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Lesson 8

Logic as a Modeling Language

Logics as frameworks for argument

- Logics are used to describe the structure of arguments but do not tell us whether the argument we are making is realistic.

Argument 1, although coherently structured, does not describe a real situation.

- The aim of formal analysis when developing blueprints is to be precise about the accepted structural framework of argument so that we can more easily compare our views of reality.

Notice that we have omitted a conclusion from this argument because it is no longer clear what the conclusion might be.

- There are numerous ways in which we might structurally revised the original argument to make it with the new assertion.

It would be useful if there was some methodical way of checking whether the consequences of our arguments hold, but to do this we need to be more precise in our language.

- First we have to agree on the primitive concepts.
- martian(X) means that X is a Martian.
- author(X) means that X is an author of this book.
- green(X) means that X has green skin.
- We must then build arguments from these primitive concepts using connecting elements.

One very common set of connectives:

- A conjunction, P Q, meaning that both P and Q are true.
- A disjunction, PQ, meaning that P or Q (possibly both) are true.
- An implication, PQ, meaning that if P is true Q is also true.
- A negation, ¬P, meaning that P is false.
- Using these, we can construct the components of our argument. For example, the first sentence of Argument 1 says ‘Everyone from Mars has green skin’. Our formal version is martian(X) green(X). We put the formal arguments within curly braces and use the symbol ‘ ‘ to separate these from the consequence.

We want to analyze whether it is reasonable to derive the consequences given in Formal arguments 1 to 4 from the statements given.

- The most direct way to do this is by finding all the primitive concepts which we could conclude from each set of statements and then testing whether the consequence appears amongst them.
- Argument 1 is

We can now observe directly that green(dave) green(jaume) is a consequence of statements because both primitive concepts appear in the conclusion set.

- Applying a similar procedure to Argument 2:

In Argument 3 we were equivocal about whether Martians are green or pink skinned. Consequently, we can generate more than one conclusion set. Two of the possibilities are:

- The set on the left is inconsistent but the one on the right is consistent and allows us to conclude pink(dave) pink(jaume).

Finally, Argument 4 gives us the conclusions:

from these we can establish ¬martian(dave) ¬martian(jaume) since each part of this conjunction is in the conclusion set.

These examples demonstrate why it is attractive to use a logic.

- If we supply statements in an appropriate form then the logic commits us to the consequences which can be inferred from these according to the semantics of our logical connectives.
- Sometimes these consequences surprise us and lead us to revise our opinion of what our statements should be.
- There are technical problems in fully exploring the consequences of statements and we shall return to these .
- First, we discuss the equally important non-mechanical issue of ensuring that the statements we make in a logic are appropriately connected to reality.

The boundary problem

- When describing any problem in logic we must judge which are the primitive concepts upon which all our conclusions will depend.
- We could ,for example, have supplied statements which allowed authorship to be inferred from other information – for instance:

in which case author is no longer primitive but we have introduced two new primitive concepts – book and name_on_cover.

Would that have been worthwhile? Probably not.

- Such decision are subjective. They depend partly on the intent of those writing the statements and partly on a consensus by practitioners in the application domain over what is or is not important.
- Along with this consensus we must strive for a working agreement on the interpretation of primitive concepts.
- Of course, we cannot guarantee that everyone has exactly the same understanding of any concept so this agreement is sometimes fragile.
- The overall effect is to form a boundary between those parts of the problem which we define by statements in a logic and those which are considered outside the scope of formal analysis.

Guidelines

- Gaining expertise in drawing boundaries, and in structuring formal representations within these boundaries, takes experience.
- The following are some guidelines:
- First try to distinguish the things which are objects in the statements you want to represent. There are two objects: daves_alarm_clock and dave in ‘If Dave’s alarm clock rings Dave gets up’.
- Having identified the objects involved, try to identify the relations which apply to those objects. We have two predicates: rings(daves_alarm_clock) and gets_up(dave), rings is a predicate applying to the object daves_alarm_clock and that gets_up is a predicate applying to dave.
- Look for keyword in English which suggest use of one or more of the logical connectives.

Logical connectives

- The ‘not’ connective

Look fornegativeslike ‘doesn’t’, ‘no’ or ‘never’

- The ‘and’ connective

Look forphraseslike ‘also’ or ‘as well as’

Logical connectives (Cont.)

- The ‘or’ connective

be careful to distinguish between the use of inclusive and exclusive or connectives.

- The ‘ ’ connective

Look forphraseslike ‘If X then Y’ or ‘Whenever X then Y’

It is also possible to find implication statements from English statements which do not, on first glance, look like implications. For example, ‘Either Dave can’t see properly or Jaume is losing hair’ could be represented as the formula:

By using the general principle that implies for any two statements a and b gives us the expression:

Quantifiers

- If variables appear in the expression then we must indicate whether they refer to all objects or only to some object. We use symbols called quantifiers:

The universal quantifier ‘All politicians are devious’ becomes:

The existential quantifier ‘There is a politician who can be trusted’ might be represented using the formula:

It is quite common to have a mixture of universal and existential quantifiers in a statement. ‘If all politician are devious then none of them are trustworthy’:

This example can be used to illustrate several important points:

- It is sometimes necessary to restrict quantification to parts of the formula, rather than to the whole thing.
- Care must be taken to distinguish variables which can represent different objects.
- The meaning of a formula can be changed just by altering the scope of the quantifiers.

which means that for any X, there is some Y such that X is devious because he/she is a politician then Y is not a trustworthy politician.

- This would allow a single untrustworthy politician whose lack of trust was dependent on the deviousness of all the others.
- Compare this to the meaning of the original example, which says that if being a politician implies deviousness then none of the politician are trustworthy.

It is sometimes possible to change the quantification of a formula without altering its meaning. For instance there is a proven equivalence between the statement and the statement

, thus:

- The precedence of the quantifiers may be important in determining the meaning of the formula. We might have a predicate of the form: votes_for(X,Y). Suppose that we quantify this as follows:

The normal convention is that, in the absence of explicit bracketing, a quantifier further to the left in a formula will dominate those quantifiers to its right.

The search problem

- In the example, we were able to identify the consequences of an argument by generating all the primitive concepts which could be concluded from it and simply checking if the consequences we need were amongst them.
- Unfortunately, this is not possible for all formal argument.

Imagine that we now attempt to generate all possible conclusions from Argument 5.

- There is no limit to the number of paths we can generate so it is impossible to generate them all.
- For this reason it is necessary to be more specific about the rules of inference which allow us to derive our conclusions, and to be precise about the strategy used to apply these rules.

Proof strategies

- The expressions of Argument 5 are all written in one of two ways: either as asserted facts or as conditionals of the form (where P is a single conclusion and C is a, possibly complex, precondition on it).
- Many of the examples in this book are similarly structured.
- This is because there is a simple but effective strategy for performing proofs using these types of expression

We can now apply this strategy to find out if particular statements follow from Argument 5. For example, let’s try to find a path to barcelona.

- Figure 10.1 gives one example, described using a tree diagram.
- The nodes on the tree are expressions which have been proved.
- Each cluster being labeled with the number of the chosen sub-strategy from Informal proof strategy 1.
- Some parts of the proof identify variables with terms. Where this occurs we have shown the instantiation of the variable next to the number of the sub-strategy responsible for it.
- The boxes on the diagram surround expressions which supplied with the argument.

Infinitely many statements of the form tour(barcelona, V) which we could establish if weapply our proof strategy exhaustively, so in the worst case we could still hit the search problem described.

- To search selectively then we can obtain those conclusions which interest us without having to explore all of the consequences of an argument.

The four most common ways of doing this in automated systems:

- To use a simple search strategy which can be borne in mind when defining arguments. This encourages those defining expressions to express themselves in ways which they already know will be handled efficiently by the search mechanism.
- By restricting the syntax allowed in making arguments,so that they are well suited to a particular search strategy.
- By restricting the things we are allowed to prove to those which the engineers of a particular argument feel will easily be found by a given search strategy.
- By defining sophisticated proof strategies which avoid exploring fruitless avenues of investigation when attempting to establish a conclusion.
- It’s necessary to apply some blend of Approaches 1 to 3, depending on the type of application.

Describing proof strategies formally

- Informal proof strategy 1 is precise enough for us t follow but is not so precise that we could easily get a machine to follow it.

We can now apply those formal proof rules to produce proof trees similar to the one we given in Figure 10.1.

- The method which we have just used is known to logic programmers as meta-interpretation because we have built a description of the strategy with which we may interpret a logic.
- The practical value of meta-interpretation is that we can use it to tailor strategies to particular types of problem.

Distinguishing proof rules from selection strategies

- The meta-interpretation conflates two different types of strategic information:
- The conditions which must hold in order for a given step in a proof to be allowed.
- The order in which we choose to prove required conditions.
- We can construct a set of proof rules directly corresponding to the sub-strategies in Formal strategy 1. These are known below, where rule(G,S) denotes that a proof is allowed for goal G by establishing all of the conditions in set S.

This is actually more general than we want because our definition does not say in which order should choose each element from .

- One way of ensuring that our strategy always terminates is to place a limit on the depth of our proof tree (measured by the maximum number of nodes from the top to any leaf node).

For instance, the goal:

where A is instantiated to the set of expressions in Argument 5, will permit only two results for X: start or visited(edinburgh,start).

- It is alsopossibleto change the set of proof rules to which our search strategy has access.
- Using the extended set of proof rules from Figure 10.2, we can perform a wider variety of proofs.

Non-deductive patterns of inference

- All of the inference methods presented in earlier sections are deductive: they allow us to conclude only those things which are necessarily logical consequences of given arguments.
- Sometimes we need to reason about arguments in ways which are not deductive.

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