I am reading a book about logic and they construct the full first order algebra as the free algebra on the elements r(x1,x2,x3,....,xn) where xi belongs to a V, and r belong to R and each r has a specifed n for the elements it takes in. the operations on the free algebra are the 0-anry F which represent the contradiction. The implication, a binary operation and the for all operator a 1-anry operation (and you have one for all operation for every x in V).

I have been attempting to prove that the relation given in definition 1.4 is transitive but I have not been able to get through even the first step given for it. I have come to the conclusion that if I have w1= (for all x)a and w2 =(for all y)b I can ignore the cases in which a and b are of the type a= a1 => a2. And directly treat it is as a= (for all x1)a1 and b= (for all x2)b1 but I dont really know where to go from here, because what unites a and b is the existence of a c(x) so that a is related to c(x), and b to c(y). so c(x)= (for all x3)c1(x) and I cannot apply the induction to c(x) and a because I cannot asure that z doesnt belong to V(c). And I do not know what to try. (I tagged it as algebra since it seems more to do with algebra than logic).

hi there! i’m currently taking a philosophy class for introduction to logic and i’m finding it extremely difficult as someone who’s not at all a math person and can’t grasp the concepts as easy! i was wondering if anyone would be able willing to help and or explain in a more simple manner! thank you :’))

The first AI program in history didn't play chess, generate images, or write poetry. It proved mathematical theorems. In 1956, Newell and Simon built the Logic Theorist — a program that could derive proofs from Principia Mathematica using formal deduction rules. It was, in essence, a syllogism machine. And it was supposed to be the beginning of something that would change everything.

Seventy years later, we are still stuck in the wreckage of that assumption.

The Logic Theorist proved 38 of the first 52 theorems in Principia Mathematica. For one theorem, it found a shorter proof than Russell and Whitehead had. The AI community celebrated. Logic was the path. Formal deduction was intelligence. If you could just encode enough rules, build a sufficiently complete axiom system, the machine would reason its way to truth.

They were wrong. Not slightly wrong. Fundamentally wrong. And the industry that grew from their mistake is still making it today — just with better marketing.

Aristotle understood something that the founders of AI did not. In the Prior Analytics, he laid out the syllogism. But he never confused the structure with the act of thinking itself. For Aristotle, λόγος was not merely formal validity. It was the capacity to give an account, to articulate reasons, to engage in the full activity of rational discourse. A syllogism is a skeleton. Thinking is the living body that moves it.

The AI pioneers looked at Aristotle's logic and saw a blueprint for machines. What they missed is that Aristotle himself treated formal logic as a tool of reasoning, not its definition. The Organon — his collected logical works — was literally named "the instrument." Logic was the instrument philosophers used. It was never the philosopher.

This confusion between the instrument and the activity it serves is the original sin of artificial intelligence.

The frame problem, identified by McCarthy and Hayes in 1969, was the first crack in the edifice. Consider a robot in a room with a red block, a blue block, and a table. The robot picks up the red block. In a syllogism machine, you need explicit axioms stating that the red block's position changed, that the blue block's position did NOT change, that the table's position did NOT change, that the color of the red block did NOT change — an infinite regress of non-change assertions.

A human child understands this instantly. Pick up one thing, and everything else stays where it is. No axioms needed. No formal deduction required. The child reasons about the world using a rich, implicit model of physical causality that no syllogism machine has ever possessed.

The frame problem is not a technical bug awaiting a clever fix. It is structural revelation: formal logic, by itself, cannot model the background understanding that makes reasoning possible. You need something beneath the logic — a world-model, a set of expectations, an embodied relationship with the environment — that the syllogisms merely operate on top of.

When deep learning displaced symbolic AI, the field congratulated itself on finally moving beyond GOFAI's limitations. But neural networks are the same mistake wearing a different costume. Modern language models are statistical syllogism machines. Instead of IF-THEN rules encoded by human experts, they use probabilistic patterns extracted from training data. But the fundamental confusion persists: they mistake pattern-matching for reasoning, correlation for understanding, fluent output for genuine thought.

Ask ChatGPT to evaluate a genuinely novel philosophical argument. It doesn't reason about the argument. It performs reasoning about the argument. It generates text that looks like philosophical analysis, uses the vocabulary of philosophical discourse, follows the structural conventions of academic argumentation. But there is no actual engagement with the logical structure of the claims. There is pattern-completion dressed in the costume of thought.

In the Nicomachean Ethics, Aristotle distinguished between five intellectual virtues: episteme (scientific knowledge), techne (craft knowledge), phronesis (practical wisdom), nous (intuitive understanding), and sophia (theoretical wisdom). Formal logic belongs to episteme. But Aristotle never claimed that episteme alone constitutes intelligence. A person who can derive syllogisms but cannot navigate a difficult conversation, who knows the formal properties of ethical arguments but cannot judge what to do in a particular situation — such a person is not intelligent. They are a syllogism machine.

Twenty-five centuries later, we have built exactly what Aristotle would have recognized as a deficient intellect: systems that excel at the narrow domain of formal pattern manipulation while lacking every other dimension of rational capacity. Our AI can prove theorems sometimes, generate grammatically correct text usually, and classify images often. But it cannot exercise phronesis. It cannot engage in genuine dialectic. It cannot bring nous to bear on a novel situation that its training data never anticipated.

The question for the logic community is not whether we can build better syllogism machines. We can, and we have, and we will build more. The question is whether we are honest about what they are — and what they are not.

Hey guys I'm just going over the Modal Ontological Argument and I'm a bit confused on how Axiom S5 works out; it doesn't seem as intuitive as other deductions.

who in this server that studies formal logic uses pure formal logic when in verbal debates but with the propositions, connectives, quantifiers, etc, translated to english because ive seen alot of people who study formal logic not use pure formal logic in verbal debates, for example: saying (sentence)p, therefore (sentence)q, which isnt a valid argument

I've been lurking around the logic world for a while as a hobbyist, but something that still strikes till today, is what would be the contents one might consider exclusive of classical propositional logic, without overlapping with other logic fields like formal proofs.

My idea is that in this subfield, it would only enter: propositional variables, metaproperties such as arity, compound vs atomicity, etc... well-formed formulae, tautologies, contradictions, contingencies, the five essential logical operators (negation, conjunction, disjunction, implication, and biconditional), and inferences, modus ponens, modus tollens, the double negation.

To me that sounds like it, but many other authors introduce lots of other concepts as if they were part of this subfield, specially many other more elaborated proofs, which to me it doesn't feel correct, as my assumption is that classical propositional logic should be a little more mutually exclusive with the subfield of proofs, and don't overlap concepts from both subfields.

But uhm, feel free to introduce what you might consider could be a good fit for it other than the already mentioned. Maybe my assumption is wrong and proofs should be taught in this subfield aswell, despite there being a dedicated subfield for these.

Can someone please explain what vacuous truth is and how the implication is true (P implies Q) if P is False.

I’m not sure if this is on topic, but are there any online resources to learn Wigmore Charts? From a logical point of view?

They seem interesting, and the fact that they were developed by a legal scholar makes them especially appealing to me.

Hi, I came upon this claim:

which I read as:

If C causes E, this implies if there does not exist any cause of E, then necessarily E did not occur.

However, why does this claim exclude uncaused events? Could it not be the case that:

If C causes E, this implies that if there does not exist any cause of E, then it is possible that E did not occur, unless it is the case that E occurred without cause.

Thanks!

Could you suggest some examples of identical concepts? I’m discussing this with my tutor, who argues that “women” and “daughters” are identical because every woman is someone’s daughter. I think that if two concepts are truly identical, they should be fully equivalent—for example, “a father’s daughter” would then have to be equivalent to “a father’s woman,” which is absurd and makes no sense

Any idea on what resources to use as a complete beginner, i used the MIT online course and found it to be confusing and unintuitive.

I don't think there any, prerequisites to starting logic but I am really confused on where to start.

any advice would be grateful!

Deduction seems to derive necessary consequences from a set of premises.

But if you keep deriving necessary consequences, the conclusions generally become weaker and less informative (e.g., "engine is running" → "fuel is being burned" → "a physical process is occurring" → "something exists").

That made me wonder:if deduction alone doesn't seem very useful for discovering explanations, how am I going to use it for practical purposes

According to Spacing Hero: My interpretation of the well-ordering theorem is incorrect.

For I interpreted the well-ordering theorem to mean that every set has the property of being well-ordered. Yet this property is not in act in all sets. In other words, all sets have this property of being well-ordered whether potentially or actually, i.e. whether not in act or in act.

But I think Spacing Hero is wrong though and this is for the following reasons: One: The axioms of ZFC are meaningless unless interpreted. An interpretation is the assignment of meaning to the symbols of a language. An interpretation often provides a way to determine the truth values of sentences in a language. If a given interpretation assigns the value true to a sentence or a theory the interpretation is called a model of that sentence or theory. Model theory is the study of the interpretation of any language, formal or natural. Two: Since the axioms of ZFC are meaningless, I or anyone else can subject them to multiple interpretations. Some of these interpretations can make the axioms turn out false. And some of these interpretations can make the axioms turn out true. And if there are interpretations that an make the axioms turn out true then they don’t have to be the standard interpretations.

A case in point is the Putnam Permutation Argument. According to the Stanford Encyclopedia of Philosophy, the Putnam Permutation Argument is the following: Putnam’s Model-Theoretic Argument is the most technical of the arguments we have so far considered. We shall not reproduce all the technicalities here. The central ideas can be conveyed informally, although some technical concepts will be mentioned where necessary. The argument purports to show that the Representation Problem—to explain how our mental symbols and words get hooked up to mind-independent objects and how our sentences and thoughts target mind-independent states of affairs—is insoluble.

According to the Model-Theoretic Argument, there are simply too many ways in which our mental symbols can be mapped onto items in the world. The consequence of this is a dilemma for the realist. The first horn of the dilemma is that s/he must accept that what our symbols refer to is massively indeterminate. The second horn is that s/he must insist that even an ideal theory, whose terms and predicates can demonstrably be mapped veridically onto objects and properties in the world might still be false, i.e., that such a mapping might not be the right one, the one ‘intended’.

Neither alternative can be defended, according to anti-realists. Concerning the first alternative, massive indeterminacy for perfectly determinate terms is absurd. As for the second, what can it mean for a mapping to be the intended mapping if not that it satisfies every conceivable operational and theoretical constraint? Yet Putnam’s Model-Theoretic Argument proves that there will invariably be interpretations of an ideal theory on which all the theory’s sentences come out true which do satisfy any constraint we might choose to impose on them, anti-realists maintain.

Now, in logic theories are treated as sets of sentences and the objects (if any) that sentences talk about appear as elements of the domain of set-theoretic entities called structures. Associated with these structures are interpretation functions that map individual constants onto individual objects of the domain and n-place predicates onto n-tuples of elements in the domain. When a structure makes all the sentences of a given theory true it is called a model of the theory. By demonstrating that there is a model of T we show theory T is consistent. If T turns out to be true in its intended model, then T is true simpliciter.

Let us call structures whose domains consist of numbers ‘numeric’ structures. The nub of Putnam’s Model-Theoretic Argument against realism is that the realist cannot distinguish the intended model for his/her total theory of the world from non-standard interlopers such as permuted models or ones derived from numeric models, even when total theory is a rationally optimal one that consists, as it must do, of an infinite set of sentences and the realist is permitted to impose the most exacting constraints to distinguish between models. This is a very surprising result if true! How does Putnam arrive at it?

Putnam uses several different arguments to establish the conclusion above. The argument of prime concern to realists, as Taylor (2006) emphasises, is the argument based on Gödel’s Completeness Theorem, GCT. For, following Lewis [Lewis, 1984], realists might concede to Putnam that they cannot single out the intended model or distinguish it from various ersatz models, but argue that this is not necessary since it suffices that an intended model exists, even if we cannot specify it. This response does not answer the GCT argument, however. For this argument purports to prove directly that an ideal theory of the world could not be false, a conclusion flatly inconsistent with realism.

Putnam has another model-theoretic argument against realism, the Permutation Argument, also designed to guarantee we can find a true interpretation of an ideal theory:

Suppose that the realist is able to somehow specify the intended model. Call this intended model W1. Then nothing the realist can do can possibly distinguish W1 from a permuted variant, W2, which can be specified following Putnam: We define the properties of being a cat* and being a mat* such that: In the actual world, cherries are cats* and trees are mats*. In every possible world the two sentences “A cat is on a mat” and “A cat* is on a mat* have precisely the same truth value.

Instead of considering two sentences “A cat is on a mat” and “A cat* is on a mat*” now consider only the one “A cat is on a mat”, allowing its interpretation to change by first adopting the standard interpretation for it and then adopting the non-standard interpretation in which the set of cats* are assigned to ‘cat’ in every possible world and the set of mats* are assigned to ‘mat’ in every possible world. The result will be the truth-value of “A cat is on a mat” will not change and will be exactly the same as before in every possible world. Similar non-standard reference assignments could be constructed for all the predicates of a language.

Suppose I want to solve a zebra puzzle instance, of which the stem is "who owns the zebra", and the puzzle additionally says the zebra is only owned by one person.

The wh question "who owns the zebra" is not an issue because an issue is the uncertainty of whether to accept or reject a claim.

Therefore, I convert the wh question to an issue that whether the Norwegian owns the zebra. Then I find that I want to accept the claim, and I give some arguments.

Up to now, have I answered the zebra puzzle?

Should I also add an argument that "because the question says the zebra is only owned by one person and I've found the Norwegian is the person, the answer to the question is the Norwegian"?

On the contrary, should I list all other issues, such as "does Ukrainian own the zebra", "does Englishman own the zebra", and retue all of them, and then confidently answer the question that the answer is only Norwegian?

1. Let P be a proposition.
2. Let P = "All eggs are white.
3. Let T(P) = "The statement 'All eggs are white' is true."
4. We are considering a logical proposition, not ordinary conversation.
5. Every assertion within a logical proposition must make a claim to truth.
6. If P does not claim truth (i.e., has no criterion of truth), then it is not a logical proposition.
7. Therefore, when P is asserted, the truth of P is already presupposed by the very act of assertion.
8. Therefore, the statement "P is true" adds no new content to the act of asserting P; it merely makes explicit what is already contained in the assertion itself.
9. Therefore, within a logical proposition, P and T(P) do not differ in content; they differ only in their form of expression.

Why Negation Is Not an Exception 1. Let P = "All eggs are white." 2. Then the statement "P is false" is itself a new assertion. 3. Let us denote it by Q. 4. Then the statement "Q is true" is a new assertion R. 5. Then the statement "R is true" is a new assertion S. 6. Therefore, every assertion about a previous assertion forms a new level.

Chain: P Q = "P is false" R = "Q = 'P is false' is true" S = "R = 'Q = 'P is false' is true' is true" ...

This chain is free of contradiction as long as no statement refers to its own truth. Each new truth predicate applies only to the immediately preceding level.

Why Uncertainty Is Not an Exception 1. Suppose someone says, "Assume that P is true. 2. Here, P is not asserted as a fact. 3. It is only conditionally accepted as a proposition that claims truth. 4. Therefore, this is not an assertion of P, but reasoning under the assumption of P. 5. Therefore, hypothetical reasoning does not refute the previous conclusions.

The Liar Paradox 1. Let L = "This sentence is false." 2. If L is true, then L is false. 3. If L is false, then L is true. 4. Therefore, a contradiction arises.

Self-Reference 1. Self-reference by itself does not create a contradiction. 2. For example: — "This sentence consists of five words." 3. This sentence is self-referential. 4. However, it contains no predicate of truth or falsity. 5. Therefore, no contradiction arises.

The Restriction 1. The application of the predicates true or false to the sentence itself is prohibited. 2. Then a sentence of the form: — "This sentence is false." cannot be constructed. 3. Therefore, the Liar Paradox cannot arise.

The Source of the Paradox 1. Self-reference by itself is not the source of the contradiction. 2. The truth predicate by itself is not the source of the contradiction. 3. A contradiction arises only when the predicate of truth or falsity is applied to the sentence itself. 4. Therefore, the cause of the paradox is the self-application of the truth predicate.

Metalanguage 1. The object language speaks about the world. 2. The metalanguage speaks about sentences of the object language. 3. The mere existence of these two levels of language does not eliminate the paradox. 4. The paradox disappears only when the application of the truth predicate to sentences of the same level is prohibited. 5. Therefore, it is precisely the prohibition of the self-application of the truth predicate that eliminates the paradox. 6. Therefore, the distinction between object language and metalanguage is not what eliminates the Liar Paradox. The sole reason is the prohibition of the self-application of the truth predicate.

(MAIN IDEA derived from all the previous premises: P and the statement "P is true" are one and the same. Therefore, within a logical proposition, the truth predicate is unnecessary, even in the metalanguage.)nvolving the truth predicate. Imagine there's a cactus sitting on a shelf. If you cut the shelf into two shelves (the object language and the metalanguage)

If even the most fundamental laws of logic aren't necessarily fixed, then what am I supposed to rely on? How am I supposed to gain knowledge about the world?

I'm a complete beginner. I'm someone who wants to find out whether God exists or not, and decide how I should live. But right now I'm just confused because I don't understand what I can actually know—or whether I can know anything at all. What am I choosing to believe, and why do I believe it?

According to Wikipedia, the following holds with regard to the well-ordering theorem: In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered.

One way to interpret this passage is the following: When it states that every set can be well-ordered it means that every set has the property of being well-ordered, where the term property means the following per the Stanford Encyclopedia of Philosophy: Properties are those entities that can be predicated of things or, in other words, attributed to them. Thus, properties are often called predicables. Other terms for them are “attributes”, “qualities”, “features”, “characteristics”, “types”. Properties are also ways things are, entities that things exemplify or instantiate. For example, if we say that this is a leaf and is green, we are attributing the properties leaf and green to it, and, if the predication is veridical, the thing in question exemplifies these properties. Hence, properties can also be characterized as exemplifiables, with the controversial exception of those that cannot be instantiated, e.g., some would say, round and square.

However having the property of being well-ordered can be understood in two different senses. In one sense it means that the property is in act. In another sense it means that the property is not in act. To illustrate what I mean when I say that a property either is in act or not in act, consider the following passage from Aristotle: Again, to be, or being, signifies that some of the things mentioned are potentially and others actually. For in the case of the terms mentioned we predicate being both of what is said to be potentially and of what is said to be actually. And similarly we say both of one who is capable of using scientific knowledge and of one who is actually using it, that he knows. And we say that that is at rest which is already so or capable of being so. And this also applies in the case of substances; for we say that Mercury is in the stone, and half of the line in the line, and we call that grain which is not yet ripe. But when a thing is potential and when not must be settled elsewhere…

Commenting on this, Aquinas says the following: Here he gives the division of being into the actual and the potential. He says that to be and being signify something which is expressible or utterable potentially or actually. For in the case of all of the foregoing terms which signify the ten predicaments, something is said to be so actually and something else potentially; and from this it follows that each predicament is divided by actuality and potentiality. And just as in the case of things which are outside the mind some are said to be actually and some potentially, so also is this true in the case of the mind’s activities, and in that of privations, which are only conceptual beings. For one is said to know both because he is capable of using scientific knowledge and because he is using it; and similarly a thing is said to be at rest both because rest belongs to it already and because it is capable of being at rest. And this is true not only of accidents but also of substances. For “Mercury,” we say, i.e., the image of Mercury, is present potentially in the stone; and half of a line is present potentially in a line, for every part of a continuum is potentially in the whole. And the line is included in the class of substances according to the opinion of those who hold that the objects of mathematics are substances—an opinion which he has not yet disproved. And when grain is not yet ripe, for example, when it is still in blade, it is said to be potentially. Just when, however, something is potential and when it is no longer such must be established elsewhere, namely, in Book IX of this work.

1. Syntax
   
2. Axioms
   
3. Definitions
   
4. Inference Rules
   
5. Theorems
   
6. Proofs
   
7. Semantics

I would probably guess its the axioms in more than 70% of systems its the axioms, I know that the question is some kind if obvious but I would like to hear your opinions on it:)

Gabriele Giannantoni wrote one of the first and important book of history of philosophy that it studied in Italy in 70th years.

The stone paradox goes like this "Can an omnipotent being create a stone it can't carry?". Was asking chatgpt and I wasn't really satisfied with it's answer. It said with predicate/classical logic an omipotent being can do anything logical so the stone paradox is considred illogical. And I asked it what would be the answer if you were to use paraconsistant logic. It said it could do both, it cancreate a stone so heavy it can carry and cannot carry it. When it start saying it's reason either I just didn't understand it or it was hallucinating. What are the answer(s) on stone paradox using paraconsisnt logic system?

Hello everyone,
I am learning to use lean theorem prover using the doc Mathematics in Lean. I am doing some basic things right now.
I am getting pretty stuck as I don’t want to use AI for it as it feels pretty interesting although intense at times.
My first question is: how do you cover the landscape of its nuances while writing proofs? Are there some rule of thumbs to break down the problems , what tactics may come useful here and things like that or is it just a muscle memory that comes up with time ?

Also if anyone wants to form a study group to want to go through it please do message me. I find it pretty amusing and want to learn new perspectives as well,

I read about Logic Theorist recently — program from 1956 that proved mathematical theorems using formal deduction. AI community celebrated it as beginning of real intelligence. Seventy years later, I think we are still stuck on same mistake.

The problem is not mechanism. Problem is assumption that mechanism is sufficient. Expert systems, neural networks, language models — all are syllogism machines wearing different costumes. They manipulate patterns (formal or statistical) but never actually reason about world.

Aristotle understood this. He built formal logic as tool of reasoning, not definition of it. He called this tool φρόνησις (phronesis) — practical wisdom that no formal system captures. Modern AI has same gap: it produces text that looks like reasoning but has no engagement with logical structure underneath.

Frame problem from 1969 was never solved. Child understands that when you pick up red block, blue block stays put. No axioms needed. No syllogism machine can do this — not because it lacks data, but because it lacks world-model beneath the logic.

What do you think — is there path from pattern-matching to genuine reasoning, or is gap fundamental?