Elementary Logic
PHIL 105-302
Intersession 2013
MTWHF 10:00 – 12:00
ASA0118C
Steven A. Miller
Day 3
Argument evaluation
What makes an argument good?
1) true premises
2) conclusion at least probable, given 1
3) premises are related to the conclusion
4) the conclusion can’t be undermined
Argument evaluation
3) premises are related to the conclusion
1) Scientists have not yet found any sign of
aliens.
2) The universe would be boring without any
aliens.
3) Therefore, there must be undiscovered
aliens.
Premise 1 is relevant to the opposite of the
conclusion; premise 2 is irrelevant to either
the conclusion or its opposite.
Argument evaluation
4) the conclusion can’t be undermined
A deductive argument is valid when
it is impossible (in the strongest sense)
for the premises to all be true
and the conclusion to be false.
Argument evaluation
it is impossible for … the conclusion to be
false
Any argument where the conclusion cannot
be false is always valid. The prime case of
this is found with logically necessary
conclusions.
1) All squares are rectangles.
2) 10 is greater than 6.
3) If it is red, then it is red.
Argument evaluation
1) Lincoln is the capital of Nebraska.
2) Barack Obama is the president.
3) Therefore, all bachelors are
unmarried.
Valid, because it is impossible for the
premises to all be true and the
conclusion false. The conclusion cannot
be false.
Argument evaluation
it is impossible for … the premises to all be
true
Any argument where the premises cannot
all be true is always valid. The prime case
of this is found with logically inconsistent
premises.
These are premises where the truth of one
guarantees the falsehood of another.
Argument evaluation
1) All dogs are brown.
2) No dogs are brown.
3) Therefore, Mark McGwire is overrated.
Valid, because it is impossible for the
premises to all be true and the conclusion
false. The premises cannot both be true
(truth of one implies the falsity of the
other).
Argument evaluation
Inductive arguments do not share this
standard of validity.
Instead, they rely on complex
standards of evidence, relevance,
and probability…
(which we’re skipping in the interest of time, see S, pp. 31-43, etc.).
Formalization
Arguments in natural language (e.g.
English, Spanish, Arabic, etc.) can be
very complicated to follow.
Formalization offers a way to simplify
and to make explicit the underlying
structure (that is, logic) of
arguments.
Argument form
Consider the following arguments:
1) The ice cream is either chocolate or vanilla.
2) The ice cream isn’t chocolate.
3) So, the ice cream is vanilla.
Argument form
Consider the following arguments:
1) John is either taller or shorter than Mary.
2) John isn’t taller than Mary.
3) So, John is shorter than Mary.
Argument form
Consider the following arguments:
1) The sauce is either ketchup or mustard.
2) It isn’t ketchup.
3) So, the sauce is mustard.
Argument form
Commonality between the three:
1) This or that.
2) NOT THIS.
3) Therefore, THAT.
This is called a “disjunctive syllogism.”
“Disjunctive syllogism” is the
argument’s form.
Argument form
All arguments have forms, though
some are easier to find than others.
All (deductive) argument forms are
either valid or invalid.
Disjunctive syllogism, for instance, is
valid.
Seventh Inning Stretch
(“…Take Me Out to the Crowd, …”)
Formalizing
Symbolization:
- simplifies arguments
- quicker than natural language
- helps in finding form
Formalizing
Symbolizing Propositional Logic
Basic unit:
the proposition
(the statement, for us)
Statement symbolization:
any capital letter
Formalizing
Statement symbolization:
1) The ice cream is either chocolate or the cream is vanilla.
2) The ice cream isn’t chocolate.
3) So, the ice cream is vanilla.
“The ice cream is chocolate” = P
“The ice cream is vanilla” = Q
“The ice cream isn’t chocolate” = Z
“It is not the case that P”
Formalizing
Statement symbolization:
1)
2)
3)
The ice cream is either chocolate or the cream is vanilla.
The ice cream isn’t chocolate.
So, the ice cream is vanilla.
1) Either P or Q.
2) It is not the case that P.
3) So, Q.
Symbol for conclusions: ∴
3) ∴
Formalizing
Negation symbolization:
“The ice cream isn’t chocolate.”
“It is not the case that P.”
Symbol for “it is not the case that”: ~
“It is not the case that P” = ~P
(You may also see ¬P, −P, NP, !P)
Formalizing
Conjunction symbolization:
“The ice cream is chocolate and cold.”
“The ice cream is chocolate and cold” = P
“The ice cream is chocolate” = P
“Cold” = Q
“The ice cream is cold” = Q
Formalizing
Conjunction symbolization:
“The ice cream is chocolate and cold.”
P and Q
Symbol for “and” = &
“The ice cream is chocolate and cold” = P&Q
(You may also see ∧, ·)
Formalizing
Conjunction symbolization:
Symbol for “and” = &
Other words that take “&” as their symbol:
but yet
nevertheless
although
moreover
Formalizing
Disjunction symbolization:
“The ice cream is chocolate or vanilla.”
P or Q
Symbol for “or” = v
“The ice cream is chocolate or vanilla” = PvQ
Formalizing
Material conditional symbolization:
“If the ice cream is chocolate, then it is cold.”
If P, then Q
Symbol for “if … then” = →
“If the ice cream is chocolate, then it’s cold”
=P→Q
(You may also see ⊃, >)
Formalizing
Material conditional symbolization:
Symbol for “if … then” = →
If I get a puppy, then I’ll be so happy.
Antecedent = the part after the ‘if’
Consequent = the part after the ‘then’
Formalizing
Biconditional symbolization:
“The ice cream is chocolate if and only
if it’s cold.”
P if and only if Q
Symbol for “if and only if” = ↔
“The ice cream is chocolate if and only if it’s
cold” = P ↔ Q
(You may also see ≡.)
Formalizing
Biconditional symbolization:
Symbol for “if and only if” = ↔
“The ice cream is chocolate if and only
if it is cold.”
is equivalent to
“If the ice cream is chocolate, then it is cold
and
if the ice cream is cold, then it is chocolate.”
Formalizing
Symbolization chart:
It is not the case =
And
=
Or
=
If … then
=
If and only if =
Therefore
=
~
&
v
→
↔
∴
Formalization tips
1) Negation applies to whatever is to
its immediate right
~P & Q is different than ~(P & Q)
(they’re also different than ~P & ~Q)
2) All other operators apply both to
their left and right
Formalization tips
3) Use parentheses, and then
brackets, to avoid confusion.
~P v Q & Z = confusing
(~P v Q) & Z, ~(PvQ) & Z,
~[P v (Q & Z)], etc. = less confusing
Formalization tips
All formulas must be well-formed
This means they must be syntactically correct
(Exactly what the ‘English sentence’ “jo.!hn wa2s
He@#rD” isn’t.)
1) All statement letters are well-formed formula.
2) All negations of statement letters are wffs.
3) All operators between two statement letters,
and their negations, are well-formed formula.
(A slightly less complex version of S, p. 52-4.)
For next time…
Read ahead, S, pp. 55-68
Not difficult, but lots to memorize / apply