PHI00005 C 2008-2009 Lecture 4 - Conjunction and negation PDF

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Reason and Argument Lecture 4: Conjunction and Negation

Lecture 4

1

Logical Negation We will symbolise negation with “~” (Tilde)

Truth Table:

A

~A

T

F

F

T

The logic of “~” is the logic of “It is not the case that” in English.

But “It is not the case that” is rarely used.

Lecture 4

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“Not” Negations in English often involve “not” inserted into a sentence:–

Barry is not a smoker

Christina Aguilera is not Patti Smith

It is not five o’clock

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Inserting “not” into a sentence does not always yield its negation Two Principles of Negation If a sentence is true, its negation must be false. If a sentence is false, its negation must be true.

Example of “not” insertion failing to delivering a negation: (i)

Some of Tom’s friends are philosophers

(ii)

Some of Tom’s friends are not philosophers

(ii) is not the negation of (i), because it might be that both were true. The negation of (i) is: (iii)

It is not the case that some of Tom’s friends are philosophers.

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Paraphrase Re-phrasing a natural language expression to allow it to be dealt with in terms of our defined logical language.

Example:

Tom is not easygoing.

It is not the case that Tom is easygoing.

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On Checking Paraphrases One task we’ll have to do regularly is check that a supposed paraphrase of a sentence doesn’t actually mean something different.

It’s important to make sure that there’s no way for a supposed paraphrase to differ in truthvalue from the original sentence.

If there’s a situation in which one is true and the other’s false, they obviously can’t mean the same.

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Example (i)

Harry wants Bill not to leave.

(ii)

It’s not the case that Harry wants Bill to leave.

These differ in meaning. (i) can be false while (ii) is true.

Question: Under what sorts of circumstances?

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Contraries Two sentences are CONTRARIES if they cannot both be true.

e.g. There are more than 100 people in this room There are fewer than 100 people in this room

Contradictories Two sentences are CONTRADICTORIES if they cannot both be true and cannot both be false.

For example, ‘There is someone in this room’ and ‘No one is in this room’.

Logical negations are always contradictories.

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People came, but no-one ate pasta People came, and no-one did not eat pasta Can these both be true? No. So they’re at least contraries. Can they both be false? Yes, if someone ate pasta and someone did not. So, they are not contradictories. The winner ate pasta The winner did not eat pasta Can these both be true? No. So they’re at least contraries. Can they both be false? Yes, if there was no/no single winner So, they are not contradictories. Lecture 4

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One Kind of Ambiguity Consider: It is not the case that I ate all the pies and I drank all the beer This could be understood as a simple denial, or as part-denial, part-confession.

In English, we disambiguate in this kind of case by using commas (in speech, pauses): It is not the case that, I ate all the pies and I drank all the beer. It is not the case that I ate all the pies, and I drank all the beer. In propositional Logic, we use brackets…

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With:– A: I ate all the pies B: I drank all the beer

A

B

T

T

T

F

F

T

F

F

~(A & B)

(~A & B)

The denial is: ~(A & B)

And the part-denial, part-confession is: (~A & B)

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Structural Ambiguity Ambiguity of this kind is called structural ambiguity, because it depends on how we see the sentence as being built up structurally from basic components. The relatively simple denial, for instance, has as a component the conjunction (A & B). The part-denial, part-confession, on the other hand, has as a component the negation ~A. [Aside: We can have structural ambiguity in English which isn’t a matter of sentential clauses. E.g. ‘Pembleton is a violent crime investigator’ can mean either that Pembleton is a violent investigator of crimes, or that Pembleton is an investigator of violent crimes.]

Lecture 4

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Ruling Out Ambiguity In constructing our logical language we rule out structural ambiguity by making sure that each string of symbols which counts as a sentence of our logical language has only one meaning.

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Grammar for Propositional Logic Symbols: Statement Letters: “A”, “B”, “C”, etc. Connectives: “&” and “~” Scope indicators (Groupers): “(“ and “)” Rules: Statement letters are well-formed formulae (wffs) If “A” is any wff, then “~A” is a wff. If “A” and “B” are wffs, then “(A & B)” is a wff (Brackets may be left off if not needed for disambiguation) Nothing not ruled a wff by the other rules is a wff

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WFF Exercise A&B

~A

~~A [Note: This is called a “double negation”. Under what conditions is it true?}

~A & B

A&B&C

A & ~B (& C)

~(A & ~B) & C

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Scope The Scope of an expression is the smallest wff in which it occurs In ~(A & B) we say that the tilde has wider scope than the ampersand. In (~A & B), the ampersand has wider scope than the tilde. In every complex statement one connective (token of either ~ or &) has wider scope than any other. The connective with widest scope determines the kind of a complex statement, e.g. whether it is a conjunction or a negation. The connective with widest scope is the main connective of the sentence.

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Truth Tables Again We can construct truth tables to determine under what sorts of circumstances a complex statement is true. e.g. ~(~(A & ~B) & C) If we’re going to construct a truth table for this statement, how many rows must it have?

A

B

C

T

T

T F

F

T F

F

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T F

F

T F

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A

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Lecture 4

~

(~ (A

&

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&

C)

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(~ (A

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Lecture 4

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(~ (A

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(~ (A

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~

&

C)

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A

B

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(~ (A

&

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B)

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Lecture 4

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&

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A

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(~ (A

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Lecture 4

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A

B

C

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(~ (A

&

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&

C)

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Testing for Validity Is the following form of argument valid? ~(~(A & ~B) & C) ———————— ~(~A & C)

A B C ~(~(A & ~B) & C)

~ (~ A

& C)

T T T

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T

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T T F

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Summary ~A is true iff A is false Testing paraphrases/representations. Two sentences are contraries iff they cannot both be true Two sentences are contradictories iff it must be that exactly one of them is true Sentences of English can be structurally ambiguous. The grammar of PL rules out ambiguity. Truth tables can be constructed to show the truth values of complex PL sentences in all logically possible circumstances.

Truth tables can be used to test for validity. Lecture 4

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