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Problems of translation - especially 'if' Formal logical systems provide a way of studying inference within a purely formal context, enabling us to make precise the ideas behind our informal use of logical principles in everyday language. They take certain aspects of logical thinking and formalise them by putting them into a simplified language that is precise enough to express statements and inferences more clearly than they would be in ordinary language, and contain rules that enable them to assign truth-values to formulas in a complete and consistent way. And standard formal logics are truth-functional - meaning that the logical connectives in them are such that the truth-value of a formula is fixed by the truth-values of its constituent propositions. Truth-functions are especially amenable to formal treatment because they allow the possibility of a mechanical decision procedure. Precision and truth-functionality are desirable features of formal logics from the point of view of simplicity and rigour. However, these features also limit formal logics, and can lead to problems of translation between ordinary language and formal language. This problem crops up with regard to the logical relations between propositions that can (or cannot) be expressed by the standard logical propositional connectives. For one thing, there are some sentence connectives in ordinary language whose meanings cannot be captured truth-functionally: for instance 'but', which truthfunctionally means 'and' but carries the extra suggestion that the second clause is surprising in the light of the first. And then there is 'because', which carries the suggestion that the subordinate clause expresses a fact that is responsible for that expressed in the main clause. More problematically, there are also ways in which it is alleged that certain logical connectives do not completely capture the meaning of their supposed ordinary language counterparts. '~' is not exactly 'not' because while '~' operates on whole sentences, 'not' can negate a sentence or just a predicate. Also, often in natural languages double negations do not 'cancel out', but may simply be emphatic. 'And' is sometimes used to mean 'and then', whereas '^' does not say anything about temporal order. Even more tricky is disjunction, or 'v'. It is often claimed that 'or' has two
senses, an inclusive one and an exclusive one. In some logics, exclusive or (or 'XOR') has its own symbol, such as ' ', although this is not strictly necessary because XOR can be expressed as '(A v B) ^ ~ (A ^ B)'. Another alleged discrepancy between 'or' and 'v' is that in ordinary discourse you would hardly ever state a disjunction just because you knew just one of the disjuncts was true, but in formal logic, '(A v B)' on the grounds that A, would be perfectly valid. But by far the connective that has caused the most difficulties is 'if' - the connective of implication. The simplest and most standard way of capturing 'if' in logic is what is known as the material conditional - a controversial character and the subject of much debate, coming in various guises such as '?', ' ' or '
'. In this essay I will
use '?'. This is its truth-table: A T T F F
C T F T F
A?C T F T T
That is, 'A ? C' is only false when its antecedent is true and its consequent is false. The material conditional can be expressed in a propositional calculus using just '^', 'v' and '~' in the following ways: since 'A ? C' rules out having A true while C is false, you could say ~(A ^ ~C), or (~A v C). So to what extent does '?' capture our ordinary language 'if'? To start with, because it is truth-functional it can only capture 'indicative' conditionals, as opposed to possible-world conditionals - those including counterfactual conditionals, and arguably future-tense conditionals too. In possible-world conditionals, assigning truthvalues to the antecedent and consequent involves considerations about hypothetical or other-worldly states of affairs rather than their actual this-worldly truth values, which is what ordinary propositional calculus is equipped to deal with. This is not always a very clear distinction as it is often not obvious which category a given conditional falls into; the superficial grammatical mood or tense of the sentence can be misleading. Still, whatever the exact scope of 'indicative' conditionals, other logics
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