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Reason and Argument

Lecture 3:

The Project

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Review Logic: The systematic study of (good) reasoning Statement: What is expressed by an indicative sentence that is apt to be true or false Argument: A set of statements, divided into premisses (0 up to infinitely many), and conclusion (1 only), with premisses considered as support for the conclusion. “Support”: Premisses support conclusion where, were one to believe them, that would give one reason to believe the conclusion. Validity: An argument is valid if and only if If the premisses are true, then its conclusion must be true An argument is valid if and only if It is not possible for it to be the case that: both (i) the premisses are all true, and (ii) the conclusion is false.

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Formal Validity Some arguments seem to be valid just on the basis of their logical form (their logical shape) Logical form: Roughly, the pattern of components making up statements and arguments. Examples: Tom is easygoing or Tom is a philosopher It is not the case that Tom is easygoing —————————— Tom is a philosopher There will be talks or hostilities will break out It is not the case that there will be talks —————————— Hostilities will break out α or β It is not the case that α ————————— β

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Our Project To investigate the logic of some expressions in English which seem to play a key role in a wide range of formally valid arguments. Our main targets: “and”, “not”, “or”, “if… then…” Aims: —to be able to represent arguments in a systematic way —to be able to evaluate arguments for validity in a systematic way

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Simple Statements and Sentence Connectives Key Point: Some sentences are built of from one or more sentences which could express statements in their own right, plus some additional words. (1) “Tom is easygoing or Tom is a philosopher”

Components: “Tom is easygoing” “or” “Tom is a philosopher” (2) “It is not the case that Tom is easygoing”

Components: “It is not the case that” “Tom is easygoing” The additional words here are often called sentence connectives, because they can be used to build bigger sentences by having smaller sentences plugged into them. Simple statements are, roughly, ones which lack this kind of internal logical structure.

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“Propositional Logic” Our main targets, “and”, “not”, “or” and “if…

then…” are sentence connectives.

The logic of these expressions is often called

Propositional Logic, because its basic

components — the simple statements —

express propositions (roughly, claims).

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Logic and Meaning Arguments of the form α or β It is not the case that α ————————— β are valid because of the meaning of “or” and “it is not the case that”. Our investigation is, in part, an investigation into the meaning of the logical words.

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Our Strategy To give some sentence connectives particular

logical powers by stipulative definition,

then see how far we can capture the logic of the

English expressions in terms of these

connectives.

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Logical Conjunction

Symbolised with “&” (“ampersand”).

Where α and β express claims/statements ‘α & β’ is the conjunction of α and β

Definition: A statement θ is a logical conjunction iff it has component statements α and β, and

if α is true and β is true, then θ must be true, and if θ is true, then it must be that α is true and β is true.

[Note: Lepore’s formulation uses “just in case”. “Just in case” is US philosopher’s English for “if and only if”.]

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Logical Conjunction (Continued) The definition of logical conjunction guarantees that the following forms of argument are valid: α & β. Therefore, α.

α & β. Therefore, β.

α. β. Therefore, α & β.

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Logical Conjunction

It is true that α & β if, and only if, it is true

that α and it is true that β.

Alternatively (if you don’t like the fact that ‘and’ turns up on the right-hand-side):

It is true that α & β if, and only if, both of the

following are true: α, β.

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‘Truth Value’ If a statement is true, we say it has the truth-value true. If a statement is false, we say it has the truth-value false. The expression ‘truth value’ is mainly useful

because it allows us to talk about the condition

of a statement with respect to truth and falsity

without having to use the expression

‘the condition of a statement with respect to truth

and falsity’.

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Truth Tables We can show, in tabular form, how the truth or

falsity of a conjunction depends on the

truth/falsity (the “truth value”) of its

components.

A B A & B T T T T F F F T F F F F

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Truth Functions

Logical conjunction is a truth function:

That is, the truth-value of a conjunction is

(always) fixed by the truth-values of its

component statements (claims).

Equivalently, it has a complete truth-table.

Once the truth-value of the components is

determined, no further information is needed to

determine the truth-value of the logical

conjunction.

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Not all sentence connectives are truth functions Example: “It is possible that…”

A It is possible that A T F

“Barry is wearing a pink shirt” is false “2 + 2 = 3” is false “It is possible that Barry is wearing a pink shirt” is true “It is possible that 2 + 2 = 3” is false

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Finding Logical Conjunctions in Natural Language Christian smokes and Barry drinks Christian smokes & Barry drinks Stephen smokes and drinks Stephen smokes & Stephen drinks Christian and Stephen smoke Christian smokes & Stephen smokes Barry, Christian and Stephen drink Barry drinks & (Christian drinks & Stephen drinks)

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More Conjunctions in Natural Language He (Oliver) was poor but he (Oliver) was honest Although it was raining, he did not take an umbrella I enjoy philosophy, however, I find it hard work Stephen drank; Barry cleared away the bottles.

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Testing Our Theories

We can test theories about the meanings of

sentence connectives.

The stipulative definitions of our invented

connectives tell us, e.g., what the truth-value of

an ‘&’-sentence is in a particular situation.

(e.g. Where two sentences A, B are both true,

‘A & B’ is true.)

Our ordinary language skills tell us whether the

corresponding ‘and’-sentence is true in that

situation.

Test: Are there any cases where ‘A & B’ is true,

and ‘A and B’ is not?

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A Difficulty for Understanding “and” as “&”? Consider:

(1) Dave bought Barry a Rolex and Dave got

first class marks for Reason and Argument.

If someone said this, they might seem to say

more that that both of the following are true:–

(i) Dave bought Barry a Rolex (ii) Dave got first class marks for Reason and

Argument But is (1) false if there’s no connection? There may be something wrong with saying (1) if there’s no connection, but we need to be careful…

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Logic and Conversation Don’t confuse what’s merely suggested by the

fact that someone uses particular words with

what those words (and sentences) mean.

Classic Example:

Ann: Is Ben good at logic?

Caz: His writing’s really neat, and he buys me

a coffee after every lecture.

Caz has suggested that Ben is not good at logic.

But the sentence ‘His writing’s really neat …’

etc. does not mean that (even here).

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‘Implicature’

This kind of suggestion in conversation is often

called implicature or conversational implicature

by philosophers.

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Logic and Conversation (Continued) Implicatures and ‘Cancellation’

(Mere) implicatures can be cancelled:–

Dave bought Barry a Rolex and Dave got

first class marks for Reason and Argument,

but there’s no connection between the two

events.

If connection was part of the meaning of “and”,

the attempt at cancellation would result in

contradiction:–

Dave bought Barry a Rolex and (i.e. this is

connected with the fact that) Dave got first

class marks for Reason and Argument, but

there’s no connection between the two

events.

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Summary

Some arguments are formally valid

Some arguments are formally valid because of

simple statement/sentence connective structure

Some sentence connectives are truth-functional

The significance of sentence connectives in

natural language is a matter for investigation

Strategy: Invent connectives, stipulate their

meanings, test hypotheses about NL sentence

connectives

Beware implicature effects!

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Aside: ‘If, and only if’

An argument is valid if, and only if,

it is not possible for all of its premisses to be true

and its conclusion false

2 parts to this:

The ‘if’ part:

If an argument is such that

it is not possible for all of its premisses to be true

and its conclusion false,

then it is valid.

The ‘only if’ part:

An argument is valid only if

it is not possible for all of its premisses to be true

and its conclusion false