Chapter 10 Flashcards
truth functional logic
also known as propositional or sentential logic: prominent since the late 19th uses truth values in functions as input and produces truth values (Wikipedia)
the truth of the compound claim results entirely from the truth values of the smaller parts
T-F
Truth- functional
What do uppercase letters represent in categorical logic and what do they represent in T-F logic?
in categorical: terms
in T-F: claims
claim variables
statements that can be true or falls, can be linked to words like “not”, “and”, “or”.
truth-table
displays possible truth values
negotiation or contradictory
~P. Which truth value P has, the negotiation has the other: “change the truth value from T to F or from F to T, depending on P’s values.”
A conjunction (compound claim)
A conjunction is true if and only if both of the simpler claims that make it up (its conjuncts) are true. Sign: “and” (Die komische Seite will mich keine “and” Zeichen machen lassen -.-)
Some conjunction symbol words
and, but, even though, while
A disjunction (compound claim)/ disjunctive conjunction
A disjunction is false if and only if both of its disjuncts are false. Sign: ∨ (“wedge”) words like “or”, “either.. or”, “unless”
conditional claim (compound claim)
A conditional claim is false if and only if its antecedent is true and its consequent is false: Sign: P→Q, words like “if…then…”, “provided that”
antecedent
first claim in a conditional claim, e.g. the P in P→Q
consequent
second claim in a conditional claim, e.g. the Q in P→Q
the four types of truth-functional claims
negation, conjunction, disjunction, and conditional
Every time we add another letter to a truth table, the number of possible combinations of T and F-
doubles, and so, therefore, does the number of rows
in our truth table. ( r=2^n, where r is the number of rows in the table and n is the number of letters
Two claims are truth-functionally equivalent if they
have exactly the same truth table—that is, if the Ts and Fs in the column under one claim are in the same arrangement as those in the column under the other.
What is the difference between the word “if” and the phrase “only if” in a conditional claim? What does “if and only if” mean for a compound claim?
“if” introduces an antecedent
“only if” a consequent
What does “if and only if” means for a claim?
makes both antecedent and
consequent out of the claim it introduces. e.g. P: (P→Q)”and”(Q→P)
the phrase”provided”/ “provided that” can introduce what kind of claim?
An antecendent of a conditional claim
necessary conditions
the necessary condition becomes the consequent of a conditional:
e.g.: “If we have combustion (C), then we must have oxygen (O).” C→O
sufficient conditions
guarantees whatever it is a sufficient condition for. Being born in the United States is a sufficient condition for U.S. citizenship—that’s all one needs to be a U.S. citizen. Sufficient conditions are expressed as the antecedents of conditional claims. We would say, “If Juan was born in the United States (B), then Juan is a U.S. citizen (C)”: B→C
the word “if” introduces a … condition, the word “only if” introduces a … condition
sufficient
necessary
Unless -> How to symbolize the claim: “Paula (P) will foreclose unless Quincy (Q) pays up.”
~Q→P, but better: P∨Q, because they are truth-functionally equivalent
The word “either” in a claim symbolizes..?
Where a disjunction begins.
E.g: Either P and Q or R= (P & Q) v R
P and either Q or R= P& (Q v R)
Once more please: The word “if” tells us where a … begins
Where a conditional claim begins
E.g: P and if Q then R= P&Q->R
If P and Q then R= (P&Q)->R
An argument is valid, if and only if
the truth of the premises guarantees the truth of the conclusion—that is, if the premises were true, the conclusion could not then be false. (Where validity is concerned, remember, it doesn’t matter whether the premises are actually true.)
The truth-table test for validity
Just take the claims of the argument, make the truth-table and compare with your argument if it must be right if the premises are right
the short truth-table method underlying principle
If an argument is invalid, there has to be at least one row in the argument’s truth table where the premises are true and the conclusion is false. With the short truth-table method, we simply focus on finding such a row.
“if the premises are all true in one row and so is the conclusion, the conclusion follows from the premises”
right or wrong?
Wrong, every case of right conclusion must fit right premises, no wrong conclusion can have right premises
deduction
deduce (or “derive”) the conclusion from the premises by means of a series of basic, truth-functionally valid argument patterns.
Rule 1: Modus ponens (MP): affirming the antecedent
If you have a conditional among the premises, and if the antecedent of that conditional occurs as another premise, then by modus ponens the consequent of the conditional follows from those two premises. The antecedent can be just one letter or some compound claim
annotation
notes about the logical steps that were made
modus ponens rule and all other Group I rules can be used only on whole lines.
Oui E.g: (P → Q) ∨ R P \_\_\_\_\_ Q ∨ R is not a valid deduction
Rule 2: Modus tollens (MT)
or denying the consequent: If you have a conditional claim as one premise and if one of your other premises is the negation of the consequent of that conditional, you can write down the negation of the conditional’s antecedent as a new line in your deduction. E.g.: P → Q ~Q \_\_\_\_\_\_\_ ~P
Rule 3: Chain argument (CA)
allows you to derive a conditional from two (conditionals) you already have, provided the antecedent of one of your conditionals is the same as the consequent of the other. E.g: P → Q Q → R \_\_\_\_\_\_ P → R
Rule 4: Disjunctive argument (DA)
From a disjunction and the negation of one disjunct, the other disjunct may be derived. E.g.: P ∨ Q ~P \_\_\_\_ Q
Rule 5: Simplification (SIM)
If the conjunction is true, then of course the conjuncts must all be true. You can pull
out one conjunct from any conjunction and make it the new line in your deduction.
E.g:
P & Q
____
P
Rule 6: Conjunction (CONJ)
This rule allows you to put any two lines of a deduction together in the form of a conjunction. P Q \_\_\_\_ P & Q
Rule 7: Addition (ADD)
no matter what claims P and Q might be, if P is true then either P or Q
must be true. The truth of one disjunct is all it takes to make the whole disjunction true.
P
_____
P ∨ Q
Rule 8: Constructive dilemma (CD)
The disjunction of the antecedents of any two conditionals allows the derivation of the disjunction of their consequents. P → Q R → S P ∨ R \_\_\_\_\_\_ Q ∨ S
Rule 9: Destructive dilemma (DD)
The disjunction of the negations of the consequents of two conditionals allows the derivation of the disjunction of the negations of their antecedents. P → Q R → S ~Q ∨ ~S \_\_\_\_\_\_\_ ~P ∨ ~R
The 9 group 1 rules
1) Modus ponens (MP),
2) Modus tollens (MT),
3) Chain argument (CA),
4) Disjunctive argument (DA),
5) Simplification (SIM),
6) Conjunction (CONJ),
7) Addition (ADD),
8) Constructive dilemma (CD)
9) Destructive dilemma (DD)
