Week 2 - Validity and Formalisation

Recap

In week 1, we looked at some of the complexities of natural language. Following the standard tripartite distinction, we distinguished between syntax, semantics, and pragmatics.

(i) Syntax

About syntax, we found that it has a recursive property, i.e., the rules which determine how words should be put together to form sentences (of English) apply to their own products to generate further sentences. This feature explains how it is that we can be competent with an unbounded number (an ‘infinity’) of sentences, even though, given we are finite creatures, we would never get around to using an infinite number of sentences (our performance is bounded). A finite number of words and a finite number of recursive rules allow to us generate an infinity of sentences.

One way of thinking about this is to consider novelty. With notable exceptions - ‘I love you’, ‘Pass the salt’, etc. - the sentences we utter and hear are new to us. It would be quite mysterious how our linguistic understanding could be continuously novel if we didn’t have the equipment to generate any of an infinity of sentences as the conversational situation demands.

As we shall see, recursiveness is a property shared with formal languages, such as logic.

On the other hand, natural language syntax is irregular, ‘messy’. Rules don’t apply universally (e.g., the passive), and words of the same grammatical category (verb, noun, etc.) don’t behave alike. Further, even synonymies (e.g., likely/probable, told/reported) don’t behave alike. A neat way of capturing this idea is to say that syntactic well-formedness doesn’t generalise over substitution. This is a mouth-full. All it means is that if Fa is a sentence of English, then it doesn’t follow that Fb or Ga are also sentences, even if F and G and a and b belong to the same respective categories, even if they are synonymous pairs. This feature has the following consequence: syntactic relations don’t track semantic (meaning) relations. Otherwise put: the rules of syntax don’t care about what words mean (at least not up to substitution).

As we shall see, such irregularity is not a property of logic.

(ii) Semantics

About semantics, we reached the same conclusion as above, except from the opposite direction, as it were. There are complex semantic relations which are not enshrined in the syntax. Most clearly, sentences can be ambiguous, i.e., the same syntactic structure can support distinct meanings. We also saw that individual words have aspects of meaning that don’t show up in each sentence they occur in. Bill kept the car implies ownership. Bill kept the crowd happy, doesn’t.

Again, we have a mismatch between syntax and semantics.

(iii) Pragmatics

Pragmatics is perhaps the most confusing area of language. This is because pragmatics concerns the use of language, and so involves a whole range of factors: the context of the conversation, the psychology of the speakers, what has been previously said, etc.

We saw that aspects of pragmatics, such as metaphor, irony, and implicature, appear to involve a distinction between sentence (literal) meaning and speaker meaning. That is, we often use a sentence to communicate some thought which is not the literal meaning of the sentence used. In such cases, we rely on the hearer to know the literal meaning and figure out what we mean (our speaker meaning) by her reasoning why we should utter a sentence with such a literal meaning. For example, if I say ‘Bill is a lion’, I’m saying that Bill is brave or something, but the hearer must know what ‘Bill is a lion’ literally means to be able to understand the metaphor.

We also saw, however, that pragmatic effects are quite pervasive; that is, the very notion of literal meaning as something determined independent of pragmatics is not so stable. It might be that what we think of as the literal meaning itself involves pragmatic factors. If I say, ‘I want a red car’, I only want a car whose body is red, not its whole surface, still less its interior. If I say, ‘I want a red ball’, I only want a ball whose exterior surface is red. If I say ‘I want red paint’ I want something is wholly red.

As we shall see, logic wholly ignores pragmatics. This might or might not be a problem.

Formalisation

A formal language is often spoken of as a formalisation, where what is formalised is a set of intuitive concepts (or perhaps just one). This may or may not involve a departure from our commonsense understanding of the concepts, more often than not, it does. We may think of the formalisation is an idealisation of a certain core aspect of our understanding of the concepts. For example, probability theory (a branch of mathematics) is a formlisation of our notions of chance and probability. Decision theory is a formalisation of our notion of arriving at a good decision, given our preferences and beliefs about outcomes. Logic is a formalisation our notion of a valid argument.

Two key points:

(i) None of these formlaisations are (necessarily) intended to be accounts of how we in fact reason about probability or validity.

(ii) Once we have a formalisation, we can ignore the relation it has to the intuitive concepts and simply investigate the properties of the formal system. That said, it remains an interesting question to ask what the relation is between the formal and intuitive concepts.

A formalisation is, in effect, an invented language designed to expresses the formal analogues of the intuitive concepts. The syntactic rules of the language are designed so that they mirror or track semantic relations which hold between the targeted concepts. In other words, the language is designed so that semantic/conceptual notions can be reasoned about in a rule governed, mechanical way.

This might seem terribly obscure. The fog will only clear when we look in some detail at logic.

Useless Formalisations

As we saw in week 1, it is easy to design a formal language. All one needs is an alphabet of symbols - ‘a’s and ‘b’s, say - and rules to combine them. Thus:

L = {ab, aabb, aaabbb,…}

(i) ab is a sentence of L.

(ii) If X is a sentence of L, then aXb is a sentence of L.

(iii) These are all the sentences of L.

This is called a recursive definition of ‘a sentence of L’; the rules generate all and only the L sentences. Note that ‘X’ is a variable ranging over sentences of L, it is not a symbol of L. So, neither ‘X’ nor ‘aXb’ are sentences, but (ii) remains fine - it can be read as saying, ‘If you have any sentence of L, then putting a at the front of the sentence, and b at the end of the sentence, results in another sentence’. The variable ‘X’ is just a way of saying this more simply. For example, the equation x0 = 0 doesn’t say that the 24th letter of the English alphabet times 0 equals 0. It says that any number times 0 equals 0. ‘x’ is not a number, just as ‘X’ is not a symbol of L.

L is not an interesting formal language. Why not? Well, because nothing is being formalised: we have the simple syntactic rules, but there are no concepts which are being expressed - formalised - by the rules. What we want is a formal language that actually succeeds in capturing some concepts within its mechanical rules. Logic seeks to capture validity.

Arguments

Logic is about arguments. Arguments here are not disputes but lines of reasoning from premises or assumptions to conclusions. We say that an argument is a good one, if its conclusion follows from its premises. Logic is about formalising what ‘good’ means here.

There are, of course, many ways in which an argument can be deemed good. Logic is concerned with a special way in which an argument can be good. The three arguments below might well be considered good ones.

(1)a. All bachelors are unhappy.

Bill is an unmarried man.

Therefore, Bill is unhappy.

b. The Sun has risen every morning since the dawn of time.

There is no reason to think tomorrow will be different.

Therefore, the Sun will rise tomorrow.

c. One hundred ill people saw the healer.

After seeing the healer, 90% of them recovered.

Therefore, the healer has some power to cure the sick.

These arguments might be persuasive, but they are not logically valid. On the other hand, this argument is logically valid:

(2) All Frenchmen are fish.

Tony Blair is a Frenchman.

Therefore, Tony Blair is a fish.

What’s good about a logically valid argument doesn’t depend on either its premises or conclusion being true. A logically valid argument need not persuade you of its conclusion at all.

Validity

Logic is concerned with validity: logically good arguments are valid ones. There are a number of ways to express validity. Here is a general way:

Validity: An argument is valid if and only it is not logically possible for all its premises to be true, and for its conclusion to be false.

Three immediate things to note:

(i) The validity of an argument can’t be determined by just looking at its premises or just at its conclusion; it is necessary to look at both.

(ii) To criticise an argument in terms of its validity, it is not enough to establish that either its premises or conclusion is false.

(iii) An argument might be invalid, even if its premises and conclusion are true.

(iv) An invalid argument can have some true premises, and a false conclusion.

Consider argument (2). Its two premises are false, as is its conclusion. Validity, however, is a property not based upon what is true or false given the way the world is, but on what is true or false given an assumption about some other sentences being true or false. Thus, to test for validity, we ask, if the premises were true, would it be possible for the conclusion not to be true. Otherwise put, Is it possible for the conclusion to be false, and the premises true? Okay, well:

(i) If all Frenchmen are fish, then any given Frenchmen is a fish.

(ii) The second premise tells us that Tony Blair is a given Frenchman.

(iii) Thus, Tony Blair must be a fish, if the premises are true.

Going the other way:

(i) Assume that it is false that Tony Blair is a fish.

(ii) If Tony Blair isn’t a fish, then he can be a Frenchman (i.e., the second premise can be true) only if not all Frenchmen are fish (i.e., the first premise is false).

(iii) Thus, it is not possible for the conclusion to be false and for both premises to be true.

We may say: an argument is valid just if the assumption that the conclusion is false generates an inconsistency between its premises. If no inconsistency is generated, then the argument isn’t valid.

Consider argument (1)c. Assuming that the healer has no powers (i.e., the conclusion is false) generates no inconsistency. It is perfectly possible for the premises to be true and for the healer to have no powers. How? Simply by assuming that 90% of the sick people recovered from their colds independent of the healer’s intercession.

Consider argument (1)b. Assuming that the conclusion is false (i.e., the Sun will not rise tomorrow) doesn’t generate an inconsistency in the premises. Of course the Sun has always risen in the past, and I have no reason to think tomorrow will be different, but the Sun might still blow up at any moment. That is logically possible, and it doesn’t contradict the premises.

Argument (1)a is an interesting case. As so far presented, it looks to be valid. We shall return to it. In the meantime, think about the difference between it and (2).

Here is yet another way to understand validity:

An argument is valid if and only if the negation of its conclusion conjoined with its premises creates a contradiction.

Here is another, more picturesque way:

Imagine a world (actual or not) which is described by the premises of an argument. Can the negation of the conclusion also consistently describe this world? If it can, the argument is invalid. If it can’t, the argument is valid.

Glossary

Sainsbury defines his terms quite clearly, but here are a few things which might help you.

Possibility: Validity is about what is logically possible, not what is actual or physically possible. We can think of these three notions as contained in each other. This may be represented as below, where ‘x Í y’ means ‘x is included in y’:

Actual Í Physically possible Í logically possible

The actual covers what is in fact true. The physically possible covers what could be true, given the laws of physics (whatever they might be). The logically possible covers that which is non-contradictory. This is a vast category. Anything which isn’t self-contradictory is logically possible. It doesn’t follow that it is straightforward to tell what is logically possible. Consider this mathematical claim:

(3) For all numbers n, n + 1 > n.

Would the negation of (3) be contradictory? If anyone explains to me why (3) is in fact false, I’ll buy them a drink of their choice.

Consistency: A set of sentences is consistent if they can be all true or false together. In other words, a consistent set is one which is not contradictory. Note, consistency doesn’t imply truth.

Negation: The negation of a sentence is its contradictory. If P is a sentence, and not-P is its negation, then P and not-P is a contradiction. Contradictories are such that if one is true the other must (logically) be false. With this in hand, we can say that a consistent set is one which doesn’t contain P and its negation. Inconsistent sets need not include negations, but we must be able to derive a contradiction.

Soundness: A sound argument is one which is valid and has true premises. Thus, a sound argument is one which proves that its conclusion is true. Validity doesn’t imply soundness, as example (2) shows. Here’s a sound argument: All Englishmen are Europeans; Tony Blair is an Englishman. Therefore, Tony Blair is European.

Truth: Quine has said that “Logic chases truth up the tree of grammar”. What this means will be explored below. But what is truth? Er, well, ah, yes. Logic doesn’t tell us what truth is. It tells us how not to deviate from the truth on the assumption that we begin with truth. For our purposes, we can think of truth as a simple property which all sentences either have or lack (here we ignore questions, exclamations, and the like).

Take a Breath

This is a fair amount of jargon. If you find it confusing, try to apply each of the notions to a given argument rather than try to figure out what the definitions say in the abstract. Rest assured, you are not supposed to get this stuff in one sitting, as it were. If you do, great; if you don’t, then great - you have something to look forward to.

Some Notation and Some Properties of Validity

All notation is conventional; the only restriction is that we be consistent in our use. So, let us adopt the following convention. When we are talking about an argument, we shall use uppercase English ‘A’ for assumptions and uppercase ‘C’ for conclusions. The double turnstile ‘†’ is used to connect premises to conclusions. Sainsbury (p.24) reads ‘†’ as ‘is valid’. This is slightly odd. It is better to read it as:

‘A₁,… A_n † C’ means ‘A₁,… A_n entail C’ (or C follows from A₁,… A_n), i.e., it is not possible for C to be false and A₁,… A_n all to be true.

‘A₁,… A_n † C’ means ‘A₁,… A_n don’t entail C’ (or C doesn’t follow from A₁,… A_n), i.e., it is possible for C to be false and A₁,… A_n all to be true.

Validity is transitive (LF, Ex. 1.15, p.27)

This means:

(4) If [A₁,… A_n † C] and [B₁,… B_k, C † D], then [A₁,… A_n, B₁,… B_k † D]

Explanation: For (4) to be false, [A₁,… A_n † C] and [B₁,… B_k, C † D] must be true, and [A₁,… A_n, B₁,… B_k † D] must be false. Assume, then, that [A₁,… A_n † C] and [B₁,… B_k, C † D]. It follows that it is not possible for C to be false and A₁,… A_n to be true, and it is not possible for D to be false and A₁,… A_n, C to be true. Is it possible for [A₁,… A_n, B₁,… B_k † D] to be false, i.e., for D to be false and A₁,… A_n, B₁,… B_k to be true? Let us assume so and then generate a contradiction. The contradiction would show that (4) must be true.

(i) Assume, then, that D is false, and A₁,… A_n, B₁,… B_k are all true.

(ii) By the assumption of [B₁,… B_k, C † D], if D is false, then some of B₁,… B_k, C must also be false.

(iii) We have assumed in (i) that B₁,… B_k are all true; so that leaves C to be false.

(iv) But if C is false, then by the assumption of [A₁,… A_n † C], some of A₁,… A_n must also be false.

(v) But this contradicts our assumption in (i) that, A₁,… A_n, are all true.

(vi) Therefore, it is not possible for D to be false, and A₁,… A_n, B₁,… B_k to be true. QED.

Validity is Reflexive (LF, Ex. 1.16, p.27)

This means:

(5) If C is among A₁,… A_n, then [A₁,… A_n † C].

Explanation: Let C be A₁. For purposes of argument, assume that [A₁,… A_n † C] is false, i.e., it is possible for C to be false and A₁,… A_n to be true. Now let’s generate a contradiction.

(i) Assume, then, that C is false and A₁,… A_n are true.

(ii) It follows that A₁ is true, but A₁ is C, and so C is true.

(iii) This contradicts the assumption that C is false and A₁,… A_n are true.

(iv) Therefore, it is not possible for , A₁,… A_n to be true and C to be false. QED.

If A₁,… A_n † , then A₁,… A_n, B † , for any B (LF, Ex. 1.17, p.28)

(i) Let A₁ be the negation of A₂. This suffices for A₁,… A_n †.

(ii) It follows that any set S containing A₁ and A₂ will be inconsistent: S †.

(iii) Therefore, in particular, A₁,… A_n, B †. QED.

Necessary truths are redundant as premises (LF, Ex 1.18, p.29)

If † B, then [A₁,… A_n † C] if and only if [A₁,… A_n, B † C]

(i) ‘† B’ means that B is logically true (necessary); it follows from any premises.

(ii) ‘X if and only if Y’ (abbreviated to ‘iff’) means that if X, then Y and if Y, then X. Thus, to prove anything of the form ‘X if and only if Y’, we prove left to right (Y from X), then right to left (X from Y).

We want to show that ‘† B’ can’t be true, while ‘[A₁,… A_n † C] if and only if [A₁,… A_n, B † C]’ is false. Let us assume, then, that ‘† B’ is true and ‘[A₁,… A_n † C] if and only if [A₁,… A_n, B † C]’ is false, and generate a contradiction.

Step 1

(A) We prove that if [A₁,… A_n † C], then [A₁,… A_n, B † C].

(i) For purposes of argument, assume [A₁,… A_n, B † C] is false, i.e., it is possible that C is false and A₁,… A_n, B are true.

(ii) But if C is false, then at least one of A₁,… A_n must also be false, if [A₁,… A_n † C]. (iii) If one of A₁,… A_n must be false, then this contradicts our assumption (i).

(iv)Therefore, it is not possible for [A₁,… A_n, B † C] to be false and [A₁,… A_n † C] to be true.

(v) Therefore, if [A₁,… A_n † C], then [A₁,… A_n, B † C], which was to be proved. QED. (Incidentally, this proves that validity is monotonic; it doesn’t make any difference if B is logically true or not.)

(B) We prove, on the assumption that † B, that if [A₁,… A_n, B † C], then [A₁,… A_n † C].

(i)For purposes of argument, assume that [A₁,… A_n † C] is false, i.e., it is possible that C is false and A₁,… A_n, are true.

(ii) But if C is false, then at least one of A₁,… A_n, B must also be false, if [A₁,… A_n, B † C].

(iii) If one of A₁,… A_n must be false, then this contradicts our assumption (i).

(iv) Therefore, if C is false and A₁,… A_n are true and [A₁,… A_n, B † C], then B must be false.

(v) But B, by assumption, is logically true.

(vi) Therefore, it is not possible for [A₁,… A_n † C] to be false, and [A₁,… A_n, B † C] to be true.

(vii) Therefore, on assumption that † B, if [A₁,… A_n, B † C], then [A₁,… A_n † C], which was to be proved. QED.

Step 2

(i) We have proved If [A₁,… A_n † C], then [A₁,… A_n, B † C].

(ii) We have proved If † B, then If [A₁,… A_n, B † C], then [A₁,… A_n † C].

(iii) Therefore, we have proved If † B, then If [A₁,… A_n, B † C], then [A₁,… A_n † C], and If [A₁,… A_n † C], then [A₁,… A_n, B † C].

(iv) Therefore, we have proved If † B, then [A₁,… A_n † C] if and only if [A₁,… A_n, B † C], which was to be proved. QED.

LF, Ex. 1.18(b)

Because validity is monotonic (see Step 1(A), above):

If [B†], then [A₁,… A_n, B † C], whatever A₁,… A_n, C may be.

Now, we are asked whether if [A₁,… A_n, B † C], then [B†]. The answer is yes.

(i) Assume [A₁,… A_n, B † C] is true, for any choice of A₁,… A_n, C.

(ii) If, in particular, [A₁,… A_n, B † C] is true, for any choice of C, then A₁,… A_n, B †.

(ii)a Sub-proof: If A₁,… A_n, B were all true, then [A₁,… A_n, B † C], for a choice of false C.

(ii)b Therefore, it is not possible for A₁,… A_n, B all to be true.

(iii) But, by assumption, A₁,… A_n can all be true; in particular, there is an instance where A₁,… A_n are true and C is false and [A₁,… A_n, B † C].

(iv) Therefore, it is not possible for B to be true.

(v) Therefore, B †.

(iv) Therefore, if [A₁,… A_n, B † C], then [B†]. QED

(Try the other exercises for yourself, using the above examples as models.)

Notes on Proof

(i) Always be clear and precise; make no jumps, even if they are obvious. Behave like a machine, as it were.

(ii) For each formula, consider what follows from assuming that it is true/false.

(iii) All of the above proofs are via reductio ad absurdum. If we want to prove P, we assume not-P, and then generate a contradiction based on not-P as assumption. Truths don’t lead to contradiction; so, not-P must be false. Therefore, P is true.

(iv) If your head hurts, don’t worry, it is supposed to.

Formal Validity

So far, we have only been considering validity. Logic, however, is concerned with a strict understanding of validity: formal validity. The two notions largely coincide.

An argument is formally valid iff it is valid in terms of its form.

‘Form’ here means essentially a generalisation over the words which occur in the sentences of an argument. We keep certain words constant, and generalise over others. When we generalise over the words of a given sentence, we can replace them with letters. Why? Because it doesn’t matter what the words are.

Once we have replaced all of the words we can generalise over with letters, we then have the logical form. This might seem quite obscure. Let us consider the example from above.

(2)a All Frenchmen are fish.

Tony Blair is a Frenchman.

Therefore, Tony Blair is a fish.

b. All Gs are F.

a is G.

Therefore, a is F

Note that replacing G and F with any predicate, and a with any name, then we still have a valid argument. We can say, (2)a is formally valid because any substitution instance of (2)b ((2)b with its letters uniformly replaced by words) is valid.

Think of the letters in an argument form as essentially generalisations over all of the words which, if inserted into the form, would result in a valid argument, i.e., all the words which would make the same contribution to determining whether the containing argument is valid.

All formally valid arguments are instances of a valid argument form.

The logical form of a sentence is essentially a generalisation of the sentence in terms of its structure that is relevant to determining its truth, i.e., the contribution the sentence may make to determining the validity of an argument in which the sentence occurs - think about the relation between (2)a and b.

With all this in mind, let us return to the argument which appeared to be valid:

(1)a. All bachelors are unhappy.

Bill is an unmarried man.

Therefore, Bill is unhappy.

This seems valid because we know that bachelors are unmarried men, and so if Bill is an umarried man, then he is a bachelor, and so is unhappy. But the argument is not formally valid. Consider:

All As are B

a is C

Therefore, a is B

This is not a formally valid argument form, for it has the instance:

All whales are mammals.

Goldie is a fish.

Therefore, Goldie is a mammal.

This immediately tells us that the generalisation involved in finding the logical form of a sentence is not sensitive to what individual words mean. Logic doesn’t tell us that bachelors are unmarried men. Logic is only sensitive to the categories to which words belong - names, predicates, etc. - and we replace each word of such categories by a type of letter.

When Quine says that “logic chases truth up the tree of grammar”, he means that logic tracks how validity relations between sentences are determined by relations between sentences in terms of their logical form.

A valid argument is one which preserves truth. That is, if you start with truths, then, reasoning logically, one can only conclude with truths. Logic doesn’t tell us what is true, still less does it tell us what we ought to believe. But it does tell us, in a mechanical way, how we can guarantee truth form truths, as it were. The way is mechanical precisely because truth is preserved in terms of form, not in terms of what individual words, or, indeed, sentences, mean. Logic is a way of chasing - preserving - truth via manipulation of form - grammar.

Natural Language Syntax and Logical Form

We can think of logical form as the syntax of logic: logic is not concerned with the truth or validity of natural language sentences directly; it is only concerned with the logical form of such sentences, and it is the logical form which decides validity.

We saw that natural language syntax is ‘messy’, logical form isn’t. Logic is designed not to be messy. Each logical form is a generalisation over a class of sentences, such that substituting one word for another of the same ‘type’ produces another well-formed sentence with the same logical form (witness the Tony Blair example above). Natural language syntax is not like this.

Further, logical form is designed to express or track certain semantic or truth relevant relations between sentences (validity); natural language syntax isn’t so designed.

Logical form, in virtue of being a generalisation over words, abstracts away from individual differences in the meaning of words: there is lexical ambiguity or polysemy. As we progress, we shall also see how logical forms are not structurally ambiguous.

Logic is a formalisation of validity.

Logic as a Formal Language

Logic is a formal language which seeks to formalise validity in terms of logical form. Logic turns sentences into their logical forms, as it were, then assesses arguments for validity in terms of such forms.

But if logical form is essentially a generalisation away from particular words, then can we generalise away from all words in a sentence? No. There are some words which remain constant; we cannot generalise away from what they mean. These are the so-called logical constants, and the so-called propositional calculus is the logic which is concerned with the role they play in determining validity. In the propositional calculus, we generalise over whole sentences, not just words, but we leave the logical constants intact as words which uniformly make the same contribution to validity whatever the context. We’ll come to them next.