Rules of Replacement

LogicA very elementary introduction

Rules of Replacement

Demorgan’s Theorems

Some basic laws of logic The law of non-contradiction

A is not ~A The law of identity

A=A The law of excluded middle

statements that have a truth value are either possibly true or possibly false not half true or half false

Types of Logic Symbolic Logic Modal Logic Propositional Logic 1. Propositional logic is also called propositional

calculus is it includes the following A) consonants, sentential connectives such as if,then,and B) existing rules of inference

What is a proposition? A proposition is a statement of thought that

is expressed in language. [Can be any human language] This statement has a truth value.

For example: Boiling water is hot. This is either true or false.

Not all sentences are propositions Sentences such as: Go outside and play

ball have no truth value.

Sense and Reference Sense: The meaning of a statement Reference: The state of affairs of the

universe to which my utterance points.

Antecedents What goes before. In an if, then statement

the antecedent would be “if” portion.

For example: If it rains then wear a jacket.

Consequent What follows after. The consequent is the

then portion. Using our last example If it rains then wear a jacket

Jacket here is the consequent.

Soundness vs. Validity Valid arguments contain true premises

therefore the conclusion that follows must also be true. It is possible for an argument to be factually untrue but logically valid.

Soundness on the other hand refers to a valid argument that contains factually true premises.

Truth Functional connectives

Truth functional connectives link propositions together. For example V or vel stands for “or” the dot . Stands for “and” these truth functional connective link together logical statements.

Causation and Logical relations Logical relations do not account for

contingencies. For example if we were to look at the causal relationship between my throwing a rock and it breaking a window we would have to examine the force of my throw, the thickness of the window, the distance, the thickness of the rock, the timing of my throw, the arm I am using, etc.

Deductive Nomological account The logician Carl Hempel argued that for

every antecedent cause x, the consequent y must by necessity happen.

Implication or Material Equivalence

P implies Q is always true except when the antecedent [P] is true and the consequent is false

A table for truth Truth tables are logical diagrams so that

every possible truth value can be examined.

Constructing truth tables 2 times the number of variables gives us the

possible number of truths. 2(n)For example p v q contains two variable p and q so

for this truth table we would construct it like this: p q p v q t t t t f t f t t f f f

Rules of Inference Modus Ponens

P -> Q

P

:. Q Modus Tollens

P->Q

~Q

.:~P

Rules of Inference #2 Hypothetical Syllogism

P->Q

Q->R

.: P->R Disjunctive Syllogism

P v Q

~P

.:Q

Rules of Inference Constructive Dilemma

(P->Q) & (R->S)

P v R

:. Q v S

Destructive Dilemma

(p->q) & (r->s)

~q v ~s

.: ~p v ~r

More Rules of Inference

Simplification

P & Q

.:P

Conjunction

P

Q

.: P & Q

One last one

Addition

P

.: p v q