Logic pt. 2 semantics quiz 09/23/2022
~~R ∨ (~P ∧ ~P) is logically equivalent to: (a) P ⊃ ~R (b) R ⊃ ~P (c) ~R ⊃ P (d) ~P ⊃ R (e) ~R ∨ P (f) ~P ∨ (~R ∧ P)
(a), (b), and (f)
Choose all that apply: which sets of things serve as the domain of an interpretation function? That is, which kinds of things can an interpretation function take as input? (a) wffs (b) truth values (c) sentence symbols (d) valuations
(c) and only (c) because an interpretation function only works for sentence symbols
VERY IMPORTANT: ((P ∧ Q) ⊃ (R ∧ R)) is logically equivalent to (USE REPLACEMENT RULES): (a) R (b) ~R ⊃ (~P ∨ ~Q) (c) (~P ∨ ~Q) ∨ ~~Q (d) (~Q ∨ ~P) ∨ R (e) (((P ∧ Q) ⊃ (R ∧ R)) ∨ (S ∧ ~S)
(d), (e), (b)
What can we use (double-turnstile) to express (for purposes of this course)
(double-turnstile) α (in which case that illustrates a tautology) (gamma) (double-turnstile) α (in which case that illustrates that a conclusion is entailed by a set of wffs serving as premises (interpretation I) (double-turnstile) α (in which case that illustrates that a wff is true relative to an interpretation I.
How to define the notion of a contradiction from that of a valid formula
A wff α is a contradiction iff ~α is a valid formula.
True or false: For any wff α and a set of wffs T (gamma): if there exists an I, such that both (i) I (double-turnstile) α and (ii) I (double-turnstile) β for all β ∈ T (gamma), then it follows that T (gamma) α.
False, T (double-turnstile) α, also it is supposed to be for ALL interpretations
True or false: α is a contradiction just in case ∅ (not-double-turnstile) α
False, an atomic wff need not be entailed by the empty set but it is not a contradiction
T/F: The recursive definition of wff and the definition of interpretation function define the truth functions associated with the connectives of PL.
False, it is the recursive definition of VALUATION FUNCTIONS and the definition of the interpretation function define the truth functions associated with the connectives of PL (and it is important to note, that interpretation functions are only applicable for atomic wffs so if that requires a scrupulous alteration in the above definition do so)
True or false: The domain of any interpretation function is the set of wffs of PL.
False, it must be only on sentence symbols
T/F: It's generally the case that, if an interpretation I, makes both a set of wffs T (gamma) true, and a conclusion α true then (gamma) (double-turnstile) σ
False, it must be the case for all interpretations I, furthermore, we should not use 'generally the case that' for we want exact truth-conditions for our definitions and in order to have that exactitude we ought to use iff or just in case which both express a biconditional
What does the double-turnstile signify
For all wffs α and interpretations I, α is true relative to I, (written: I (double-turnstile) α) iff VI (α) = 1, the idea of any wff relative to I, is equivalent to taking about α relative to a valuation
What is the new definition of validity, and how do you check if it is true
In any argument if it is valid, our new definition of validity is in terms of interpretations, if all of the premises are true relative to an interpretation then the conclusion has to be true
What does the contradiction contained in (((P ∧ Q) ∨ ((R ∧ A) ∧ (S ∧ ~S))) do to the whole complex wff
It automatically falsifies the wff because for a conjunction ALL conjuncts need to be true in order for the conjunction to be true and since one is always false then the entire conjunction is thereby falsified.
VERY IMPORTANT: T/F: Whenever we have a complex wff such as ((P ∨ Q) ∨ (S ∧ ~S)), what determines the truth-value of the wff, does the contradiction embedded in the wff affect its truth-value
NO, the truth-value of such wff is completely determined based on the wff that is not a contradiction, since when it comes to disjunctives, only one disjunct needs to be true and we know that the entire left disjunct or contradiction is false so the truth-value of the entire disjunct is based on the one on the right or the non-contradiction.
If I (double-turnstile) P ⊃ ((Q ∨ R) ⊃ ~S), then: (a) I(P) = 0 (b) I(P) = 0, or I(Q ∨ R) = 0 or I(S) = 0 (c) I(P) = 0 or I(S) = 0 or I(Q) = I(R) = 0 (d) If I(P) = 1 and I(S) = 1, then I(Q) = 1 (e) I(P) = 0 or I(S) = 0 or I(Q) = 0 or I(R) = 0
NOT (a), remember an interpretation is only one interpretation THERE CAN BE AN INTERPRETATION WHERE THE ANTECEDENT IS TRUE AND THE WHOLE CONDITIONAL IS TRUE NOT (b) It is not (b) because I(Q ∨ R) does not make sense, ONLY valuation functions can work for complex wffs CORRECT ANSWER: (c)
What is the significance of iff in the recursive rules for semantic derivations
So an interpretation or valuation function that claims iff, tells you the specific conditions in which a complex wff is true and it also gives the conditions when they are false.
True or false: For any α, if (double-turnstile) α, if {A} (double-turnstile) α
True, since a valid formula is entailed by anything
True or false: The last line of a semantic derivation for a wff of PL will contain no uninterpreted object-language truth-functional connectives
True, the last line is supposed to be the interpretation function the atomic wffs of PL, if we were to create a semantic derivation of only sentence symbols then it would just be an interpretation function
T/F: Whenever we have a complex wff such as ((P ∨ Q) ∧ (S ∧ ~S)), what determines the truth-value of the wff, does the contradiction embedded in the wff affect its truth-value
Yes, it will always be false because a conjunction is only true when both conjuncts are true and since one is always false it will always be false as well
T/F: Whenever we have a complex wff such as ((P ∨ Q) ∧ (S ∨ ~S)), what determines the truth-value of the wff, does the tautology embedded in the wff affect its truth-value
Yes, the truth-value of the entire complex wff will be determined based on the truth-value of (P ∨ Q), since we know that one conjunct is always T, we need to know that the other conjunct is always T, in order for the conjunction to be true
T/F: Whenever we have a complex wff such as ((P ∨ Q) ∨ (S ∨ ~S)), what determines the truth-value of the wff, does the tautology embedded in the wff affect its truth-value
Yes, the truth-value will always be dependent on the tautology since whenever a disjunct contains a tautology it is always true because only one disjunct needs to be true in order for the entire disjunct to be true
How to address questions of validity in relation to semantics
You can provide a proof to determine whether ~B (double-turnstile) A ⊃ (~B ∨ (C ∧ (~D ↔ E))) is valid, by definition of the valuation function, we know that for any VI, VI (~B) = 1 exactly when VI(B)= 0, by the earlier semantic derivation, we know that for any VI, if VI (B) = 0, then VI (A ⊃ (~B ∨ (C ∧ (~D ↔ E)))= 1. Any interpretation that makes ~B true also makes (A ⊃ (~B ∨ (C ∧ (~D ↔ E))) true. So the argument is valid. The meaning of the validity of the argument turns on the meaning of the truth-functions of the connectives. A semantic derivation tells you WHAT AN INTERPRETATION MUST LOOK LIKE IN ORDER TO RENDER THE ARGUMENT TRUE. So an interpretation or valuation function that claims iff, tells you the specific conditions in which a complex wff is true and it also gives the conditions when they are false. If you do a derivation if B is false then the conclusion is true, and that is what the validity claim is and it is true.
Suppose we use ↓ as a short-hand for a contradiction and ↑ as s short-hand for a tautology, then for any α of PL, if α is a wff of PL: (a) ↑ is logically equivalent to (α ∧ ↑) (b) (α ∨ ↑) is logically equivalent to α (c) α implies ↑ (d) (α ∧ ↓) is logically equivalent to ↓ (e) α is logically equivalent to (α ∨ ↓)
answers: (c), (d), (e), (c) is true because anything implies a tautology (d) is true because the truth- value of (α ∧ ↓) depends on the contradiction, because any conjunction that contains a contradiction will always be false (e) is true because the truth-value of the complex wff depends on α
What do you get at the end of a semantic derivation
at the end of a semantic derivation, you get a summary of what assignment of truth values to the sentence symbols is necessary and sufficient for the whole wff to be true, a complex wff is a function of its simplest parts
T/F: A successful semantic derivation will supply the truth conditions of a wff α by saying what the valuation function has to be like in order for α to be true
false, it is by saying what the interpretation function has to be like
T/F: Adding a premise to a valid argument can make it invalid
false, principle of monoticity holds for classical propositional logic
What does the contradiction contained in (((P ∧ Q) ∨ ((R ∧ A) ∨ (S ∧ ~S))) do to the whole complex wff
it does not affect the truth-value of the whole complex wff, you can actually ignore the row entirely since the main connective is the disjunction for that reason the complex wff will be logically equivalent or have its entire truth-value determined based on the disjuncts that are not contradictions
What is the conceptual significance of a semantic derivation
it is meant to decompose a complex wffs to determine its truth-functional outputs depending on the inputs provided for the valuation function, we take a complex wff and decompose it to its most simple constituent parts, and that provides us the specific truth-values that determine what is required for the complex wff to be true
For the complex wff: ((P ∧ Q) ∨ (R ∧ A)) why is the step in a semantic derivation I(P ∧ Q) = 0 false?
it is nonsensical we only use interpretation functions for sentence symbols, therefore, it is false to use an interpretation function for a complex wff since complex wffs only correspond to valuation functions
For any interpretation I for PL, what is the valuation for I
it is the function VI (I is meant to be subscripted) assigning wffs either 1 or 0 such that, for any sentence symbols a and all wffs α and β
What is semantics always doing
it takes a certain entity in the object-language and defines in terms of the metalanguage and that determines how it functions within the object-language
Does a contradiction follow from any set of wffs
no a contradiction only follows from inconsistent set of premises, since contradictions cannot possibly be true they must only follow from premises that cannot all be true together
when you replace a sentence symbol for a letter in the metalanguage does it have to be an exact replacement
no, as long as the truth-functional essence of a semantic derivation remains then it is fine
Why does the setting of truth-values for sentence symbols fixes the truth-values for all of the truth-functional connectives
relative to the interpretation that fixes the truth-value for all sentence symbols, then from there that fixes the truth-value outputs for all valuation functions
In the syntax and semantics of PL, then the truth-functional connective ∧ is specified in
the recursive definition of valuation
the symbol (double-turnstile) can be used to express
the relation of entailment between a set of wffs and a wff the validity of a wff (i.e. the status of a wff as a tautology) the truth of a wff relative to an interpretation function NOT: a relation between the interpretation and valuation function
Why must we extend the notion of an interpretation function, and what is a valuation
to give truth-values or truth-functional outputs for a given complex wff, an intepretation only works for sentence symbols (it is a choice of sentence symbols) whereas a valuation, tells the truth-value of all the composite parts of a complex wffs, it assigns meanings to the complex wffs in a systemic fashion
T/F: For any wff α and set of wffs T: if there exists an I such that both (i) the conclusion is true relative to an interpretation (ii) all the premises are true relative to an interpretation then T and the conclusion α are consistent
true, since consistency is defined as any set of sentences such that they are true relative to an interpretation I, relative to at least one interpretation I, then if the conclusion is true relative to an interpretation and so are all the premises of the argument, then they are consistent since there is a possible state of the world or truth at a row of a truth-table where they are all true, IT IS NOT A VALID ARGUMENT HOWEVER, it is not valid because the conclusion is not entailed by the premises as would follow ONLY IF IF AND ONLY IF it held for ALL interpretations
what is the analogue for interpretations and valuation functions
truth-tables
What do the recursive rules embedded within the semantic rules of propositional logic aim to tell us
what must an interpretation look like in order to render it true
Can you use ≠ in a semantic derivation
yes if it is tantamount to the same truth-functional output of a given valuation function, for example if we have (~D ↔ E) we can use VI (D) ≠ VI (E) since ~D and E must have the same truth-value, so in order for ~D to be true D must be false, so if D is 1 and E is 1 then the biconditional will be false, so E must be true and D must be false so D ≠ E.
Does a valid formula follow from any set of wffs
yes, because it is necessarily true under any and all interpretations
Can sentence symbols be conceived as switches that result in a certain output truth-value
yes, we have programmed the connectives to have a certain truth-functional output, the sentence symbols reflect settings of switches of true or false and an interpretation corresponds to a given switch and on the valuation side the valuation claims what is necessary to make the complex wff true
how to write the validity of an argument in our new technical notation ~B ∴ A ⊃ (~B ∨ (C ∧ (~D ↔ E)))
~B (double-turnstile) A ⊃ (~B ∨ (C ∧ (~D ↔ E)))
COMPLETE THE DEFINITION: α is a contradiction iff....
α is a contradiction iff for all interpretations I, V1 (α) = 0
