Elementary Logic
PHIL 105-302Intersession 2013
MTWHF 10:00 – 12:00ASA0118C
Steven A. MillerDay 4
Formalizing review
Symbolization chart:
It is not the case = ~And = &Or = v
If … then = → If and only if = ↔ Therefore = ∴
Logical semantics
Our interpretations are concerned with statements’ truth and falsity.
Principle of bivalence: Every statement is either true or false (and not both).
Logical semantics
Negation semantics
“The Cubs are the best team” is true, then … what’s false?
“It is not the case that the Cubs are the best team.”
Logical semantics
Negation semantics
Likewise, if:“The Cubs are the best team”
is false, then … what’s true?“It is not the case that the Cubs are
the best team.”
Logical semantics
Negation semantics (truth table)
P ~PT FF T
Logical semantics
Conjunction semantics
“My name is Steven and my name is Miller.” is true when
“My name is Steven Miller.”
Logical semantics
Conjunction semantics
“My name is Steven and my name is Miller.” is false when
“My name is not Steven or Miller, or both.”
Logical semantics
Conjunction semantics (truth table)
P Q P & QT T TT F FF T FF F F
Logical semantics
Disjunction semantics
“My name is Steven or my name is Miller.” is true when
“My name is Steven or Miller, or both.”
Logical semantics
Disjunction semantics
“…or both”:
“Soup or salad?”
Logical semantics
Disjunction semantics
Inclusive disjunction:this, or that, or both
Exclusive disjunction:this, or that, but not both
Logical semantics
Disjunction semantics
For our purposes, unless stated otherwise, all disjunctions are inclusive:
“or” means:this, or that, or both
Logical semantics
Disjunction semantics (truth table)
P Q P v QT T TT F TF T TF F F
Logical semantics
Disjunction semantics
Exclusive disjunction symbolization:
(P v Q) & ~(P & Q)
Logical semantics
Exclusive disjunction semantics (truth table)
P Q (P v Q) & ~ (P & Q)T T T TT F T FF T T FF F F F
Logical semantics
Exclusive disjunction semantics (truth table)
P Q (P v Q) & ~ (P & Q)T T T F TT F T T FF T T T FF F F T F
Logical semantics
Exclusive disjunction semantics (truth table)
P Q (P v Q) & ~ (P & Q)T T T F F TT F T T T FF T T T T FF F F F T F
Logical semantics
Exclusive disjunction semantics (truth table)
P Q (P v Q) & ~ (P & Q)T T T F F TT F T T T FF T T T T FF F F F T F
Logical semantics
Material conditional semantics
Follows the rules of deductive validity (in fact, every argument is an if-then statement).Is false only when antecedent (premises) is true and consequent (conclusion) is false.
Logical semantics
Material conditional semantics
This can be counter-intuitive, see:
If there are fewer than three people in the room, then Paris is the capital of Egypt.
Logical semantics
Material conditional semantics
If there are fewer than three people in the room, then Paris is the capital of Egypt.
Antecedent = falseConsequent = false
Logical semantics
Material conditional semantics (truth table)
P Q P → QT T TT F FF T TF F T
Logical semantics
Biconditional semantics
Biconditional is conjunction of two material conditionals with the antecedent and consequent reversed:
P ↔ Q = (P → Q) & (Q → P)
Logical semantics
Biconditional semantics (truth table)
P Q (P → Q) & (Q → P)T T T TT F F TF T T FF F T T
Logical semantics
Biconditional semantics (truth table)
P Q (P → Q) & (Q → P)T T T T TT F F F TF T T F FF F T T T
Logical semantics
Biconditional semantics (truth table)
P Q (P ↔ Q)T T TT F FF T FF F T
Seventh Inning Stretch
(“…Buy Me Some Peanuts …”)
Logical semantics
Combining truth tables
Always work from the operator that affects the least of the formula to that which affects the most of it.
~[(P & ~Q) v (Z ↔ Q)]
Logical semantics
Combining truth tables
P Q ~~ (P & Q)T T TT F FF T FF F F
Logical semantics
Combining truth tables
P Q ~~ (P & Q)T T F TT F T FF T T FF F T F
Logical semantics
Combining truth tables
P Q ~~ (P & Q)T T T F TT F F T FF T F T FF F F T F
Logical semantics
Combining truth tables
P Q ~~ (P & Q)T T T F TT F F T FF T F T FF F F T F
Logical semantics
Combining truth tables
P Q (~P & Q) → ~ (Q v P)T T T T T TT F T F F TF T F T T FF F F F F F
Logical semantics
Combining truth tables
P Q (~P & Q) → ~ (Q v P)T T F T T T TT F F T F F TF T T F T T FF F T F F F F
Logical semantics
Combining truth tables
P Q (~P & Q) → ~ (Q v P)T T F T F T T TT F F T F F F TF T T F T T T FF F T F F F F F
Logical semantics
Combining truth tables
P Q (~P & Q) → ~ (Q v P)T T F T F T T T TT F F T F F F T TF T T F T T T T FF F T F F F F F F
Logical semantics
Combining truth tables
P Q (~P & Q) → ~ (Q v P)T T F T F T F T T TT F F T F F F F T TF T T F T T F T T FF F T F F F T F F F
Logical semantics
Combining truth tables
P Q (~P & Q) → ~ (Q v P)T T F T F T T F T T TT F F T F F T F F T TF T T F T T F F T T FF F T F F F T T F F F
Logical semantics
Combining truth tables
P Q (~P & Q) → ~ (Q v P)T T F T F T T F T T TT F F T F F T F F T TF T T F T T F F T T FF F T F F F T T F F F
Three kinds of formulas
Tautologies – true in all cases
P P v ~PT T FF F T
Three kinds of formulas
Tautologies – true in all cases
P P v ~PT T T FF F T T
Three kinds of formulas
Tautologies – true in all cases
P P v ~PT T T FF F T T
Three kinds of formulas
Contradictory (or truth-functionally inconsistent) – false in all casesP P & ~PT T FF F T
Three kinds of formulas
Contradictory (or truth-functionally inconsistent) – false in all casesP P & ~PT T F FF F F T
Three kinds of formulas
Contradictory (or truth-functionally inconsistent) – false in all casesP P & ~PT T F FF F F T
Three kinds of formulas
Contingent – can be both true and false
Z R Z & R T T T T F F F T F F F F
Putting it all together
Either Peter or Saul went to the bar.Peter did not go.Therefore, Saul went.
1) P v S2) ~P3) ∴ S
Putting it all together
1) P v S2) ~P3) ∴ S
What’s this argument’s form?Disjunctive syllogism.
Putting it all together
1) P v S2) ~P3) ∴ S
[(P v S) & ~P] → S
Putting it all together
P S [(P v S) & ~P] → ST T T T T T
T F T F T F F T F T F T F F F F F F
Putting it all together
P S [(P v S) & ~P] → ST T T T F T T
T F T F F T F F T F T T F T F F F F T F F
Putting it all together
P S [(P v S) & ~P] → ST T T T T F T T
T F T T F F T F F T F T T T F T F F F F F T F F
Putting it all together
P S [(P v S) & ~P] → ST T T T T F F T T
T F T T F F F T F F T F T T T T F T F F F F F F T F F
Putting it all together
P S [(P v S) & ~P] → ST T T T T F F T T T
T F T T F F F T T F F T F T T T T F T T F F F F F F T F T F
This argument is valid; there is no line where the premises are all true and the conclusion is false.
Putting it all together
A truth table that has no lines where the premises are all true and the conclusion false presents a valid argument.
A truth table that has at least one line where the premises are all true and the conclusion false presents an invalid argument.
Things we’re skipping
- Truth / refutation trees, S. pp. 68-77
- identical in purpose to tables- more efficient- but no time = no need