Logic and Language

Logic:

Induction and Abduction in Pierce:

Pierce investigated the structure of scientific inquiry. Whilst deductive logic assists us in organising our knowledge, reasoning that extends our knowledge, which Pierce called ampliative inference he split into three categories.

Ampliative inference is a statement that takes a conclusion, or several and from that derives a generalised conclusion that whilst not empirical, appears to be correct.

The closest term that comes to mind when trying to explain ampliative arguments is the term ‘educated guess.’ Because whilst the overall conclusion is a priori in nature, it is itself based upon empirical knowledge.

So an example of such an argument would be: ‘Fire ants and army ants from colonies, therefore all ants form colonies.’

Thus whilst the conclusion inferred is logical, it is not valid because not all species of ant have been discovered, though from the empirical information we have, the most probable conclusion is the above.

Pierce stated ampliative inference can be split into three categories, these being; induction, hypothesis and analogy, all of which depend on sampling.

In relation to deductive arguments:

Deductive reasoning is where the conclusion is necessarily true if the premise is true. However deductive arguments may have false premises.

For example: ‘All men are mortal; Cleopatra is a man; therefore, Cleopatra is mortal.’

The premise is that all men are mortal and that Cleopatra is a man. Whilst based on knowledge of the time one might view these both as true i.e. there have been no immortal men and Cleopatra dressed as a man to gain more respect thus one could mistake her as such. They do not by themselves prove the conclusion ‘Cleopatra is a mortal’ and are instead at most equal to substantiating evidence which itself is not empirical in nature.

This statement is in fact another example of an ampliative argument whereby two known conclusions are used to derive a third which is based off of the two conclusions rather than evidence such as death which is empirical.

Pierce goes on to define the methods of scientific enquiry. The method of scientific enquiry is as follows: Scientists create a hypothesis based upon a theory or problem, acknowledging past explanations for the phenomena. From the hypothesis they create a prediction, i.e. if my hypothesis states this, then this what I expect to occur. The hypothesis is then tested and the observations should either confirm or refute the hypothesis.

From here a new hypothesis can be created and tested. So if the experiment was a success, the hypothesis will reflect the observations made and be tested again. If it refutes the hypothesis then the hypothesis will be changed to reflect that and the method of scientific enquiry begins again.

Pierce simplified this into three stages called abduction, deduction and induction:

The abductive phase is whereby the inquirer chooses the theory they wish to investigate. The theory chosen however must fall within certain parameters or as Pierce calls it ‘the rules of the logic of abduction.’ That the theory must be empirically testable and cohere to existing knowledge in an objective capacity.

The deductive phase is where the inquirer creates a way of testing this theory. Dedction follows abduction through the verification or falsification of the predications created which will either confirm or refute the hypothesis in question.

The inductive phase is where the results of the test are evaluated. Induction as described by Pierce seems to go at odds with deduction for he states that we only infer provisionally and whilst through constant experimentation our conclusions will become indefinitely close to the truth, the fact we must continue to experiment and going at odds with even our oldest theories means there is no deduction to be had but ampliative inference.

Thus I believe that when Pierce talks of deduction he talks of it strictly in the context of the experiment. That an experiment for example has proven something to be true, but when this result is placed in the context of the wider world we can only infer from it, leading to quantitative induction or simply induction whereby mathematical probability alone dictates this is the most reasonable conclusion to draw. That is until new evidence comes along that may support or refute that which commonly accepted as knowledge.

Consequently in science there is no such thing as fact but only theory, deductive reasoning only goes so far. However even without deduction we can hold something to be true due to the evidence that supports it and the probability that it is indeed correct such as theories concerning evolution, gravity and plate tectonics.

As well as quantitative induction there is also qualitative induction which is to infer characteristics of one individual to another Pierce demonstrates this with the concept of the mugwump:

‘He has high self-respect and places great value on social distinction. He laments the great part that rowdyism and unrefined good-fellowship play in the dealings of American politicians with their constituency.... He holds that monetary considerations should usually be the decisive ones in questions of public policy. He respects the principle of individualism and of laissez-faire as the greatest agency of civilisation. These views, among others, I know to be the obtrusive marks of a ‘mugwump.’’

The idea behind a mugwump being you understand what a mugwump is and the characteristics associated with it, thus when you meet someone who displays similar characteristics you believe them to be a mugwump. This is called hypothetic inference whereby you infer characteristics a person has not shown through the ones they have shown. For example let’s say a mugwump has 5 characteristics and you meet a person who displays 3 of these, you then infer that they are a mugwump and thus have the other 2 characteristics as well.

The saga of Principia Mathematica:

Before one can understand the various truth tables one must understand all the symbols that it is comprised of:

Firstly a proposition is a statement which is established and held to be true only with evidence i.e. a proposition is established through deductive logic. Thus propositions are Boolean in that they can only be true or false.

Negation:

~ is the sign used for negation (not).

Negation in logic is where a proposition is held to be true then the opposite must be false and vice versa. So the negation of a proposition can only be true when it is false.

An example of negation would be:
Proposition: This is a book
Negation: This is not a book

Disjunction:

V is the sign for disjunction (or).

Disjunction is where two propositions are joined by ‘or’ to form a proposition which is only false if both of its component parts are false. However whilst seeming contradictory both component propositions can be true at the same time.

For example:
Proposition one: The cat hasn’t been fed.
Proposition two: The cat is just hungry
Compound proposition: The cat hasn’t been fed or the cat is just hungry.

Truth functional:

A horseshoe shape or an arrow represents the truth functional (if). (I can't import the symbols from word for some reason).

A truth-functional is where the truth value of a proposition depends on the truth of its component parts. If one part is false then the overall proposition is false. Negation also falls under truth function as if something is true the negation is false.

Conjunction:

Conjunction is represented by p.q or p & q (and).

Conjunction is whereby a proposition is made up of two component propositions which mst both be true for the proposition to be true.

For example:

Proposition one: Ants are small
Proposition two: Ants are fast
Compound proposition: Ants are small and fast.

Whilst both propositions might be true, if one were false then the compound proposition is false as it is based on the predicate that both component propositions are true.

Principia is an axiomatic system in which logical truths are by rule from a select few axioms. Whereas Frege’s initial set were ‘if’ and ‘nt’ from which all others could be defined, Russell and Whitehead took ‘or’ and ‘not’ instead.

Wittgenstein developed an alternative to axiomatic systems which was the truth table, a new way to give logic a rigorous form in an easily understood format.

p             q             p & q
T             T             T
F             T             F
T             F             F
F             F             F

The table demonstrates conjunctive statements, how their component values reflect the truth or false value of the overall proposition. The compound proposition is thus a truth function of component propositions.

Of Wittgenstein’s truth tables emerge the concepts of tautologies and contradictions. A tautology is a proposition which is true for all truth possibilities of its component propositions and similarly in contrast a contradiction a proposition false for all truth possibilities.

Tautologies are the equivalent to axioms and theorems of Frege’s system and vice versa. However the truth table is superior to the axiomatic system in several ways. For example it represents all logical truths on the same level whereas Frege’s system is based on an arbitrary set of axioms.

A problem of language and the theory of descriptions:

A problem arises when something non-existent is used in a phrase, because by talking about such a concept you are giving some sort of existence to it. So for example the statement: ‘Golden frogs do not exist.’ Denotes some sort of existence to the idea of a golden frog which you deny.

Thus the theory of descriptions exists to counter such problems by removing conjunctive declarations from propositions. 

For example a sentence that does not have the word ‘and’ such as: ‘The black cat caught a mouse’ can be simplified too: ‘A Black cat exists and it caught a mouse.’

We denote the generic ‘cat’ as p and specific ‘black cat’ as q so that the statement can be simplified further as:

‘There is an entity p such that the statement; ‘q caught the mouse’ is true if q is p and false otherwise; moreover p is the black cat.

‘Moreover’ means that the ‘black cat’ exists (or existed or will exist).

So using this method we can deconstruct the statement; ‘golden grogs do not exist’ in the same way.