Logic
John Stuart Mill wrote System of Logic, which set forth a system of formal logic. Mill wanted to disassociate himself from the work of Hobbes, which he accused of nominalism. Nominalism, to Mills, was the theory that a proposition can only be true if the subject and predicate are names of the same thing.
Mill argued that this worked in situations where both predicate and subject were proper names, but was not suitable for proving other propositions, where this was not the case.
It should be noted that, unlike Hobbes, Mill believed that a name could not just be a specific name such as 'John Mill', but a descriptive term such as 'man' or 'wise'. Mill also believed that the laws of physics, arithmetic and logic only existed as well-confirmed generalisations.
Mill was in opposition to Frege on these points. Frege believed that arithmetic and logic were both known a priori. However, Frege's contributions to logic were in moving away from the simple subject-predicate method of logic that was used by Mills and those before, and his formulation of arguments and functions, concepts that will be familiar to any software developers out there.
Frege's system works as follows:
Tyson beat Bruno
In the sentence above, the term 'beat' would comprise the function, while 'Tyson' and 'Bruno' would each be arguments of the function. Frege developed an entire system of notation for illustrating logical problems in this manner.
Frege's innovations were independently arrived at by C.S. Peirce, an American philosopher. However, Peirce's contributions were largely overlooked at the time of their creation, and (like Frege) the true quality of what he had achieved was only recognised posthumously.
One of Peirce's arguments, was that our forms of inference (induction, hypothesis and analogy) all depend on sampling, therefore he argued, the mathematical laws of probability are necessary for non-deductive inference to occur. Peirce also argued that it was by the ongoing extension of our experience, that we gain proficiency at generating hypotheses.
Peirce's work, while valuable, was not closely examined by Russell and Whitehead, who instead concentrated on Frege's Begriffsschrift when creating their own three volume work entitled the Principia Mathematica. The Principia Mathematica received much greater attention than either Frege or Peirce's work. This is possibly due to the replacement of the system of symbols that Frege used with a more accessible form of notation.
Russell and Whitehead's work drew largely from Frege's work, but made use of a different set of axioms to Frege's. Where Frege used 'if and 'not' as his primitive connectives, Russell and Whitehead used 'or' and 'not'. Their term for these connectives was 'logical constants'.
However, it was Wittgenstein who first realised that 'axiomatic' systems were not the only (or most intuitive method for showing logic in a rigorous form. Wittgenstein was the first to use truth tables, a simple and elegant method for setting out the constituent components of a proposition and the proposition itself.
For example, if the proposition was to see under which circumstances 'p and q' were true, a table could be laid out as follows:
The work of Frege, Russell and Wittgenstein was criticised in some quarters. A group of logicians known as 'intuitionists', and founded by L.E.J. Brouwer, considered mathematics to be a construct of the human mind, and therefore they considered that only demonstrable mathematical proofs could be assigned the value of truth. That is, they did not accept that the existence of 'p' was proven by disproving 'not p'.
Frege, Russell and Whitehead had hoped, as part of their work, to establish that arithmetic was a branch of logic. However, Kurt Godel, an Austrian logician and philosopher, published a paper in 1931 entitled 'On Formally Undecidable Propositions of Principia Mathematica and Related Systems' which proved conclusively that arithmetic was not a complete system, by constructing a formula within the system used by Principia Mathematica that can be shown to be true, but is not provable within the system itself. This paper ended attempts to prove that arithmetic was a complete system in itself.
Focus then switched to a previously neglected form of logic known as Modal Logic, which had largely been neglected since the Middle Ages. Interest in this form of logic was revived by C.I. Lewis who was interested in implication. Lewis believed that Russell and Whitehead's work regarding 'material implication' was flawed, and suggested that the only genuine implication was known as strict implication. However, critics considered Lewis's strict implication to be no less flawed than the work carried out by Russell and Whitehead.
Further study has taken place in logic in the years since Russell, Frege, Wittgenstein et al. But many of the conventions and developments that they were responsible for are still used by modern logicians today.
Peirce's work, while valuable, was not closely examined by Russell and Whitehead, who instead concentrated on Frege's Begriffsschrift when creating their own three volume work entitled the Principia Mathematica. The Principia Mathematica received much greater attention than either Frege or Peirce's work. This is possibly due to the replacement of the system of symbols that Frege used with a more accessible form of notation.
Russell and Whitehead's work drew largely from Frege's work, but made use of a different set of axioms to Frege's. Where Frege used 'if and 'not' as his primitive connectives, Russell and Whitehead used 'or' and 'not'. Their term for these connectives was 'logical constants'.
However, it was Wittgenstein who first realised that 'axiomatic' systems were not the only (or most intuitive method for showing logic in a rigorous form. Wittgenstein was the first to use truth tables, a simple and elegant method for setting out the constituent components of a proposition and the proposition itself.
For example, if the proposition was to see under which circumstances 'p and q' were true, a table could be laid out as follows:
| p | q | p & q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
The work of Frege, Russell and Wittgenstein was criticised in some quarters. A group of logicians known as 'intuitionists', and founded by L.E.J. Brouwer, considered mathematics to be a construct of the human mind, and therefore they considered that only demonstrable mathematical proofs could be assigned the value of truth. That is, they did not accept that the existence of 'p' was proven by disproving 'not p'.
Frege, Russell and Whitehead had hoped, as part of their work, to establish that arithmetic was a branch of logic. However, Kurt Godel, an Austrian logician and philosopher, published a paper in 1931 entitled 'On Formally Undecidable Propositions of Principia Mathematica and Related Systems' which proved conclusively that arithmetic was not a complete system, by constructing a formula within the system used by Principia Mathematica that can be shown to be true, but is not provable within the system itself. This paper ended attempts to prove that arithmetic was a complete system in itself.
Focus then switched to a previously neglected form of logic known as Modal Logic, which had largely been neglected since the Middle Ages. Interest in this form of logic was revived by C.I. Lewis who was interested in implication. Lewis believed that Russell and Whitehead's work regarding 'material implication' was flawed, and suggested that the only genuine implication was known as strict implication. However, critics considered Lewis's strict implication to be no less flawed than the work carried out by Russell and Whitehead.
Further study has taken place in logic in the years since Russell, Frege, Wittgenstein et al. But many of the conventions and developments that they were responsible for are still used by modern logicians today.
