CH 6 (Deductive Reasoning: Propositional Logic)
Syntax (logical connectives) for propositions (logic)
&-ampersand v-wedge/vel ~ tilde --> arrow <--> double arrow parenthesis ( )
Under what circumstances is a disjunction true (or false)--inclusive sense? (truth table definition of the connective)
(always assume inclusive sense) -for a disjunction to be true, only 1 of the disjuncts must be true (A disjunction is true even if one of the disjuncts is false) -The disjunction is false only if both disjuncts are false. (p v q is false only if both p and q are false)
How would you symbolize the statement:If Billy shot the sheriff but he didn't shoot he deputy, then he's guilty of only one crime. But it's not true that he short sheriff but didn't shoot the deputy. Therefore, it's not the case that he's guilty of one crime.
(p & ~q)-->r ~(p & ~q) therefore ~r
How would you symbolize the statement:Either Jay Leno is funny or the show is rugged, or the network has made a bad investment.
(p v q) v r
propositional logic
-AKA truth-functional logic -the branch of deductive reasoning that deals with the logical relationships among statements. -uses symbols to stand not just for the statements but also for the relationship between statements--that is, to indicate the form of an argument. These relationships are specified & made possible by logical connectives (if-then,and,or, not). Propositional logic gets this work done by using the symbol language of symbolic logic. -helps us assess the validity of an argument w/o being distracted by nonformal elements such as the language used to express the content
In what situation is a conjunction false (or true)?(truth table definition of the connective)
-If just one statement in the conjunction is false, the whole conjunction is false -Only if both conjuncts are true is the whole conjunction true -p & q is true on when p is true and q is true. p & q is false whenever at least one of the component statements is false.
If you've converted an argument into its symbolic form and know all the possible truth values of the argument's variables (statements)--in other words, if you know under what circumstances a statement is true or false due to the influence of the logical connectives, how would this info help us?
-It can help us quickly uncover the validity/invalidity of the whole argument -Given the possible truth values of some statements in the argument, and given the statements' relationships with one another as governed by logical connectives, we could infer the possible truth values of all the other statements. Then we would just need to answer ''Is there a combo of truth values in the argument such that premises are true & the conclusion is false? Is there a counterexample instance?''. If yes, argument is invalid. If there's no such circumstance, argument is valid
What is the short method of argument evaluation?
-a more efficient technique for calculating validity when arguments have more than 2 or 3 variables -the best strategy for the short method is based on the same fact we relied on in the truth table test: it's impossible for a valid argument to have true premises and a false conclusion -So we try to discover if there's a way to make the conclusion false and the premises true by assigning various truth values to the argument's components. That is, we try to prove that the argument is invalid. If we can do this, then we'll have the proof that we need.
statement (or claim)
-an assertion that something is or is not the case, is or is not actual. It can be either true or false -a sentence that makes a claim--either true or false (has a truth value). -every statement has a truth value. That is, a statement is either true of false. A true statement has a truth value of true and a false statement has a truth value of false -questions/commands don't have truth values
Conditional
-basic form is ''if...then..'' but it can be expressed in other ways than the if-then configuration (/w words like only if, provided, unless, whenever, given p, q/ q given p, assuming that etc.) -symbolized as : p-->q, where the arrow represents the connective -a conditional asserts that if the antecedent is true, then the consequent must be true. (It doesn't assert that the antecedent is actually true or that the consequent is actually true, but only that under specified conditions a certain state of affairs will be actual)
How can we use truth tables to test for validity?
-devising truth tables for arguments can reveal the underlying structure--the form of the argument -the big question is ''Does the truth table show (in any row) a state of affairs in which the premises of an argument are true and the conclusion is false? That is, does the table show any counterexamples'' If we can find even one instance of this arrangement (counterexample), we'll have shown that the argument is invalid.
If p is Leo sings the blues and q is Fat sings the blues, what distinction is made involving parenthesis? ~q v ~r
-either Leo doesn't sing the blues or Fat doesn't sing the blues -says it is not the case that Leo AND Fat sing the blues
What part of a conditional (antecedent or consequent) does "only if" introduce? "Unless"? "If"?''If only''?
-if introduces the antecedent; if A then C= A-->C -only if introduced the consequent, A only if C= A-->C -if only introduces the antecedent; C if only A= A-->C -unless introduces the antecedent and also means ''if not'', p unless q=~q-->p / p v q
How many rows are in a two-variable truth table? In a 3-variable table?
-in a 2-variable table there are 4 rows -in a 3-variable table there are 8 rows and thus 8 possible combination of truth values
Why is propositional logic called truth functional logic?
-in propositional logic, the basic unit of concern is statements. Simple statements make up compound statements joined by logical connectives -the truth value of a compound statement is a function of the truth value of component statements. This important fact is the reason why propositional logic is also called truth functional logic
What is the exclusive sense of the word ''or''?
-it can mean ''either but not both'' (This is the exclusive sense) -in this sense, p v q means ''p or q, but not both'' So if P and Q both have truth values of T, then in the exclusive sense P v Q is false.
What is the inclusive sense of the word ''or''?
-it can mean ''one or the other, or both'' -in this sense, p v q means ''p or q, or both'' -standard practice in logic is to assume the inclusive sense when dealing with disjunctions So if P and Q both have truth values of T, then in the inclusive sense P v Q is also true
If p is Leo sings the blues and q is Fat sings the blues, what distinction is made involving parenthesis? ~(q & r)
-it is not the case that Leo sings the blues and Fat sings the blues -says that it's not the case that both sing the blues concurrently. Maybe Leo sings and Fat doesn't or vice versa
If p is Leo sings the blues and q is Fat sings the blues, what distinction is made involving parenthesis? ~(q v r)
-it is not the case that either Leo sings the blues or Fat sings the blues. -says that NEITHER Leo nor Fat sings the blues
What part of a conditional (antecedent or consequent) does "provided" introduce? "whenever''? ''given'' ''assuming''?
-provided introduced the antecedent; p provided q= q-->p -whenever introduces antecedent; whenever p, q= p-->q -given introduces the antecedent; given p, q /q given p= p->q -assuming introduces the antecedent; q assuming that p=p-->q
What part of a conditional (antecedent or consequent) does ''sufficient'', ''necessary'', ''on the condition that'' and ''implies that'' introduce?
-sufficient introduces the consequent: A is sufficient for C= A--C -necessary introduces the antecedent C is necessary for A= A-->C -on the condition that introduces the antecedent: C on the condition that A= A-->C -implies that introduces the consequent: A implies that C= A-->C
What are the truth values in each guide column for a 3 variable truth table?
-the first column is 4Ts , then 4Fs -the 2nd column is alternating pairs of truth values beginning with TT -the 3rd column is alternating Ts and Fs starting with T
What are logical connectives?
-words, such as ''if-then'', ''or'','' and'', ''not'' -they help to specify the relationships between statements and thus shape the form of the argument
What are the symbols for, and the meaning of, the 5 logical connectives?
1) conjunction (and), &, p& q 2.) disjunction (or), v, p v q 3.) negation (not), ~, ~p 4.) conditional (if-then), -->, p -->q 5.) biconditional (if and only if/ just in case/ is necessary/sufficient for/ if P, then Q;and if Q, then P) <-->, p<-->q
Describe the short method: step by step.
1.) Write out the symbolized argument in a single row 2.) assign truth values to the VARIABLES in the conclusion to make the conclusion false. (Write the appropriate Ts and Fs below the row).Assign these truth values to the same variables elsewhere 3.) consistently assign truth values to variables in the premises (to make the premises true). Assign truth values first to premises where specific truth values are ''locked in'' 4.) try to make the assignments that yield a false conclusion and true premises. If you can, the argument is invalid. If not, the argument is valid.
2 types of symbols used to express an argument
1.) variables=they're the letters or symbols used to represent /express a statement 2.) the symbols for the logical connectives that indicate relationships between statements (&, v, ~, -->)
conjunction
2 simple statements joined by a connective to form a compound statement (symbolized as p & q) -make sure the connective really is joining two distinct statements and not a set of compound nouns
What is the overall goal of the short method?
=to see if we can prove invalidity in the most efficient way possible. -best strategy for doing this is to look for truth value assignments that cannot be any other way given the truth value assignments in the conclusion.. That is, focus on premises /w assignments that are ''locked into'' the argument by the truth values you've given in the conclusion. Make the assignments in those premises first, regardless of which premise you start with.
Under what circumstance is a conditional false? (truth table definition of the connective)
A conditional is false if and only if its antecedent is true and the consequent is false. -in all other possible combinations of truth values (FF, FT, TT), a conditional is true
~(P &Q) is equivalent to ~P v ~Q ~(P v Q) is equivalent to ~P &~Q
DeMorgan's Law
Even if the order of the conjuncts and disjuncts are reversed in a conjunction or disjunction, respectively, truth is still preserved. However, truth IS NOT PRESERVED when the order is reversed for a conditional. P-->Q does not equal Q--->P
Even if the order of the conjuncts and disjuncts are reversed in a conjunction or disjunction, respectively, truth is still preserved. However, truth IS NOT PRESERVED when the order is reversed for a conditional. P-->Q does not equal Q--->P
A valid argument with twenty true premises and one false premise is more sound than an argument with three true premises and one false one
False
All unsound arguments are invalid
False
Every invalid argument has a false conclusion
False
If an argument has a false conclusion it is invalid.
False
If the conclusion of a valid argument is true, the premises must be true as well.
False
No invalid arguments have a false conclusion
False
No unsound arguments have a false conclusion
False
No unsound arguments have a true conclusion
False
True or False. A valid argument can have true premises and a false conclusion
False
True or False. A valid argument must have a true conclusion
False
True or False. If an argument has all true premises and a true conclusion, then it is valid.
False
True or False. If an invalid argument has all true premises, then the conclusion must be false.
False
True or False. You can have a sound argument that is invalid.
False
True or False. You can have a sound argument with a false conclusion.
False
True or False. You can have a sound argument with a false premise.
False
True or False. You can have a valid argument with all true premises and a false conclusion.
False
Every invalid argument has a true conclusion
False --some not all
If the conclusion of a valid argument is false, then all of its premises are false as well
False--at least one not all
In a 2-variable truth table, the first two columns of Ts and Fs represent the 4 possible sets of truth values for the variables. In other words, the table shows that there are only 4 combinations of truth values for the pair of variables p and q: TT, TF, FT, FF. These are the only combinations possible for a 2-variable compound.
In a 2-variable truth table, the first two columns of Ts and Fs represent the 4 possible sets of truth values for the variables. In other words, the table shows that there are only 4 combinations of truth values for the pair of variables p and q: TT, TF, FT, FF. These are the only combinations possible for a 2-variable compound.
If p is Leo sings the blues and q is Fat sings the blues, what distinction is made involving parenthesis? ~q & ~r
Leo does not sing the blues and Fat doesn't not sing the blues -Neither of them sing the blues
Which types of statements are truth functional?
Negation Conjunction Disjunction Conditional Biconditional
material implication
P-->Q is equivalent to ~P v Q
contraposition
P-->Q is equivalent to ~Q --> ~P
Several terms can express a logical conjunction: and, but, yet, however, moreover, while, even though, nevertheless. In propositional logic, all these are logically equivalent; they are therefore properly symbolized by (&).
Several terms can express a logical conjunction: and, but, yet, however, moreover, while, even though, nevertheless. In propositional logic, all these are logically equivalent; they are therefore properly symbolized by (&).
In the short method, sometimes a conclusion can be made false in more than one way. In such cases, strategy should be to try each possibility-each way that the conclusion can be false-until you get what you're after: an argument /w true premises and a false conclusion. As soon as you get it, stop. You've proven that the argument form is invalid and there's no reason to continue making assignments. If you try all possibilities and still can't prove invalidity, the argument is valid.
Sometimes a conclusion can be made false in more than one way. In such cases, strategy should be to try each possibility-each way that the conclusion can be false-until you get what you're after: an argument /w true premises and a false conclusion. As soon as you get it, stop. You've proven that the argument form is invalid and there's no reason to continue making assignments. If you try all possibilities and still can't prove invalidity, the argument is valid.
What is the truth table definition of the connective (<-->)?
The biconditional statement is true when both of its parts have the same values. So if the truth value of P is T and Q is T, then P<-->O is true. Or if the true value of P is F and Q is also F, then P<-->Q is true -If both of the parts have different values (TF/FT), then P<-->Q is false
The steps you use to check validity of a 3-variable argument are the same ones you apply in a 2-variable argument. You devise a truth table, calculate truth values and check for true premises /w a false conclusion. The truth table, of course, has an additional guide column for the 3rd variable and there are more rows to accommodate the larger number of possible true-false combinations
The steps you use to check validity of a 3-variable argument are the same ones you apply in a 2-variable argument. You devise a truth table, calculate truth values and check for true premises /w a false conclusion. The truth table, of course, has an additional guide column for the 3rd variable and there are more rows to accommodate the larger number of possible true-false combinations
The truth table can tell us definitively whether an argument is invalid b/c the table includes every possible combination of truth values. If the truth table doesn't reveal a situation in which the argument has true premises and a false conclusion, then the argument is valid.
The truth table can tell us definitively whether an argument is invalid b/c the table includes every possible combination of truth values. If the truth table doesn't reveal a situation in which the argument has true premises and a false conclusion, then the argument is valid.
The truth value of a compound statement depends on the truth value of its components. That's why it's a good idea to start out /w guide columns in a truth table., The truth value of these variables (letters) determine the truth values of the statements that are comprised of variables. The truth values of these compound units in turn determine the truth value of any larger compound units
The truth value of a compound statement depends on the truth value of its components. That's why it's a good idea to start out /w guide columns in a truth table., The truth value of these variables (letters) determine the truth values of the statements that are comprised of variables. The truth values of these compound units in turn determine the truth value of any larger compound units
Every sound argument has a true conclusion
True
No sound arguments have a false conclusion
True
Some invalid arguments have a false conclusion
True
Some invalid arguments have a false premise
True
Some invalid arguments have a true conclusion
True
Some unsound arguments have a false conclusion
True
Some valid arguments are unsound
True
Some valid arguments have a true conclusion
True
True or False. A valid argument can have false conclusion.
True
True or False. A valid argument can have false premises and false conclusion and still be valid.
True
True or False. A valid argument can have false premises.
True
True or False. A valid argument must have a true conclusion only if all of the premises are true
True
True or False. An argument can have all true premises and a true conclusion but still be invald
True
True or False. An argument that is sound cannot have false conclusion.
True
True or False. An invalid argument can have false premises and a true conclusion.
True
True or False. An invalid argument can have true conclusion
True
True or False. An invalid argument can have true premises.
True
True or False. If a valid argument has a false conclusion, then at least one premise must be false
True
True or False. If an argument has all true premises and a false conclusion, then it is invalid.
True
True or False. It is possible for an argument to have all true premises and a true conclusion but still be invalid.
True
True or False. It is possible for an invalid argument to have all true premises and a true conclusion
True
True or False. It's impossible for a valid argument to have true premises and a false conclusion.
True
True or False. The best clues to where to insert parenthesis come from the words: either neither, conjunction and disjunction words such as and/or, and the punctuation of the sentences.
True
True or False. Valid forms do NOT have counterexamples. INVALID forms DO HAVE counterexamples--that is, it's possible to have premises that are true and conclusion false. (TT/F).
True
True or False. You can have a valid argument that is unsound.
True
True or False. You can have a valid argument with all false premises.
True
True or False. You can have a valid argument with some true premises and some false premises.
True
True or False. You can have an invalid argument with all true premises and a true conclusion.
True
True or False.You can have a valid argument with all false premises and a false conclusion.
True
True or false. ~q-->p is logically equivalent to p v q
True
counterexample instance
When an argument is shown to have true premises and false conclusion thus making the argument invalid
When dealing with simple arguments, the first 2 columns of a truth table are guide columns in which the variables, or letters, of the argument are listed, followed by a column for each premises and then a column for the conclusion
When dealing with simple arguments, the first 2 columns of a truth table are guide columns in which the variables, or letters, of the argument are listed, followed by a column for each premises and then a column for the conclusion
Whether a disjunction is inclusive or exclusive has no effect on our evaluation of disjunctive syllogism. They would be valid regardless of whether the disjunction was construed as inclusive or exclusive. In a disjunction syllogism, if one of the disjuncts is denied, the arguments is valid in any cause. (either p or q. not p. therefore q.) But if one of the disjuncts is affirmed, the argument is invalid when the disjunction is inclusive. If the disjuncts mean ''p or q or both'', then by affirming p we can't conclude not q.
Whether a disjunction is inclusive or exclusive has no effect on our evaluation of disjunctive syllogism. They would be valid regardless of whether the disjunction was construed as inclusive or exclusive. In a disjunction syllogism, if one of the disjuncts is denied, the arguments is valid in any cause. (either p or q. not p. therefore q.) But if one of the disjuncts is affirmed, the argument is invalid when the disjunction is inclusive. If the disjuncts mean ''p or q or both'', then by affirming p we can't conclude not q.
With invalid arguments, the premises may be true or false, and the conclusion may be true or false: all combinations are possible
With invalid arguments, the premises may be true or false, and the conclusion may be true or false: all combinations are possible
disjunction
a compound statement of the form: either p or q (symbolized as p v q) -the word ''unless'' is also sometimes used in place of or to form a disjunction -the word either or neither usually signal the beginning of a disjunction
invalid argument
a deductive argument that fails to provide logical conclusive support for its conclusion -conclusion doesn't logically follow from the premises
valid argument
a deductive argument that succeeds in providing logical conclusive support for its conclusion -The conclusion logically follows from the premises -a deductively valid argument is such that if its premises are true, its conclusions must be true. That is,if the premises are true, there's no way that the conclusion can be false -can have false premises and false conclusion, false premise and true conclusion, or true premise and true conclusion. It cannot have a true premise and false conclusion -IT'S ABOUT FORM!! NOT CONTENT
simple statement
a statement that doesn't contain any other statements as constituents
compound statement
a statement that's composed of at least 2 simple statements
truth table
a table that specifies the truth values for claim variables and combinations of claim variables in symbolized statements or arguments -a graphic way of displaying all the truth value possibilities of statements/arguments
interpretation
an assignment of truth values (T or F) to the atomic components of a compound formula
What are stylistic variants of the word ''and''?
and but yet however moreover while even though nevertheless
unsound
argument is either invalid or it's valid but has at least 1 false premise
P is necessary and sufficient for Q. What type of statement is this?
biconditional statement
P-->Q is equivalent to ~Q --> ~P
contraposition
conjunct
each of the component statements of the conjunction; one of 2 simple statements joined by a connective to form a compound statement
disjunct
each statement in a disjunction; a simple statement that is a component of a disjunction
The truth-table test is based on what elementary fact about validity?
it's impossible for a valid argument to have true premises and a false conclusion
P-->Q is equivalent to ~P v Q
material implication
symbolic logic
modern deductive logic that uses symbolic language to do its work
How would you symbolize the statement: If gods intervene, then neither peace nor war can change the destiny of the nation.
p--> ~(q v r)
___may be used to join variables. ___enable us to symbolize arguments more precisely and avoid confusion.
parenthesis
deductive argument
premises are intended to provide logically conclusive support for its conclusion -that is, the premises are intended to guarantee the truth of the conclusion. If premises are true, conclusion must be true
negation
the denial of a statement, which we can indicate with the word ''not'' or a term that means the same thing. (it is false that..., it is not the case that..., fails to.., it is not true that..'' -symbolized as ~p -a negation reverses the truth value of a statement--changes the statement's truth value to its contradictory. A true statement becomes false; a false statement becomes true. (A double negation is the same thing as no negation since if you negate a negation you end up /w a positive)
sound
valid and all true premises
DeMorgan's Law
~(P &Q) is equivalent to ~P v ~Q ~(P v Q) is equivalent to ~P &~Q
What is the symbolization for the statement "It is not the case that either Alice walks home or Jan walks home"
~(p v q)
