A good way to look at this is through the concepts that Frege introduced - sense (sinn) and reference (bedeutung).
The question becomes whether the proposition
All unicorns are beautiful
has sense and reference: one can ask whether the proper names - unicorn and beautiful refer; one can argue that these names occur in the corpus of written works, that ...
This specific case is indeed a vacuous truth. A vacuous truth is "a statement that asserts that all members of the empty set have a certain property".
It takes three forms:
∀ x : P(x) → Q(x) where ∀ x : ¬P(x)
∀ x ∈ P : Q(x) where P = ∅
Your concern is sound ...
In Aristotle's Logic the inference from :
∀x (Fx → Gx)
∃x (Fx & Gx)
is legitimate. In modern logic, this is not; we say that general terms have existential import.
See the discussion of The Traditional Square of Opposition :
This representation of the four forms is now generally accepted, except for qualms ...
"Every politician in this circle will have a firefighter to their immediate right" is typically coded into predicate calculus as ∀x∃y(P(x) → F(y)∧IR(x,y)). If there are no politicians in the circle P(x) is always false, and, by the convention about the material conditional, when the premise is false the conditional is true. So it is not that the proposition ...
There is a difference between semantic consequence expressed by truth tables, and syntactic consequence in a deductive system, some authors use ⊨ for the former and ⊢ for the latter, and the corresponding difference in equivalence. The latter can be used to capture what you are describing somewhat. In Kant's theory of conceptual containment equivalence "not ...
Think of ∀xP(x) as an implicit conditional: ∀x(xϵU → P(x)), where U is the universe. In an empty universe the antecedent is always false, hence the conditional is vacuously true. In contrast, ∃xP(x) is an implicit conjunction ∃x(xϵU ∧ P(x)), so it is vacuously false. This is in line with the standard way of transcribing "all humans are liars" with a ...
I get, what you are saying, but implication in classical logic has nothing to do with the "meaning" of propositions. In particular, 3>2 and 4+6=10 are in fact equivalent statements.
The reason for having two symbols to represent logical equivalence goes roughly like this: inside a given theory of mathematics A we construct a logical system B. It (B) ...
I did my undergrad thesis on fictional characters/objects and truth value so I might be able to help out. It depends on your view of fictional objects.
If you just take a classical logic view of fictional objects (none exist), then the sentence is vacuously true simply because there are no fictional objects. The "x" in "every x" quantifies only over ...
The two different symbols on the page you link to are indeed different. The first is the turnstile symbol Ⱶ which may be read as 'proves', while the arrow → is material implication. These are very different. Material implication is a symbol in the object language defined by the truth table that you give, i.e. T/F/T/T. Turnstile is a symbol in the ...
Propositions are (usually) not linguistic entities: thus they differ from statements and sentences.
This is the meaning of:
"The term proposition is used to refer to ... the referents of that-clauses, and the meanings of declarative sentences."
Propositions are (usually) not mental entities, like thoughts or states of mind.
The are (...
4.22 An elementary proposition consists of names. It is a nexus, a concatenation, of names.
4.221 It is obvious that the analysis of propositions must bring
us to elementary propositions which consist of names in immediate combination.
4.23 It is only in the nexus of an elementary proposition that
a name occurs in a proposition.
4.24 Names ...
In his review of Peter Coffey's book : The Science of Logic (1st ed 1912), published in The Cambridge Review, Vol.34, 1913, Wittgenstein criticizes it as a representative of "old" logic, precedent to the new mathematical logic of Frege, Peano and Russell [the first volume of Principia Mathematica by Alfred North Whitehead and Bertrand Russell was first ...
There's some disagreement among professional philosophers about the relationship of the terms "sentences" and "propositions."
I would say the most important distinction is captured pretty well in the other answer but that it might be considered misleading to say that "a proposition is a form of a sentence."
The key distinction is that sentences are the ...
As the OP and the comments note, both of the results are tautologies.
Here is the truth table for the first one:
Here is the truth table for the second one:
As the OP notes the two sides of the biconditional are also tautologies.
As to why these were marked false, perhaps the answer key was in error or the problem that was intended was misstated. Perhaps ...
Assuming for the sake of discussion the fact that clouds are the cause of rain, this does not mean that "if there are clouds, then there is rain" is correct.
From the fact that clouds are the cause of rain we have:
"if there is rain, then there are clouds".
A good exercise is to replace the "if..., then___" construction with a different one using "when"....
Well semantically propositions have a definition that may be different in maths and science. So this means varying answers depending on which department did the teaching.
In philosophy I was taught all propositions have a truth value. You are now bringing up awareness of the truth value which sounds like a science approach. Because I don't know which ...
Simply put, the speaker begs the question. Their argument seems to be structured like this:
The propositional content of the two characters' knowledge can be exhaustively expressed by "Bill has been treed by a moose"
The two characters respond differently to their equivalent propositional knowledge.
Therefore there is such a thing as non-...
I'm not an expert in this area, so someone may wish to correct me.
Since you appear to be wishing to look at this formally, let's start with a formal view of the problem.
In a formal setting, a statement P is called not decidable if it is impossible to prove P and it is impossible to prove not(P).
Note that provability is a purely syntactic notion here. ...
The "meaning" of the logical connective are defined by the rules of inference governing them.
For conjunction ("and") we have :
(φ∧ψ) ⊢ φ and (φ∧ψ) ⊢ ψ;
for disjunction ("or") we have :
φ ⊢ (φ∨ψ) and ψ ⊢ (φ∨ψ).
The first couple of rules formalize the fact that :
"Asserting a conjunctive proposition is equivalent to asserting each of its component ...
Actually the two symbols in question, ≡ and ⇔, have very different behaviors in propositional logic. For 99% of situations, you can interchange them and get away with it. However, in the last 1% the difference is essential to the use of propositional logic.
The difference is that the meaning of ⇔ is formally defined in the definition of propositional ...
I think what you're picking up on is the Fregean distinction between sense and reference. The referent of a term t is the object that t picks out 'in the world', whereas the sense of t is, roughly, something like the idea associated with t. (VERY roughly. Frege calls the sense of the term the 'mode of presentation' of the term; one way to think of sense is ...
5.4733 Frege says: Every legitimately constructed proposition must
have a sense.
Thus, we may equate nonsense [unsinn] with an illegitimate grammatical combinations of words, something like an ill-formed formula of a formal language.
Only well-formed combinations of words generate proper sentences, i.e. sentences that express a thought [3.2] or ...
"A is B" and "A is not B"…. Are both of the above statements mutually
exclusive? If so, then would that not mean that the principle of
non-contradiction is self-evidently true?
Assume the truth of statements "A is B" and "A is not B". Now assume A is true. The combination of premises means that the following is true: B and not-B.
Such a conclusion is ...
See John Burgess, Philosophical logic (2009), page 84 :
let A hold at both u and v, C fail at u and hold at v, and B hold at u and fail at v.
Then the least remote A-state u is an B-state, but the least remote (A & C)-state v is not an B-state.
Thus, in u A > B holds but (A & C) > B does not, and so :
the inference from A > B to (A &...
The entry cited does not claim that a factual proposition cannot be formulated using logical terms. Rather, it makes a distinction between logical propositions which express logical truths, and factual propositions which express empirical truths (or falsehoods). Alternatively you might say that the latter express facts.
A logical truth is not just a truth ...
I am guessing that "propositional" refers to the subject-predicate-object structure of traditional grammar. However, this structure is not semantically binding, we do use predicates with only nominal subjects (to satisfy the rules of grammar). This is what the "it is" construction and participles are for in English. For example, "it is raining" or "raining ...
Answer to the first version of the question.
Because, see the truth table for →, a conditional with false antecedent is always true.
I assume that you are evaluating the truth value of the formula: (p→q) → (q→p).
If so, two cases:
1) p is TRUE
2) p is FALSE.
In this case, the formula is evaluated to: (F→q) → (q→F).
We have that (F→q) is T, for q ...
There's basically two notations for doing truth-tables. The one you're using is the harder one, so it might be best to look at how it would look with the easier one and then transition to the harder one.
The easier one uses the following format:
Columns for each variable
Columns for helper rows
Columns for the premises
A column for the conclusion
(In the ...