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Valid & Invalid ArgumentsoArgument is a sequence of statements ending in a conclusion.oDetermination of validity of an argument depends only on the form of an argument, not on its content.
“If you have a current password, then you can log onto the network.”p=“You have a current password”q=“You can log onto the network.”p → qp∴ q where ∴ is the symbol that denotes “therefore.”
Valid & Invalid ArgumentsoAn argument is a sequence of statements, and an argument form is a sequence of statement forms(have proposition var.).o All statements in an argument and all statement forms in an argument form, except for the final one, are called premises (or assumptions or hypotheses). oThe final statement or statement form is called the conclusion. The symbol ∴, which is read “therefore,” is normally placed just before the conclusion.
Valid & Invalid ArgumentsoTo say that an argument form is valid means that no matter what particular statements are substituted for the statement variables in its premises, if the resulting premises are all true, then the conclusion is also true. Conclusion q is valid, when (p1 ∧ p2 ∧ · · · ∧ pn) → q is a tautology.oTo say that an argument is valid means that its form is valid.
Valid & Invalid ArgumentsoThe truth of its conclusion follows necessarily or by logical form alone from the truth of its premises. When an argument is valid and its premises are true, the
truth of the conclusion is said to be inferred or deduced from the truth of the premises.
If a conclusion “ain’t necessarily so,” then it isn’t a valid deduction.
Testing an Argument Form for Validity1. Identify the premises and conclusion of the argument form.2. Construct a truth table showing the truth values of all the
premises and the conclusion.3. A row of the truth table in which all the premises are true is
called a critical row. If there is a critical row in which the conclusion is false, then
it is possible for an argument of the given form to have true premises and a false conclusion, and so the argument form is invalid.
If the conclusion in every critical row is true, then the argument form is valid.
Testing an Argument Form for Validityop →q ∨ ∼roq → p ∧ ro∴ p →r
Hence this form of argument is invalid
Testing an Argument Form for Validity“If 101 is divisible by 3, then 1012 is divisible by 9. 101 is divisible by 3. Consequently, 1012 is divisible by 9.”
Although the argument is valid, its conclusion is incorrect, because one of the hypotheses is false (“101 is divisible by 3.”).
If in the above argument we replace 101 with 102, we could correctly conclude that 1022 is divisible by 9.
Rules of Inference for Propositional LogicoAn argument form consisting of two premises and a conclusion is called a syllogism.o The first and second premises are called the major
premise and minor premise, respectively.oThe most famous form of syllogism in logic is called modus ponens.
Modus Ponens oThe modus ponens argument form has the following form(“method of affirming”):
If p then q.p
∴ qIf the sum of the digits of 371,487 is divisible by 3, then 371,487 is divisible by 3.The sum of the digits of 371,487 is divisible by 3.∴ 371,487 is divisible by 3.
oif a conditional statement and the hypothesis of this conditional statement are both true, then the conclusion must also be true.
Modus Tollens o Modus tollens(“method of denying” (the conclusion is a denial)) has the following form:
If p then q.∼q∴ ∼p
(1) If Zeus is human, then Zeus is mortal; and(2) Zeus is not mortal.Must Zeus necessarily be nonhuman?Yes!
If Zeus is human, then Zeus is mortal.Zeus is not mortal.∴ Zeus is not human.
Because, if Zeus were human, then by (1) he would be mortal.But by (2) he is not mortal.Hence, Zeus cannot be human.
Translating Propositions
Rules of Inference for Propositional LogicoWhen an argument form involves 10 different propositional variables, to use a truth table to show this argument form is valid requires 210 = 1024 different rows.oRules of inference. First establish the validity of some relatively
simple argument forms. A rule of inference is a form of argument that
is valid. Thus modus ponens and modus tollens are both
rules of inference.
Rules of Inference for Propositional LogicoGeneralizationThe following argument forms are valid:a. p b. q∴ p ∨ q ∴ p ∨ qif p is true, then, more generally, “p or q” is true for any other statement q.Anton is a junior.∴ (more generally) Anton is a junior or Anton is a senior.oAt some places with the name Addition It is below freezing now. Therefore, it is below freezing or raining
snow.
Rules of Inference for Propositional LogicoSpecializationThe following argument forms are valid:
a. p ∧ q b. p ∧ q∴ p ∴ q
These argument forms are used for specializing.Ana knows numerical analysis and Ana knows graph algorithms.∴ (in particular) Ana knows graph algorithms.oAt some places with the name simplification It is below freezing and snowing. Therefore it is below freezing.
Rules of Inference for Propositional LogicoEliminationThe following argument forms are valid:
a. p ∨ q b. p ∨ q∼q ∼p∴ p ∴ q
oThese argument forms say that when you have only two possibilities and you can rule one out, the other must be the case.
x − 3 =0 or x + 2 = 0.If you also know that x is not negative, then x ≠ −2, sox + 2 ≠ 0. By elimination, you can then conclude that∴ x − 3 = 0.
oAt some places with the name Disjunctive Syllogism
Rules of Inference for Propositional LogicoTransitivityo The following argument form is valid:
p →qq →r∴ p →r
If 18,486 is divisible by 18, then 18,486 is divisible by 9.If 18,486 is divisible by 9, then the sum of the digits of 18,486 is divisible by 9.∴ If 18,486 is divisible by 18, then the sum of the digits of 18,486 is divisible by 9.
oAt some places with the name Hypothetical Syllogism
Rules of Inference for Propositional Logic-Exampleo“If it rains today, then we will not have a barbeque today. If we do not have a barbeque today, then we will have a barbeque tomorrow.Therefore, if it rains today, then we will have a barbeque tomorrow.” p: “It is raining today.” q: “We will not have a barbecue today.” r: “We will have a barbecue tomorrow.”So the argument is of the following form:
Rules of Inference for Propositional LogicoProof by Division into Cases:The following argument form is valid:
p ∨ qp →rq →r∴ r
oIf you can show that in either case a certain conclusion follows, then this conclusion must also be true.
x is positive or x is negative.If x is positive, then x2 > 0.If x is negative, then x2 > 0.∴ x2 > 0.
Rules of Inference for Propositional LogicoConjunction
pq
∴ p ∧ qoResolution
p ∨ q ¬p ∨ r∴ q ∨ r
Rules of Inference for Propositional Logic-Exampleo“It is not sunny this afternoon and it is colder than yesterday,” “We will go swimming only if it is sunny,” “If we do not go swimming, then we will take a canoe trip,” and “If we take a canoe trip, then we will be home by sunset” lead to the conclusion “We will be home by sunset.”
p: “It is sunny this afternoon ” q: “It is colder than yesterday.” r: “We will go swimming .” s:” we will take a canoe trip” t: “We will be home by sunset”
Step Reason1. ¬pΛq Premise2. ¬p Simplification using (1)3. r → p Premise4. ¬r Modus tollens using (2) and
(3)5. ¬r →
sPremise
6. s7. s→t
Modus ponens using (4) and (5)Premise
8. t Modus ponens using (6) and (7)
Rules of Inference for Propositional Logic-Example
Arguments with Quantified Statements
All men are mortal.Socrates is a man.
∴ Socrates is mortal.oUniversal Instantiation If some property is true of everything in a set, then
it is true of any particular thing in the set.
For all real numbers x, x1 = x. universal truthr is a particular real number. particular instance∴ r 1 = r.
Universal Modus Ponens
o Could be written as “All things that make P(x) true make Q(x) true,” in which case the conclusion would follow by universal instantiation alone.
Universal Modus PonensIf an integer is even, then its square is even.k is a particular integer that is even.
∴ k2 is even.
oMajor premise can be written as ∀x, if x is an even integer then x2 is even.oLet E(x) be “x is an even integer,” let S(x) be “x2 is even,” and let k stand for a particular integer that is even. Then the argument has the following form:
∀x, if E(x) then S(x).E(k), for a particular k.∴ S(k).
oThis argument has the form of universal modus ponens and is therefore valid.
Universal Modus Tollens
Universal Modus TollensAll human beings are mortal.Zeus is not mortal.∴ Zeus is not human.
Solution The major premise can be rewritten as ∀x, if x is human then x is mortal.
Let H(x) be “x is human,” let M(x) be “x is mortal,” and let Z stand for Zeus. The argument becomes ∀x, if H(x) then M(x)
∼M(Z)∴ ∼H(Z).
This argument has the form of universal modus tollens and is therefore valid.
Universal Modus TollensAll human beings are mortal.Zeus is not mortal.∴ Zeus is not human.
Solution The major premise can be rewritten as ∀x, if x is human then x is mortal.
Let H(x) be “x is human,” let M(x) be “x is mortal,” and let Z stand for Zeus. The argument becomes ∀x, if H(x) then M(x)
∼M(Z)∴ ∼H(Z).
This argument has the form of universal modus tollens and is therefore valid.