LECTURE 5 - Counter-Examples and Necessary and Sufficient Conditions

Disjunctive Syllogism

(P1) Either it is day or it is night (P1) p or q (P2) It is not night (P2) not-q (C) So, it is day (C) So, p

Conjunction mode 2

(P1) It is day (P1) p (P2) It is light (P2) q (C) So, it is day and it is light (C) So, p and q

Conjunction: mode 1

(P1) It is day and it is light (P1) p and q (C) So, it is day (C) So, p

Stipulative definition

A Stipulate definition is a statement which arbitrarily fixes down the meaning of a word Generally, one is free stipulate, whatever meaning ones cares to: just say, 'Let's stipulate that X means Y' Used when a new thing is discovered, or To be totally clear on the meaning of terms; or, To be economical in expression But stipulative definitions cannot be true nor false.

A definition of 'definition'

A definition is a statement which gives the meaning of a word

useful conceptual analyses:

A is a sufficient condition for B iff if A then B A is a necessary condition for B iff if B then A A is a necessary and sufficient condition for B iff A if and only if B An ARGUMENT is VALID iff (a) if the PREMISES are true, then the CONCLUSION must be true. (a) is both necessary and sufficient for an argument to be valid. An ARGUMENT is SOUND iff it is (a) VALID and (b) has true PREMISES (a) and (b) are each necessary conditions on soundness Together (a) and (b) are sufficient for soundness. These last examples again show that you can learn something from definitions and conceptual analysis, including things about definitions and conceptual analysis!

Lexical definition

A lexical definition is a statement which records meaning of a term as it is in fact used. If you look in a dictionary, you'll find many examples of definitions like this. Here is how a dictionary might define 'human': 'human, noun: a human being, especially a person as distinguished from an animal, or (in science fiction) an alien' This is good enough to give a rough sense of how the word 'human' is used; Such definitions can be true or false, since the term may not be used in the way described.

Prescriptive definition

A prescriptive definition is a statement which claims that a certain meaning of a term is the correct meaning of that term. Common in linguistics and amongst snobs and newspaper sub-editors. But very useful in philosophy too, as they tighten up or focus on one aspect of a commonly used term. You might want only to focus on that aspect in an essay. For example: "By 'just', Socrates means 'has not broken any promises" In this case, we prescribe how Socrates uses a certain term. Can be true or false: there is a fact of the matter about whether a certain term ought to mean something

Conceptual Analysis: definition as a philosophical project

A term is a linguistic item -> We can define it A concept is a mental item -> We can analyse it A conceptual analysis gives the necessary and sufficient conditions for something to fall under that concept.

famous sorts of inductive argument: Inference to the Best Explanation

An IBE argument asks what best explains the facts expressed in the premises. That best explanation is the conclusion. Compare an inference to the non-best explanation: (P1) There are huge, humanoid footprints in the snow (C) A Yeti walked past here With an inference to the best explanation (P1) There are huge, human-shaped footprints in the snow (C) A human in big snow-shoes walked past here

famous sorts of inductive argument: Argument from analogy

An argument from analogy argues that two things, a and b are similar in one or more respects so, a and b are similar in another respect. Formally: (P1) a and b are similar is respect F (C) So, a and b are similar in respect G

Stipulate meaning

Arbitrarily fixes down the meaning of a term

famous sorts of inductive argument: Argument to future projection

Arguments to future projection take a historical trend as premises and conclude that it will continue into the future. (P1) The first raven I saw is black (P2) The second raven I saw is black (P3) The third raven I saw is black (C) So, The next raven I see will be black

Prescribe meaning

Claim that one meaning in common use is the right meaning

Circular

Clearly this analysis has not made much progress, one of the conditions on the RHS contains the notion that was supposed to be analysed. This is usually a problem, since we haven't reduced 'freezes' to other notions. But its not to say that we have gained nothing: for example, we know that only liquids can freeze. To fix it, we'd have to replace (b) with something like: (b') it becomes solid.

Record meaning

Describe how a term is actually used, without saying it is the right meaning

Hypothetical syllogism

Hypothetical syllogisms can be chained together indefinitely! (P1) If it is night, it is dark (P1) If p, then q (P2) If it is dark, it is scary (P2) If q then r (C) So, if it is night it is scary (C) So if p, then r

Why care about definitions?

One reason is that we are not 'talking past each other'. If not all arguers in a debate are using terms in the same way, then they cannot make progress. By having clear definitions we can avoid fallacies of equivocation. By defining technical or philosophically loaded terms in the thesis of an essay, we can show that we're good philosophers! Another reason might be philosophical: I can learn something about the world and how humans think about it, by giving a really good definition of a term we use. Sometimes this is called 'conceptual analysis'.

Conditional conclusions

Some arguments can have a conditional or a quantified proposition as a conclusion:

Form of an analysis:

Something is A if and only if (a) condition 1; (b) condition 2; (c) condition 3 (d)... Necessary conditions (B) are what something must have to be an A In other words, necessary conditions are conditional: if A then B Sufficient conditions (B) are enough for something to be an A In other words, sufficient conditions are a conditional: if B then A

Parts of a definition

Term to be defined (definiendum), e.g. 'human' Statement that does the defining (definiens) e.g. 'a human being, especially a person as distinguished from an animal, or (in science fiction) an alien'

How to criticise a conceptual analysis

There are three ways an analysis can be defective: it includes part of itself (it is circular) it includes too many things (it is too weak) it includes too few things (it is too strong).

Too Strong

This analysis is too strong because it includes too few objects as a human. Not every human plays tennis. To fix this analysis, we'd have to drop this condition.

Too Weak

This analysis is too weak, because it includes too many objects as a bachelor. It is not strong enough to exclude boys. We don't think that boys are bachelors, even though they are male, unmarried and human. To fix it, we'd have to add: (d) is an adult.

Modus Ponens (Mode that affirms)

This argument form has a conditional premise and a premise that affirms the antecedent. (P1) If it is night, it is dark (P1) If p, then q (P2) It is night (P2) p (C) So, It is dark (C) So, q P2 is false, but the argument is still valid

Modus Tollens (Mode that denies)

This argument has a conditional premise and a premise that denies the consequent. It concludes that the antecedent must be denied. (P1) If it is night, it is dark (P1) If p, then q (P2) It is not dark (P2) not-q (C) It is not night (C) So not p

An argument is valid

just in case if the premises are true, then the conclusion must be true.

An argument is invalid

just in case the premises could be true and the conclusion be false

A counter-example

set of premises and conclusion with the same form as an invalid argument, but where the premise are true and the conclusion false. A counter-example shows an argument to be invalid.