Philosophy: Critical Reasoning

Sentential Logic Steps

Simplify, do a truth table, and read the truth table to determine validity

Grammar Rules

1.) Any sentence letter is grammatical

2.) If P is grammatical, then so is ~P

3.) If P and Q are grammatical, then so are P&Q, PvQ, P→Q, and P ←→Q

4.) Nothing is grammatical if it can’t be constructed by way of 1-3

Main connective

The connective you would add last

Turnstyle P

The opposite of P, meaning that P is false

P ampersand Q

Both P and Q are true

P wedge Q

At least one of P and Q is true

P arrow Q

It is not the case that P is true and Q is false

P double arrow Q

P and Q have the same truth value

Truth Table for a sentence

1.) write alphabetical order of sentence

2.) set up table

3.) Read table

How to read a truth table

If the premises are true and the conclusion is false, it is an invalid argument; otherwise, it is valid

How to find the rows for a truth table

The rows are equal to n² where n is the number of letters

Arguments

Conclusion follows from premises

English to SL rules

Not P= ~P

P and Q= P&Q

P or Q= PvQ

If P then Q= P → Q

P if and only if Q= P ←→Q

English to English rules for P and Q

But

Although

Yet

Moreover

However

English to english for unless

if not

English to english for if

provided

when

Note: you switch the order in the case of only if

English to english for if and only if

just in case

when and only when

precisely when

English to english for if P then Q

P only if Q

Q if P

Note for sentences

Order matters for if then sentences

What is an argument

An argument is a set of two or more sentences; no single sentence is an argument

What is validity

An argument is valid if and only if there are no possibilities in which both the premises are true and the conclusion is false

What is invalidity

An argument is invalid if and only if it is not valid

Soundess

An argument is sound if and only if it is valid and all of its premises are true

Unsoundness

An argument is unsound if and only if it is not sound

What do you need for a good argument

We want good logic with good arguments

What about mathmatics

Mathmatical truths hold in all possibilities

Logical equivalence

has to do with sentences not arguments

Contraposition

Equivalence of sentences

Demorgan’s Law