefore somebody else did.

    In the argument, the premise m2 is asserted, and the conclusion m3 is deduced. So that the speaker acquires commitments to the truth of the premise m2 , the truth of the conclusion m3, and the validity of the deduction.

    Whereas with the hypothetical m1 the speaker acquires only a commitment to the validity of the deduction, and hence a conditional commitment to the truth of m3. (Which means that should she ever acquire a commitment to the truth of m2, for instance as she learns from a conspiracy theorist that Oswald cannot have killed Kennedy, she thereby acquires a commitment to the truth of m3.

    You will find it helpful to contrast the case of 'since'.

    Affirmation:

    The dependent message m2 is not affirmed, but merely hypothesized, and speaker acquires only a conditional commitment to the truth of the independent message m3. There is a lot more to say here, so I shall point you at it with a footnote . 7

    Formalization:

    And so we cannot give a proper formalization. The only solid commitment of the speaker is to the validity of her deduction, and that is not a notion which can be captured by anything involving only truth-values.

    Nevertheless, Propositional Calculus would be utterly crippled if we had no way of representing these hypothetical messages, and so we settle for an ersatz. We extract something weaker, which does just deal with truth-values. It's a trick, but it sort of works (some of the time).

    Here's the trick. If the deduction is correct, then either Oswald did kill Kennedy ( so that m2 is false) or he didn't, in which case it now follows that somebody else killed Kennedy, and m3 is true. Got that? Either m2 is false, or if not, m3 is true. And and so we formalize as a disjunction.

    P:    Oswald didn't kill Kennedy .

    Q:    Somebody else did .

    And we formalize as: [P Q]. Or, if you prefer to use the arrow functor, as [P Q] .

    But NEVER, EVER read this 'arrow' formulation as 'if'. ALWAYS think of it as 'arrow', and ALWAYS REMEMBER that [P Q] is just a convenient shorthand for [P Q]. Otherwise, I guarantee, you will get into all kinds of tangles.

    In the mistaken framework of the boy Hodges, we say that '--if--' does not operate as a truth-functor, but that we are going to pretend that it does.


    Even if..

    Consider the message m1, encoded by the sentence

    [1]    Even if Granny was gunned down in the street this morning, she's chirpy enough now.

    m1 is built up from the two prior messages m2 and m3, encoded respectively by the sentences

    [2]    Granny was gunned down in the street this morning

    [3]    She's chirpy enough now.

    Status:

    'Even if ' differs only marginally in its effect from 'whether or not'. It accords the dependent message m2 the status of an allowable hypothesis, but one whose acceptance will not influence the outright affirmation (often, re-affirmation) of the independent message m3. You can imagine the conversation:

    Her Majesty:    "Granny seems chirpy enough."

    Duke:    "Really? BigEars told me that she was gunned down in the street this morning."

    Her Majesty:    "Even if she was gunned down, she's chirpy enough now."

    Our Queer Dean, sorry, Our Dear Queen is prepared to allow that Granny may have been gunned down, but is not prepared to qualify her commitment to her original message, which she reaffirms in the teeth of the hypothesis that granny was gunned down..

    Affirmation:

    And so, of course, the dependent message m2 is not affirmed at all. And the independent message m3 is affirmed outright. So the speaker is committed to the truth of m3, and not committed to anything else at all.

    Formalization:

    ...and so we formalize as a single proposition.

    Q:    She's chirpy enough now.

    And we formalize as: Q.

    In the mistaken framework of the boy Hodges, we say that '--even if--' operates as a truth-functor.


     

    General Information

    It is vital to acquire a proper understanding of compound messages. Tradition in The Ego-Bloated Hovel tells a completely false story. Repeated faithfully by the boy Hodges. You may, if you wish, download a copy of Fear and Loathing in Hodges, a Word document in Rich Text Format, which fleshes out the details of the sketchy tale I here relate. But for now, just the headlines. To return to this exact spot at any time, either use your browser buttons, or click on the image of Gottlob Frege, the humourless and racist founding father of our discipline.

    It will do you no harm at all to remind yourself of Our Standard Warning. And then there will be the question of Truth for compound messages, the question of Structure, and the question of Affirmation. And to return to the main menu, click here.

     

     

    Anti-semitic, humourless, grinding bastard, who set our whole discipline off on the wrong foot.

     

     

     

     

     


    Our Standard Warning

    We cannot say it too often. It is vital to distinguish messages from the sentences which encode them. Sentences are strings of words, structured only in the way that a string of sausages is structured. Sentences do not have any semantic properties. They have no truth-values. Nor do they entail anything. Which means they are not the subject matter of Logic.

    So NEVER think of formalizing as a matter of transliterating English sentences into the formulas of the Propositional Calculus.

    Logic, properly understood, studies logical connections between messages. ALWAYS, when formalizing, think of the message. Remember:

    The Medium is the Message.

    The sentence is only the vehicle. I remind you of the words of Our Revered Leader:

     

    "True Sons and Daughters of the Revolution will never confuse the realm of messages with the realm of sentences."

    Emmanuel GoldsteinVive La Revolucion!








    Truth and compound messages

    Tradition, and Hodges, have it that compound sentences have truth-values, that they are either True or False. Well, we know that's a load of nonsense. No sentence has a truth value. But what about the corresponding messages? Could they have truth-values? The answer is a resounding

    NO!

    And this for a very simple reason, which you will find richly confirmed when you turn to Structure. Compound messages are just too damn complicated to have anything as simple as a truth-value. Remember: there are two messages involved, not one. The independent message, encoded by the independent sentence, and the dependent message, encoded by the dependent sentence. And certainly, these prior messages may have truth-values. Indeed, if the prior messages are propositions™, and not judgments™, they will have truth-values. But the compound cannot and will not. It places two separate messages in a special relation to one another. And that relation varies, case by case.

    As an allegory, imagine a pair of semi-detached houses, side by side in a street. Each has its own front door, its own wiring system, its own separate plumbing. But the complex formed by the two of them does not have any of these things. It has no front door, no wiring system, no plumbing system.

     











    The Structure of compound messages

    Again, Tradition and Hodges have it all arse-over-tip. See Fear and Loathing for full details. As always, they treat mistakenly of sentences, and diagnose a ternary structure, with two sentences and a 'sentence-functor' whose role is to build bigger sentences from smaller ones. Wicked waste! But even at the level of messages, tradition discerns a ternary structure, with three immediate factors: the independent message, the dependent message, and the meaning (whatever it is, case-by-case) of the subordinating conjunction. Key to understanding this area is to realise that compound messages have only two immediate factors: the independent message, and the complex encoded by the subsidiary string. Watch:

    Let us take, for definiteness, the first sentence in my menu. You can check for yourselves that the same point works for all the others. Consider

    [1]     Although it was very hot, Her Majesty wore a cardigan.

    And reflect on the point of the comma, which divides the compound sentence into two discrete parts. And notice that here, the subsidiary string 'although it was very hot' precedes the independent sentence. It doesn't have to go there, for: it can also follow the independent sentence:-

    [2]     Her Majesty wore a cardigan, although it was very hot

    Indeed, it can even interrupt the independent sentence, like this:-

    [3]     Her Majesty, although it was very hot, wore a cardigan.

    Or even like this:-

    [4]     Her Majesty wore - although it was very hot - a cardigan.

    Check to be sure that each of [1] - [4] encodes exactly the same message. And notice that the 'although' sticks like glue to the dependent sentence 'it was very hot'. They never come apart. Whereas if you do let them separate, so that the 'although' comes just before the independent sentence, you get a quite different message:-

    [5]     It was very hot, although Her Majesty wore a cardigan.

    All of which confirms that the subsidiary string is an unbreakable unit. And that it doesn't belong anywhere in particular in the final compound sentence.

    So now we can actually write down the meta-programme for generating compound sentences. Here it is:

    R1. Encode the independent message as the independent sentence

    R2. Encode the dependent message as the dependent sentence

    R3. Select a subordinating conjunction from the usual list, and prefix it to the dependent sentence to form the subsidiary string.

    R4. Loosely amalgamate the independent sentence and the subsidiary string (in the manner desribed above).

    And now for the semantics of it all. What is the role of the subordinating conjunctions? What is it that they do? And the answer is that they accord the dependent message a certain status:

    'Although' gives the dependent message the status of a conceded, but discounted truth.

    'If' accords the dependent message the status of an hypothesis, announcing that the dependent message is being treated as true, whether or not it really is.

    And similarly for the other subordinating conjunctions. You can check them out from the main menu.








    Affirmation and compound messages

    According to DesCartes, the ability to speak a language is the mark of mind. It is what distinguishes humans, who have minds, from other species, whose members are mechanical automata. Wrong about animals, right about mind. Language has evolved as our practical solution to the Other Minds Problem. Through its offices, we can convey messages to distant minds. And these distant minds can signal back their understanding of our messages. When we thus communicate, we broadcast sentences over the airwaves, sentences which encode our messages.

    This is not just a matter of floating words on the air. We stand in a special relation to the messages thus conveyed: we affirm them. We not only stand behind them as authors, we do something further: we add our own personal imprimateur. And thereby commit ourselves to (being accountable for) their truth (in the case of propositions™), or their justification (in the case of judgments™).

    For simple (i.e. non-compound) messages, encoded in simple (subject-predicate) sentences, the question of what is affirmed has a correspondingly simple answer:-

    The whole message is affirmed as a unit

    But when we have a compound message, built up from two prior messages, what happens to affirmation? Are both prior message affirmed? Or just one? Or perhaps none? And how do the two messages bear on one-another?

    The answers, you will find, very tremendously case by case. So check out each one carefully. If anyone wants to try and be a clever-clogs, I shall reveal the master rule:

    The independent message is presumptively affirmed, which means that it always is affirmed, unless there is rational reason to think otherwise.

    It will be a very good test of deep understanding of this material if you can work out, case by case, how the master-rule plays out. And with that dark hint, I shall bow out, and say no more.






    1. Or better, not committed to anything else of logical import. Nothing that could serve as either premise or conclusion in an argument. Which is all that matters in formalization.

    Of course, she is further committed, dialectically, to a justification or explanation of her selection of ‘although'. She will have grounds for her choice, and those grounds are further commitments. But they are not signalled in her output sentence, nor are they part of the message she broadcasts by it.


    2. In fact these grounds, where they occur, are further commitments of the speaker, additional premises necessary for, and so justifying, the deduction. Sometimes, there will be no further grounds, and the conclusion will be deduced from the affirmed premise on its own. Here are a few samples:-

    Since 7 is an odd number, it cannot be divided by 2 without remainder.

    Since John is a bachelor, he is unmarried.

    Since Latzy is a Hungarian, and there are no Hungarians in Balliol, (I can assure you that) you will not find him here.

    But more often than not, such inferences will need further grounds. Check these, and see if you can work out the obvious ground of the inference:-

    Since you have a room on staircase 20, you will have a delightful view.

    Since no-one here smokes, you will have to go elsewhere to find cigarettes.

    Since Sir Jasper is a scouser, we can expect a coarsening of the conversation.




    3. Of course, there are, as with 'although' other dialectical commitments. We will expect the speaker to have her reasons for selecting 'unless', reasons for withdrawing commitment in the circumstances specified. But just as with 'although', there is nothing here of logical import. Nothing that could serve as either premise or conclusion in an argument.

    Now, people sometimes think that there is something else of logical import conveyed by an 'unless'-message. And it is important to get this straight, so I shall go into detail. Take, for instance, the sentence

    [4]    Unless the tabloids catch Blair in a three-in-a-bed romp, he will be re-elected

    Many people will casually assume that the message m4 also means and/or entails the message m5 encoded by

    [5]    And if they do so catch him, he won't be re-elected.

    This is an error. Understandable, but an error nonetheless. m4 neither means nor entails m5. No doubt the speaker of [4] will have in mind, as her reason for withdrawing commitment in the case of a sex scandal, the thought that a sex-scandal might well bear on the election result. But she is not committed to the claim that said scandal would ruin Blair's chances of yet one more dismal period in office. For notice that there is nothing at all amiss with saying

    [6]    Unless the tabloids catch Blair in a three-in-a-bed romp, he will be re-elected. Indeed, he might even survive that. God knows, it might enhance his chances, for all I know. The moral world is going to hell in a handcart nowadays.

    And if m4 did entail m5, this performance would be logically contradictory.

    While you are here, store this trick away as a useful piece of technique. If you are not sure whether something is or is not an entailment of a message, see if you can cancel it (just as above) without logical oddity. And if you can, it ain't an entailment. The technique is sometimes known in the profession as the Cancellability Test. But that's just a piece of overblown and unnecessary jargon. Don't bother memorising it.


    4. With the usual caveat that she is not committed to anything else of logical import. Nothing that could serve as either premise or conclusion in an argument. Of course, she acquires further dialectical commitments. To justify her selection of 'despite the fact that' she must have reasons for thinking that Tyson's conviction should be relevant to the administration's decision. But these extra matters are nowhere encoded in her output sentence, nor are they part of the message she broadcasts.


    5. As I said, I have been a bit loose with the grammar. 'Despite the fact that' does the same work as the other subordinating conjunctions, but the true subordinating conjunction here is 'despite' and not the longer phrase 'despite the fact that'. The longer phrase is not a genuine syntactical unit. Observe:-

    Despite Tony's opposition, Cherie continued with aromatherapy.

    Despite the stench of corruption attaching to the deal, Tony gave Ecclestone what he wanted.

    Despite all his previous misdemeanours, Peter once again rose without trace.

    The actual grammatical pattern of the subordinate clause here is (DESPITE + NOUN PHRASE), and not (DESPITE THE FACT THAT + SENTENCE)

    So that in our actual example sentence

    Despite the fact that Mike Tyson is a convicted rapist, this administration let him in.

    the true structure has 'the fact that Mike Tyson is a convicted rapist' as a syntactic unit. None of this, of course, affects any of the points made in the main body of the text. But I thought I would add a footnote in case anyone wants to get picky.


    6. Nota Very Bene Indeed

    This is an important footnote. I suggest you make sure you master it.

    The message discussed in the main text is (what I call) a Logical 'if'-message. Goldstein himself calls such messages 'hypotheticals', but I have chosen a name to remind you that these are the only 'if'-messages that propositional logic has any legitimate interest in.

    It would be nice to be able to tell just by looking, i.e. on purely grammatical grounds, whether a sentence encoded a Logical message. Unfortunately, it can't be done. There are many 'if'-sentences, grammatically indistinguishable from hypotheticals, which do not encode Logical 'if'-messages. And we shall eventually be looking at some. First, we need a way of pinning down the logical ones. Unsurprisingly, the crucial test is a logical one. It is called the contraposition test.

    Logical 'if'-messages are exactly parallel to arguments, and so they contrapose. I will explain. Let m1 be the natural interpretation of

    [1]    If Tony is richer that Peter, he's richer than Gordon.

    m1 is a condensed argument, premissed on the obvious ground that Peter is richer than Gordon. The argument itself would be encoded as

    Tony is richer than Peter. So he's richer than Gordon.

    Now any such piece of reasoning can be upended, and used - as it were - the other way round. Just as we can argue from the truth of the premiss to the truth of the conclusion, we can argue from the falsity of the conclusion back to the falsity of the premiss. So here we could argue, using the same obv ious ground,

    Tony is not richer than Gordon. So he's not richer than Peter.

    In general, if P entails Q, then not-Q entails not-P. Or, in terms of Logical 'if'-messages, 'If P then Q' is logically equivalent to 'If not-Q then not-P'.

    This equivalence is known as the Principle of Contraposition, and it yields a wonderful, failsafe test for Logical 'if'-messages. Here's how the test works. m2, the natural interpretation of

    [2]    If Tony is not richer than Gordon, he's not richer than Peter

    is logically equivalent to its contrapositive, m1. And when a message has thus an intuitively equivalent contrapositive, we say it contraposes. And when a message contraposes, it is thereby certified as a Logical 'if'-message.

    Check it out with a few other examples. Notice how in each case the messages encoded by these sentences contrapose:

    If Oswald didn't kill Kennedy, someone else did    -    If no-one else killed Kennedy, Oswald did

    If Sheila is my aunt, I am her nephew    -    If I am not her nephew, Sheila isn't my aunt

    If John eats cheese he is no true vegetarian    -    If John is a true vegetarian, he doesn't eat cheese.

     

    Now for the payoff. Here are some sentences which encode messages which do not contrapose.

    [3]    If the Mayor is married, his wife did not accompany him

    [4]    The dog, if it was a dog, ran off

    [5]    It's a matter of principle, if you have any.

    These three are grammatically indistinguishable from our other, logical, compound messages. There is an independent sentence, and there is a dependent sentence prefixed with 'if'. But their messages do not contrapose. Compare:

    [3??]    If the Mayor's wife did accompany him, he isn't married

    [4??]    If the dog didn't run off, it wasn't a dog

    [5??]    If it isn't a matter of principle, you do have some (principles).

    So what has happened? Why are such messages different? The answer is, as hinted in the main text, that there are all kinds of reasons why you might want to give a dependent message the status of a hypothesis. And doing so in order to deduce a conclusion is just one of them. Take [3], for instance. If the Mayor is not married, there would be something infelicitious about saying simply 'His wife did not accompany him', for then the expression 'his wife' has nothing to refer to. So the hypothesis that he is married is invoked to waive away this scruple of detail.

    Similarly with [4]. What the message amounts to is something like 'The creature, dog or not, ran off'. The hypothesis that it was a dog is invoked, not to deduce anything, but just to avoid the problem that the word 'dog' might not be appropriate.

    [5] is rather different. Here the whole point is to affirm straight out that it's a matter of principle. The hypothesis that the audience has principles is of course irrelevant. If it's a matter of principle, then it's a matter of principle whether or not any individual is a principled person. And adding in the hypothesis is just taking the chance to have a sly dig at the audience while you are there.

    So WATCH OUT for messages like these. They may look just as if they are logical ones, and so formalizable with ' ', but they are not. And if you unthinkingly formalize with the arrow, you will get into all kinds of tangles.


     

    7.More to come...


    Grammatical Terminology

    The dependent sentence is so called because syntactically it sticks like glue to (depends on, we say) the subordinating conjunction prefixed to it. And the independent sentence is so called because it is independent of any such appurtenance. It just stands there on its own.

    English Grammar, recommended revolutionary accomplishment, recognises two classes of conjunction. The co-ordinating conjunctions, 'and' and 'or', which co-ordinate two items, nouns, pronouns, verbs, sentences, adjectives, whatever:

    Cherie and Peter had an affair last summer

    Between you and me, it was a bit of a shocker

    They tried to keep it secret and private

    But once the tabloids were on to it the story ran and ran

    Either Tony found out or the dog blabbed

    And so had to be killed or in some way silenced

    and the subordinating conjunctions, which prefix (and thus subordinate to their will) various items (often sentences) to form a single clause, called (wait for it) a subordinate clause. The main menu lists eight of them, but that's only a representative sample. There are many others. And now a farewell from Noam Chomsky himself:-

     

    Farewell. I hope you never forget that English Grammar, recommended revolutionary accomplishment, is the most wondeful intellectual discipline, full of delights and surprises

     

     

     


    And a farewell from Francis....

    Goodbye, everyone. Francis is shutting down now.