HCJ Seminar paper

Lee Jarvis- Modern mathematics, logic and language: Philosophy in the modern world chapters 4,5 and 6 and The history of Western Philosophy chapter 31: seminar paper

John Stuart Mill; a British philosopher born 1906 wrote ‘system of logic’ which is composed of two parts; firstly formal logic and then the methodology of the natural and social sciences.  Mill was the first British empiricist to take formal logic seriously and was anxious to dissociate himself from the nominalism of the empiricists since the time of Hobbes. Nominalism is the theory that a proposition is true if subject and predicate are names of the same thing.  The word ‘name’ is used broadly by Mill; he felt that proper names, pronouns, general terms like man and wise, abstract nouns and definite descriptions all count as names. All names connote things; they connote the things that they signify. In logic connotation is prior to denotation.  Because ‘name’ according to Mill, covers so many terms he can accept the nominalist view that every proposition is a conjunction of names but does not commit him to the Hobbesian view. Unlike Hobbes, Mill’s can use connotation in setting out the truth conditions of propositions; a sentence joining two connotative terms such as ‘all men are mortal tell us that certain attributes are  accompanied by the attribute of morality.

Mill then talks about inference which he puts into two kinds:

Verbal inference: Brings no new knowledge about the world, knowledge of the language alone is enough to allow us to derive the conclusion from the premiss. E.g. the inference from ‘no great general is a rash man’ is that ‘no rash man is a great general’; both premiss and conclusion tell us the same thing.

Real inference:  Is when we infer to a truth, in the conclusion, which is not in the premises.

Mill found it hard to explain how new truths could be discovered by general reasoning, accepting that all reasoning was syllogistic.  ‘Induction’ was the name given to the process of deriving a general truth from particular instances. There is more than one kind of induction however.

He felt that deductive argument involves confusion between logic and epistemology. He felt that an inference could be deductively valid without being informative; validity is a necessary but not a sufficient condition for an argument to produce true information.

Syllogism is not the only form of inference however, and other non-syllogistic arguments are also capable of conveying information. But Mill argues that we could never make connections such as ‘all men are mortal’ if it is not by induction.

Mill attempted to set out five rules or canons of inductive discovery to guide researchers in the inductive discovery of causes and effects. The first two are called the method of agreement and the method of disagreement.

The method of agreement: States that if phenomenon F appears in conjunction of the circumstances A,B and C and also in the conjunction of the circumstances C,D and E then we conclude that C, being the only common feature, is casually related to F.

The method of disagreement: States that if F occurs with A,B and C but not in the presence of A,B and D then we are to conclude that C, being the only feature not in both cases in causally related to F.

Mills believed that we are always applying these canons in daily life but we are not always conscious of applying them.

Mill was an epistemologist and therefore derives all knowledge from experience. He felt that mathematics was also derived from experience. He understood that the senses found it hard to distinguish between two larger numbers such as 102 and 103 but could easily distinguish between two and three. Mill’s felt that arithmetic is essentially an empirical science.

Frege contrasted to Mill. He felt that arithmetic was a priori and analytic. He organised logic in a whole new way. Frege devised a system to overcome the difficulties and weaknesses of syllogism; such as not being able to cope with inferences with words such as ‘all’ or ‘some’. His first step was to replace the grammatical notions of subject and predicate with new logical notions which Frege called ‘argument’ and ‘function’. This was an introduction of a more flexible method of analysing a sentence, brining out logically relevant similarities between sentences.  Frege called the first fixed part of a sentence a function and the second part is the argument. Next he tired to introduce a new notion to express generality expressed in a word like ‘all’. In the sentence Socrates is mortal’ if it is a true sentence, we can say that the function ‘is mortal’ holds true for the argument ‘Socrates’. In a formula is would be, ‘for all x, x is mortal’

Frege believed that all objects were nameable, including numbers which were named by numerals. In Frege’s system of propositional logic, the most important element is a sign for conditionality known as

The sign can be viewed as a version of the word if, in his terminology the sign is a function that takes sentences as its arguments: its values are sentences too. This sign helped to build a complete system of propositional logic.  

Frege was not interested in epistemology for its own sake but felt that it had been given a fundamental role in philosophy that should be assigned to logic, but philosophers in the empiricist tradition had confused logic with psychology.

Frege’s work was carried forward and his successors Russell and Whithead with Principa Mathematica continued work on logic, taking over from Giuseppe Peano and his axioms. There were as follows:

1.    0 is a natural number.

2.    For every natural number x, x = x.

3.    For all natural numbers x and y, if x = y, then y = x.

4.     For all natural numbers x, y and z, if x = y and y = z, then x = z.

5.     For all a and b, if a is a natural number and a = b, then b is also a natural number.

This system allowed logical truths to be derived from by a handful of axioms. Principa Mathematica tried to find the logical basis of numbers. Russell said that numbers were infinite.

Throughout history in philosophy there has been a split between those whose thought was inspired by mathematics and those who’s are inspired by empirical sciences:  Plato and Kant are examples of those whose thought was swayed by mathematics.  A school of philosophy has arose which tried to eliminate pythagoreanism from mathematics, and tries to combine empiricism with the deductive parts of human knowledge; its achievements have been as solid as those who’s thought is swayed by science according to Bertrand Russell.

The origin of this type of philosophy is a result of the achievements of mathematicians; Leiniz believed in actual infinitesimals (objects which are too small to be measured, Weierstrass showed how to establish the calculus without infinitesimals, Georg Cantor developed the theory of continuity and infinite number. Cantor gave precise significance to the word continuity which he felt had a vague meaning; he said that it was the concept needed by mathematicians and physicists.  Cantor also showed that numbers were clearly not finite. He defined the infinite collection as one which has parts containing as many terms as the whole collection contains; this mathematical theory of infinite numbers before Cantor was confusing and given over to mysticism.

Bertrand Russell could not accept Mill’s view of mathematics as an empirical science and his empiricism was always blended with an element of the Platonism that he had shared with Frege. Russell warned us that sense data brings no assurances because sense data is private and personal and we cannot guarantee that everyone sees the same thing, a table may to look like on thing to one person and then different to another.  He also said that there was no proof the life is not just one dream. Our lives are instinctive rather than reflective.  Russell felt that every proposition that we understand must be composed wholly of constituents with which we are acquainted.

Frege was important in this school of philosophy with his definition of ‘number’. Before Frege every definition of number that had been suggested contained logical blunders; it was nonsense and didnt make sense, but from Frege’s work it followed that arithmetic and pure mathematic is nothing but a prolongation of deductive logic.

Some felt that It became clear that a great part of philosophy can be reduced to ‘syntax’ and that all philosophical problems are syntaxical; when syntax errors are avoided philosophical problems are solved or shown to be insoluble

For the philosopher the substitution of space time for space and time in the theory of relativity is important. Philosophy and physics developed the notion of ‘things’ into material substance containing particles; each small and persisting throughout all time. Einstein substituted events for particles; each event had to each other a relation called ‘interval’ which could be analysed in various ways into a time element and a space element.  Events seem to be the ‘stuff’ of physics.  A particle is to be known as a series of events.  Matter is a convenient way of collecting events into bundles.  This conclusion is reinforced by quantum theory.

 In 1918 C.I. Lewis approached modal logic with the theory of implications. Modal logic is the logic is the logic of the notion of necessity and possibility. What is it for proposition p to imply a proposition q. Russell and Whitehead treated their truth-functional ‘if’ as a sign of implication: If p  and q=q then q was a valid inference. But they noticed that it was an odd form of implication and that any false proposition implies every truth proposition and therefore called in ‘material implication.  Lewis however insisted the only genuine implication was strict implication: p=implies only if it is impossible that p should be true and q false. He felt that p strictly implies q was the same as q follows logically from p.

He created axiomatic systems which were the first formal systems of modal logic where the sign for material implication was replaced by a new sign to represent strict implication. But many critics felt that it was hardly less paradoxical than material implication, as an impossible implication strictly implies every proposition and therefore a proposition such as, ‘if cats are dogs then pigs can fly’ is true.

Lewis offered five different axiom systems numbered S1-S5, showing that each set was consistent and independent.  They each vary in strength. In S4; reading ‘if’ as a material, not strict, implication with the flowing additional axioms:

1.       If necessarily p, then p.

2.       If necessarily p, then 9if necessarily[if p then q] then necessarily q)

3.       If necessarily p, then necessarily necessarily p.

The system exploits the interdefinability of necessity. The relative merits of S4 and S5 as a system of modal logic remain a matter of debate today for logicians and philosophers. Philosophers of religion for instance have argued about the existence of god with S4 and S5.

If quantification is introduced into modal logic, and modal operators and quantifiers are used together, the system resembles double quantification.  In quantified modal logic it is important to mark the order in which the operators and quantifiers are placed. It is important to distinguish between, for example:  ‘For all x, x is possibly F’ is not the same as ‘it is possible that for all x, x, is F’ because in a fair lottery, everyone has the chance of winning but there is no chance of everyone being the winner. Sentences where the modal operator precedes the quantifier, like the second sentence in the example are called modals de dicto, and sentences where the quantifier comes before are called modals de re.

There are also important differences between modal logic and quantification theory; when we introduce identity.  Modal logics are referentially opaque but quantificational contexts are not. The example given is that if P is a sentence containing A, and Q is a sentence resembling P in all respects except that it contains B where P contains A, then P is referentially opaque  if P and E do not together imply Q; and therefore modal contexts are seen as opaque. Some logicians rejected modal logic all together because of its opacity, but it was made respectable in the 1960’s.

One of the ideas of modern modal logic is to look at similarities between quantification and modality by defining necessity as truth in all possible worlds.  Talk of possible worlds does not need to involve any metaphysical implications. One possible world may not be accessible from another. The iteration of modalities is now explained in terms of relationship to be defined between different possible worlds.  It shows that the different world’s accessibilities are the same.

Philosophers have extended the notion of modality into many different contexts, such as the logic of time where always corresponds to ‘necessary’ and ‘sometimes’ to ‘possible’. Problems with referential opacity can be dealt with by making a distinction between two different kinds of reference.  To be a genuine name a term must be a ‘rigid designator’; it must have the same reference in every world.